Random And Vector Measures

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Random and Vector Measures

8162hc.9789814350815-tp.indd 1

7/15/11 10:33 AM

SERIES ON MULTIVARIATE ANALYSIS Editor: M M Rao

ISSN: 1793-1169

Published Vol. 1: Martingales and Stochastic Analysis J. Yeh Vol. 2: Multidimensional Second Order Stochastic Processes Y. Kakihara Vol. 3: Mathematical Methods in Sample Surveys H. G. Tucker Vol. 4: Abstract Methods in Information Theory Y. Kakihara Vol. 5: Topics in Circular Statistics S. R. Jammalamadaka and A. SenGupta Vol. 6: Linear Models: An Integrated Approach D. Sengupta and S. R. Jammalamadaka Vol. 7: Structural Aspects in the Theory of Probability A Primer in Probabilities on Algebraic-Topological Structures H. Heyer Vol. 8: Structural Aspects in the Theory of Probability (2nd Edition) H. Heyer Vol. 9: Random and Vector Measures M. M. Rao

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S e r i e s

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Multivariate Analysis Vol. 9

Random and Vector Measures M M Rao University of California, Riverside, USA

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RANDOM AND VECTOR MEASURES Series on Multivariate Analysis — Vol. 9 Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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To the memories of Professors K. Itˆo and S. Kakutani whose deep insights into stochastic integration have been decisive, and to the memory of my beloved sister Jamantam the first to complete Bhasha Praveena and the first to teach it among our people

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Preface

The aim of this work is to present an analysis of vector valued measures with several important applications when the values are taken in complete metric spaces. If these range spaces are Banach, then one calls them vector measures and if they are Lp (P ), p ≥ 0, based on a probability measure, so that they are Fr´echet (and not necessarily Banach) spaces then they are termed random measures where the underlying probability structures enhance the value of the subject and often compensate for the lack of local convexity of their range space. This feature also provides several new and useful special properties of these measures leading to important analyses. For an insight and appreciation of this work, which includes both the general and geometric aspects, I shall outline the coverage included here and motivation for it. More detailed summaries will be found at the beginning of each chapter indicating the main results and its contents. The reader is assumed to be familiar with a standard Real Analysis as in Royden or my Measure Theory book, and some aspects of Probability Theory such as that given in Feller, Lo´eve or my text on this subject. The material included here can be devided into three closely related parts in the sense that the first four chapters emphasize the way random measures arise in many familiar forms in stochastic or other applications and the treatment is leisurely. On the other hand the last four chapters deal with problems in which the vector measures play a very significant part leading to research topics and somewhat advanced ideas are employed, analogous to those found in expositions aiming to include several key applications without enlarging the volume beyond a reasonable size. The middle part consists of Chapter 5 which connects the other two parts and has a key role to play in the transition. Here the basic ideas will be outlined by the following synopsis of the above parts. For a convenient reading and reference and to aid the reader, I reiterate some key concepts and ideas at different chapters in the book. This may annoy some but I feel that it can help many others. After observing the distinction between scalar and vector measures of finite variation in the scalar, and semi-variation in the vector cases (these are also called Vitali and Fr´echet variations), the existence of a positive finite control-

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ling measure is detailed. The general result here is the Bartle-Dunford-Schwartz theorem on the existence of a control measure for all vector measures valued in Banach spaces. For the general Fr´echet space valued set functions no such general result is available, but for Lp (P ) spaces p ≥ 0, using some essential probability results including those obtained with stable or infinitely divisible and certain other classical results, and another due to the combined work by Talegrand-Kalton-Peck-Roberts, such controls are produced. Indeed the classical work by Kakutani on metric topological groups, sharpened by Klee in early 1950’s produces such a control (cf., pp. 445–6) for a large class of nonlocally convex Orlicz spaces, which has been overlooked in much of the current stochastic analysis. In any case, it is only then these integrals can be and are defined. In particular for the L2 (P )-ranges, the random measures give rise to bimeasures and the resulting analysis exploits the Hilbert space geometry in establishing some key results some of which are extended in later chapters. The most important and unique feature of the present work is in showing the central role played by Bochner’s Lp,q -boundedness principle as well as Grothendieck’s result on the representation of bilinear forms in obtaining all the known types of stochastic integrals. This has not been done in any other publication as far as I know. Both these applications are somewhat overlapping in Hilbert space but are different in other spaces since Bochner’s principle, being the most general, covers larger collections. These approaches are effectively used in getting integral representations of many classes of random processes or fields, exhibiting the nonlinear structures of the problems. Some of these lead to analysis leading to solutions of “local functionals” on continuous (and other) function spaces introduced by Gel’fand and Vilenkin (1964). Several related characterizations are also included. Consequently the Bochner boundedness principle and Grothendieck’s inequality receive a detailed and comprehensive treatment, perhaps for the first time in a book of this type. Thus the first four chapters contain the primary analysis of random measures which may be used for a second year graduate course or for a collateral text on probability theory, even with selected omissions. The next part is Chapter 5 which as indicated already is a sort of a bridge between the first part discussed above and the one that follows afterwards. Here Grothendieck’s analysis on bilinear forms and related random measure algebras leading to aspects of random convolutions with resulting harmonic analysis are studied. Special topics when the measures are positive, bounded or satisfying certain order relations are discussed, showing new possibilities to be treated in the next part consisting of the last four chapters. Here some relatively specialized and new tools are used in which general vector measures are prominent. The last part begins with Chapter 6 which discusses the Itˆ o integral, as an extension of Wiener’s method centering on the set of bounded continuous stochastic integrands which was generalized by Meyer and his associates to all square integrable (sub)martingales and the integrands satisfying only certain “non-anticipating” conditions. The treatment in this chapter contains a fairly

Preface

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general account based largely on Bochner’s Lp,q -boundedness principle which is also shown to be essentially the most general condition known. About this formulation Bochner states, in his third Berkeley Symposium paper, that “it is a promising concept, it subsumes the so-called ‘Wiener process’ more readily than any other stochastic process sufficiently general, ... and we shall deal with it briefly here.” This is utilized fully in this book where it is shown that Grothendieck’s inequality and Bochner’s principle complement each other playing basic roles in random measure (and integration) theory. In fact Bochner’s concept, slightly extended, is applicable to all known integrals including quasimartingale measures as well as McShane’s approach to the subject. All of this is in Chapter 6 where stopping time calculus is observed as a modern alternative to L.C. Young’s generalized method of Stieltjes integration. Moreover it also includes Stratonovich’s approach and many other ‘anticipative’ integrands. Multiple random measures and integrals present new challenges and opportunities. They are treated in Chapter 7, which is the longest one of the book. Here the first half is on two-parameter random integrals starting with the pioneering approach of the subject for the planar Brownian motion (BM) by R. Cairoli and J. B. Walsh. This work is extended to two-parameter quasimartingales leading to nonsymmetric versions of the Fubini and Gauss-GreenStokes theorems with BM in the background. For an extension to n-parameters (n > 2) a general partial ordering of the ambient space is needed. Here the approaches by H¨ urzeler, and Rota-Wallstrom, both using combinatorial methods, are considered in some detail which opens up new areas for future investigations. The next item is the multidimensional parameter space Rn , as in Whitney and Federer, using key ideas of chains and cochines with their new forms of vector spaces. The emphasis here is on general vector measures, random measures being secondary at first but the analysis is inspired by the stochastic analysis so as to employ it to the work on random integrals. This analysis is detailed in Chapter 8 which is also a fairly long one. Some ideas of Abstract Analysis play a prominent role here. Also there is an interesting open problem on the integral representation of flat chains which was raised by Whitney after presenting his primary work on ‘sharp chains’ and a corresponding solution of the ‘flat case’ was studied and a solution obtained by Noltie (1975) in his thesis, so far unpublished, and it is included here. His work on the (multiple) sharp chains was already published. [The main difficulty is that the supports of flat chains are just of positive Hausdorff measures in Rn , but are usually of negligible Lebesgue measure!] The work of Chapter 8 is also needed for the structural analysis of isotropic random currents already considered by Itˆ o (1956) with some elaboration and interesting Engineering Applications discussed by Wong and Zakai. Both of these are included here. Consequently, this chapter contains some of the fundamental results of this book. Finally the last chapter is devoted to an analysis of multilinear forms together with their integral representations. Also some work on the subject due to Blei (1985) is utilized, and possibilities of generalizing the earlier study on bimeasures with these tools are discussed along with some aspects of Dobrakov’s

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ideas of multidimensional vector integration, called ‘polymeasures’ by him. The analysis in this part shows many new directions opening up both for probability problems with refined structural analysis and also those of geometric integration benefited with stochastic considerations, thereby showing several new avenues of research. To limit the volume, some details are only sketched. Each of the chapters has an Exercises section with copious hints and references which quite often complement the main text. Many aspects here can be used in research and graduate seminars leading to new research and theses works. In fact the book contains a large amount of new material that appears in such an advanced work for the first time. The contents of Chapter 8 should also be of special interest for workers in Geometric Integration [Measure] Theory inviting them to study stochastic integration for interesting applications. In executing the above ideas that are essentially a culmination of my studies over many years, I have been fortunate in having some of the best graduate students to work with. Particularly the material in Chapters 7 and 8 is substantially based on the UCR theses of M.D. Brennan, M.L. Green and S.V. Noltie. The unpublished representation of the key flat chains problem solved in Noltie’s thesis is sketched in Chapter 8 with his consent. Some of the details that could not be included for space reasons and the omitted steps can be found in the cited sources. I am indeed happy that I could include the general versions of these results in comparison with other monographs. The only works touching on some of the problems considered here are the one presented by N. Dinculeanu (2000) and the other by M. M´etivier and Pellaumail (1980) but the overlap between these and what follows is small. They generally complement each other. The numbering system followed in the book is standard. For instance, 5.2.9 refers to the 9th item in Section 2 of Chapter 5 and Section 5.2 denotes the second section of Chapter 5. In a given section often only the item number is noted. Sometimes the AMS-TeX and LaTeX are not in accord (may change sometimes for indexing page numbers), but I hope readers are not much inconvenienced with this problem. I have tried to give all references and due credits for the work presented, and no result should be attributed to me unless stated explicitly. Some well-known results are not traced to original sources if it takes too much effort. The preparation of this material took more than five years with a couple of UCR sabbatical leaves. My handwritten and difficult manuscript of all chapters has been patiently and efficiently transformed into the LaTeX version by Ms. Ambika Vanchinathan from a long distance over this period for which I am deeply indebted to her. Some computer problems are resolved with the help of Marc Mehlman. For all this assistance of these colleagues I am deeply grateful. I earnestly hope that this monograph helps to stimulate graduate students and other research workers in extending the frontiers of the areas connecting random valued integration and its abstract vector valued counterpart. Riverside, CA January, 2011

M.M. Rao

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1

Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introducing Vector Valued Measures . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Additivity Properties of Vector Valued Measures . . . . . . . . . . 10 1.4 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2

Second Order Random Measures and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Structures of Second Order Random Measures . . . . . . . . . . . . 2.3 Shift Invariant Second Order Random Measures . . . . . . . . . . . 2.4 A Specialization of Random Measures Invariant on Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Random Measures Admitting Controls . . . . . . . . . . . . . . . . . . . 3.1 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Controls for Weakly Stable Random Measures . . . . . . . . . . . . 3.3 Integral Representations of Stable Classes by Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Integral Representations of Some Second Order Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 25 38 49 54 58 61 62 74 83 94 113 122

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5

6

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Contents

Random Measures in Hilbert Space: Specialized Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Bilinear Functionals Associated with Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Local Classes of Random Fields and Related Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Bilinear Forms and Random Measures . . . . . . . . . . . . . . . . . . . . 4.4 Random Measures with Constraints . . . . . . . . . . . . . . . . . . . . . . . 4.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 126 133 140 150 157 164

More on Random Measures and Integrals . . . . . . . . . . . . . . . 5.1 Random Measures, Bimeasures and Convolutions . . . . . . . . . . 5.2 Bilinear Forms and Random Measure Algebras . . . . . . . . . . . . 5.3 Vector Integrands and Integrals with Stable Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Positive and Other Special Classes of Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 177

Martingale Type Measures and Their Integrals . . . . . . . . . 6.1 Random Measures and Deterministic Integrands . . . . . . . . . . . 6.2 Random Measures and Stochastic Integrands . . . . . . . . . . . . . . 6.3 Random Measures, Stopping Times and Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Generalizations of Martingale Integrals . . . . . . . . . . . . . . . . . . . . 6.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 218 221

Multiple Random Measures and Integrals . . . . . . . . . . . . . . . 7.1 Basic Quasimartingale Spaces and Integrals . . . . . . . . . . . . . . . 7.2 Multiple Random Measures, Part I: Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Multiple Random Measures, Part II: Noncartesian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Random Line Integrals With Fubini and Green-Stokes Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Random Measures on Partially Ordered Sets . . . . . . . . . . . . . 7.6 Multiple Random Integrals Using White Noise Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269 270

189 201 207 215

236 250 261 267

288 303 317 335 350 362 369

Contents

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9

Vector Measures and Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Vector Measures of Nonfinite Variation . . . . . . . . . . . . . . . . . . . 8.2 Vector Integration with Measures of Finite Semivariation, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Vector Integration with Measures of Finite Semivariation, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Some Applications of Vector Measure Integration, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Some Applications of Vector Measure Integration, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random and Vector Multimeasures . . . . . . . . . . . . . . . . . . . . . . 9.1 Bimeasures and Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Bimeasure Domination, Dilations and Representations of Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Spectral Analysis of Second Order Fields and Bimeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Multimeasures and Multilinear Forms . . . . . . . . . . . . . . . . . . . . . 9.5 Complements and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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373 373 378 386 403 420 436 443 447 447 457 468 478 489 494

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

1 Introduction and Motivation

This chapter presents a brief motivation for a comparative analysis of random measures as both a class of general signed measures and a subclass of vectorvalued mappings whose range space is based on a probability space, endowed with a norm or invariant metric, arising from numerous applications. Here probabilistic concepts play a fundamental role along with the basic structures of vector spaces employed. The mappings are σ-additive on a ring of sets. The work indicates the need for a detailed study of these objects together.

1.1 Introducing Vector Valued Measures Let (Ω, Σ0 ) be a pair where Ω is a (nonempty) point set, Σ0 a ring of subsets of Ω, and X a (real or complex) vector space. A mapping Z : Σ0 → X is called an additive function if it is additive on disjoint sets so that A, B ∈ Σ0 , A ∩ B = ∅, implies Z(A ∪ B) = Z(A) + Z(B). This is well-defined since X is a vector space, whence S has an additive operation. If An ∈ Σ0 , n ≥ 1, An ∩ Am = ∅ for n 6= m and ∞ n=1 An ∈ Σ0 , imply P S (1) Z( n An ) = n Z(An ),

then Z(·) is called a vector measure provided the right side sum is endowed with a meaning in X. This demands that one must have a topology for X. In case X is a Banach space, e.g., X = Lp (S, S, µ) on some measure space (S, S, µ) and 1 ≤ p ≤ ∞, then the series is assumed to be convergent in one of its topologies—usually this convergence is taken to be in the norm topology or at least in its “weak topology”. The latter means, S is a σ-algebra and P S (2) (x∗ ◦ Z)( n An ) = n (x∗ ◦ Z)(An ),

for each continuous linear functional x∗ on X, and S then the numerical series on the right of (2) converges unconditionally since n An is the same set for all permutations {Aik , k ≥ 1} of the sequence {An , n ≥ 1} so that the right

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1 Introduction and Motivation

side of (2) must have the same value. This assumes that there exist sufficiently many continuous linear functions x∗ on X, distinguishing its points, a fact valid for all Banach spaces but not for all (topological) vector spaces. In fact if X = Lp (S, S, µ), 0 ≤ p < 1, it is a metric space on which there need not be such x∗ . Yet the convergence of this series in (1) can be demanded. Such (translationinvariant) metric spaces are important in many applications. In fact the space X = L0 (S, S, µ) with µ(S) < ∞ is metrized by d : (f, g) → d(f, g), f, g ∈ X, where Z |f − g| dµ, (3) d(f, g) = S 1 + |f − g|

and has the following property. It is not difficult to see that d(f, g) = d(f − g, 0) [also written as d0 (f − g) of one variable for ease] is a metric so that {X, d0 (·)} is a complete linear metric space which is translation-invariant but is not a Banach space, since with k · kX = d0 (·) it is clearly not true that kaf kX = |a|kf kX, a ∈ R (or C), a necessary condition for the metric to be determined by a norm. However, the metric given by (3) with µ taken as a probability measure (µ(S) = 1) for convenience, is equivalent to convergence in measure (see Proposition 1.3.2 in Section R 3 below). But it will be seen that such spaces as Lp (S, S, µ), with dp (f, 0) = S |f |p dµ, 0 < p < 1 or p = 0 in (3), will be of interest in probabilistic applications. When X is a Banach space, Z(·) is called a vector measure which is of great interest in abstract analysis and if X = L0 (µ), with µ(S) = 1, Z(·) satisfying (1) with the metric topology (3), is called a random measure. The vector measure, with X a general Banach space, is used for numerous integral representations and other problems. Thus it is evident that both these vector-valued set functions are of considerable interest and they demand different methods of analysis. These will be studied in some detail in the following pages from several angles motivated by important applications. In the published proceedings of a symposium on the “Mathematical Consequences and Developments arising from the Hilbert Problems” sponsored by the American Mathematical Society in 1974, the well-known French mathematician P. Cartier states (Part I, p. 76 on Random Measures and Point Processes) as follows: “We know barely more than a few basic facts and scattered results. Recent work by Lenard for instance indicates that even the foundations (on point processes ?) need re-examination.” Keeping this admonition in view, and basing the work on the well-understood and familiar (vector-valued) measure theory that is employed by the mathematical community, it will be our aim to present a detailed treatment of this subject towards meeting the above concerns.

1.2 Basic Structures

3

1.2 Basic Structures Analysis of measures starts on domains which are nonempty collections of sets (of some fixed nonempty set) having some algebraic structure. Thus a family A of subsets of a nonvoid point set S is a ring if it has the empty set ∅ ∈ A, and is closed under finite unions and differences so that A, B ∈ A ⇒ {A∪B, A−B} ⊂ A and it is a δ-ring if it is somewhat richer, namely it is a ring containing also countable intersections of its members. This will be sufficient for most of our work, and for convenience as well as future applications some further restrictions are also considered. Thus a ring is called an algebra if the whole space S ∈ A and hence is closed under complements. An algebra A, if it is also a δ-ring, is called a σ-algebra since it is then also closed under countable unions and intersections. For most of our work, it is actually enough to begin with even a weaker concept. Namely, a nonempty class A0 of subsets of Ω is termed a semi-ring ifSit is closed under finite intersections, ∅ ∈ A0 , and with n A, B ∈ A0 , A − B = i=1 Ci , Ci ∈ A0 , disjoint, i.e., the difference need not be in A0 , but is a finite disjoint union of members of A0 . For most of our work on random measures, δ-rings play a significant role, while generally σ-rings are preferred for vector measures, i.e., those valued in a Banach space. In all these cases, the basic couple, (S, A) is called a measurable space since all set functions (later ‘measures’) are defined on A which is formed of subsets of a nonempty point set S. Thus some familiarity with the basic aspects of the classical measure theory is essential and the reader is assumed to be acquainted with it. Let us formalize the above concepts used in the introduction. Definition 1.2.1 (a) Let (S, A) be a measurable space with A as a ring, and let X be a (real or complex) vector space. A mapping µ : A → X is an additive set function if for A, B ∈ A, A ∩ B = ∅, one has µ(A ∪ B) = µ(A) + µ(B),

(∈ X).

(1)

(b) If A is a σ-ring and X is a topological vector space, then µ of (1) is σ-additive whenever, An ∈ A, An ∩ Am = ∅, m 6= n, ⇒ [ X ∞ ∞ µ µ(An ), (∈ X) (2) An = n=1

n=1

where the series on the right is convergent in the topology of X, and µ is then termed a vector(-valued) measure. Since the set union is commutative, (2) implies that the series there, if convergent in X, is independent of the order of summation, so that the convergence is unconditional. The properties of the space X and of the ring play important roles in the following work. The vector valued Z(·) defined above can be either a random or a vector measure. The distinction comes from the type of the range space which may be a Banach space or a metric space based on a probability triple (Ω, Σ, P ) where Ω is a set of points representing all the possible outcomes

4

1 Introduction and Motivation

of an experiment and Σ is a σ-algebra (or a σ-ring), of subsets of Ω, called events, that are of any interest for an experiment, enlarged into a σ-algebra for mathematical convenience and P : Σ → [0, 1] is a probability measure attached to each of the events. Thus P (Ω) = 1, 0 ≤ P (A) ≤ 1, A ∈ Σ and P (·) is σ-additive on Σ. Now X = L0 (P ), the space of all scalar (= real or complex) functions on Ω is made into a metric space by the function d : (f, g) 7→ d(f, g) given by (3) of Section 1 above for f, g ∈ L0 (P ). Then X = L0 (P ) is a metric, but not a Banach space and µ : A → X is a random (but not a vector) measure. If X = Lp (Ω, Σ, ν) on some measure space (Ω, Σ, ν) with 1 ≤ p ≤ ∞, then µ is a vector measure and if ν(Ω) = 1, it becomes a random measure as well. Thus random measures can be (a subclass of ) vector measures but need not be so for all applications. A familiar example of a δ-ring that plays a significant part in the following work is as follows. If the domain S is a topological space let A be the collection of all of its bounded sets so that A ⊂ S, A ∈ A if and only if there is a compact set C ⊂ S such that A ⊂ C. In particular if S = Rn , the Euclidean space, then its bounded sets are the familiar ones that can be enclosed in rectangles of finite volumes. Similarly A0 is a semi-ring of Rn if it is the collection of all half-open rectangles {x = (x1 , . . . , xn ) ∈ Rn : ai < xi ≤ bi , i = 1, . . . , n} for all −∞ < ai ≤ bi < ∞, ai , bi ∈ R. The smallest σ-ring (or σ-algebra) that contains a semi-ring A0 (or a δ-ring of bounded open sets) is termed a Borel σalgebra. These ensemble concepts are used constantly in the following work and are fundamental in all of the standard measure theory, even without topology in S but with suitable substitutes. For most of our analysis S will be a Hausdorff topological space and usually also a locally compact one. If X is a one-dimensional vector space (so X = R or C by identification) and µ : A → X is σ-additive where A is a σ-ring, then µ is often called a signed measure. It has the following property which plays an important part when X is an infinite-dimensional Banach space so that µ is a vector measure. Let us present the definition of (Vitali) variation of a (vector or scalar) measure. Definition 1.2.2 Let µ : A → X (Banach space) be additive on a measurable space (S, A). Then its variation on each A ∈ A, denoted |µ|(A) is given by |µ|(A) = sup

n nX i=1

o kµ(Bi )k : Bi ∈ A(A), disjoint n ≥ 1 , A ∈ A,

(3)

where A(A) = {A ∩ B : B ∈ A}, the trace of A on A. This notion of trace A(A) will often be used. If |µ|(S) < ∞ then µ is said to be of bounded variation on the algebra A. So S ∈ A, and this inclusion is always assumed unless stated otherwise. When X is the space of scalars, the signed measure µ has the following useful property. Proposition 1.2.3 Let µ : A → R (or C) be a signed measure on the σ-ring A. Then |µ|(S) < ∞ so that µ is bounded. Moreover

1.2 Basic Structures

5

|µ|(S) ≤ 2 sup{|µ(A)| : A ∈ A}, (and replace 2 by 4 in the case of C) implying that µ has bounded variation automatically. Proof. Taking µ to be real-valued, we assert that sup{|µ(A)| : A ∈ A} is finite. For an indirect argument, if the assertion is false, then for some A ∈ A, sup{|µ(B)| : B ∈ A(A)} = ∞. To derive a contradiction we find a sequence An ↓, An ∈ A, with the property that |µ(An )| > n − 1 for each n and sup{|µ(C)| : C ∈ A(An )} = ∞. This will imply that A0 = limn An = T ¯ n An ∈ A and |µ(A0 )| = ∞ which is impossible because µ(A) ⊂ R (not R), by hypothesis. Thus let ν(A) = sup{|µ(B)| : B ∈ A(A)}. Since ∅ ⊂ A, and µ(∅) = 0 it is clear that ν(·) ≥ 0, and for A1 , A2 ∈ A, ν(A1 ∪ A2 ) ≤ ν(AS 1 ) + ν(A2 ). In fact ν(·) is countably subadditive, since if Bn ∈ A, disjoint and n Bn ∈ A, let C ∈ A so that C ∩ Bn ∈ A, disjoint and ∞ X |µ(C)| = µ(C ∩ Bn ) , since µ is σ-additive n=1

≤

∞ X

|µ(C ∩ Bn )| ≤

n=1

from which one has ν

[ n

∞ X

n=1

ν(C ∩ Bn ),

X Bn ≤ ν(Bn ).

(4)

n

SThere is actually equality here. To see this, excluding a trivial case suppose ν( n Bn ) < ∞, so that ν(Bn ) < ∞, n ≥ 1. By definition of ν, given ε > 0, one can find Cn ∈ A(Bn ) such that ν(Bn ) < µ(Cn ) +

ε . 2n

But the Bn , hence the Cn (⊂ Bn ), are disjoint, so that (by the σ-additivity of µ) ∞ ∞ X X µ(Cn ) + ε = µ(∪∞ (5) ν(Bn ) < n=1 Cn ) + ε ≤ ν(∪n (Bn )) + ε. n=1

n=1

Thus (4) and (5) imply that ν is σ-additive. Next we construct the desired decreasing sequence An on the supposition that {µ(A), A ∈ A} is not bounded. The construction is detailed since it will be used again later. By the unboundedness supposition, there exists an A ∈ A such that, using the trace A(A) of A on A, ν(A) = sup{|µ(B)| : B ∈ A(A)} = ∞.

Taking A1 = A here, we use induction and so can assume that A1 ⊃ A2 ⊃ · · · ⊃ An have already been found such that ν(An ) = ∞ and |µ(An )| ≥ n − 1. Using

6

1 Introduction and Motivation

the definition of ν, ν(An ) = ∞ implies the existence of a B ∈ A(An ) such that |µ(B)| is larger than |µ(An )|, to satisfy the inequality |µ(B)| ≥ |µ(An )| + n. Let An+1 be the B in this case, since ν(An ) = ν((An −B)∪B) = ν(An −B)+ν(B) = ∞, by the additivity of ν. If ν(B) < ∞, then ν(An − B) = ∞ and in this case let An+1 = An − B. Consequently An+1 ⊂ An and |µ(An+1 )| = |µ(An − B)| = |µ(An ) − µ(B)| ≥ |µ(B)| − |µ(An ) ≥ n since by choice |µ(B)| ≥ |µ(An )| + n. Thus in both cases our An+1 ⊂ A and |µ(An+1 )| ≥ n. The desired sequence An ↓ is constructed. T Let Cn = An − An−1 ∈ A, which are disjoint, and limn An = n An ∈ A. The σ-additivity of µ implies that limn µ(An ) = µ(limn An ) and µ

[ n

n X X µ(Cn ) = lim µ(Ck ) = lim µ(An ) = +∞ Cn = n→∞

n

k=1

n→∞

T by our construction of the An -sequence. But since limn µ(An ) = µ( k Ak ) ∈ R and ∞ ∈ / R, we arrived at a contradiction. Consequently µ(A) must be bounded. For the last assertion, consider any disjoint collection B1 , . . . , Bn from A and let Bi1 , . . . , Bik be those elements for which µ(Bij ) ≥ 0, 1 ≤ j ≤ k and let ˜j ˜j ) < 0, so that ˜j1 , . . . , B be the rest for which µ(B B n−k ℓ n X i=1

|µ(Bi )| =

k X j=1

µ(Bij ) −

n−k X ℓ=1

˜j ) = µ(B) − µ(C), (say). µ(B ℓ

Then (3) becomes |µ(S)| = sup

n nX i=1

o |µ(Bi )| : Bi ∈ A, disjoint

= sup{µ(B) − µ(C) : B, C ∈ A} ≤ 2 sup{|µ(A)| : A ∈ A}.

Finally for the complex case, let µ = Re(µ) + i Im(µ). Both terms on the right are real measures for which the above inequality holds. 2 Discussion. There are many other methods of establishing this result, but none is simpler. Since we already showed in the above proof that ν is σ-additive, and clearly ν ≥ µ, we can define ν˜ = ν − µ ≥ 0 which is also σ-additive so that µ = ν − ν˜

(6)

is a decomposition of the real µ as a difference of positive finite measures. But since one can add and subtract any positive finite measure to the right of (6), the decomposition is not unique. However, there is an efficient or canonical decomposition that eliminates this difficulty. Namely, define ν˜(A) = sup{−µ(B) : B ∈ A(A)}.

1.2 Basic Structures

7

Then as in the case of ν(·) proved above, ν˜(·) is also σ-additive, nonnegative, and satisfies (6). In fact consider for any A ∈ A µ(A) + ν˜(A) = µ(A) + sup{−µ(B) : B ∈ A(A)} = sup{µ(A) − µ(B) : B ∈ A(A)}, = sup{µ(C) : C ∈ A(A)},

since the measure here is subtractive,

which is (6). Similarly one shows that |µ|(·) = ν(·) + ν˜(·)

(7)

where ν and ν˜ are called the positive and negative variations of µ, and the resulting decomposition obtained in (6) or (7) is usually called a Jordan Decomposition of the signed measure µ. [In the complex case one has four such (positive) parts, called the Riesz components of a complex µ.] It is also possible to show that ν and ν˜ of (6) are given by ν = µ+ = sup(µ, 0) and ν˜ = µ− = − inf(µ, 0). This allows one to deal with the set of all real measures on a σ-ring A as a lattice. Remark: If a signed measure µ is understood somewhat more generally, as ¯ = R ∪ {+∞, −∞}, then the σ-additivity implies that µ can taking values in R have values +∞ or −∞, but not both. Indeed, if there are elements C, D ∈ A such that µ(C) = +∞ and µ(D) = −∞, then by the additivity of µ one has µ(C ∪ D) = µ(C − D) + µ(D − C) + µ(C ∩ D) = µ(C) + µ(D) − µ(C ∩ D) must be valid. This cannot hold if such C, D exist in A which results in an indefinite form +∞ − ∞. Thus for the following work, without explicit mention to the contrary, signed measures are allowed to take values only in R or C. An important consequence of Proposition 1.2.3 is the following result for measures in any Banach space, and it plays a significant role in our work in the following chapters. Proposition 1.2.4 Let (S, A) be a measurable space and µ : A → X be σadditive, where X is a Banach space and A is a σ-algebra. Then µ is bounded in the sense that kµ(S)k = sup{kµ(A)kX : A ∈ A} < ∞.

(8)

Moreover, µ has Fr´echet or finite semi-variation, i.e., kµk(S) < ∞ where n o n X ai µ(Ei )k : |ai | ≤ 1, ai ∈ C, Ei ∈ A, disjoint kµk(S) = sup k

(9)

i=1

but not necessarily of finite variation and only |µ|(S) ≤ ∞ as in (3) above. [Here replacing S by E and A by A(E), the trace of A on E, one gets the above concepts restricted to E.]

8

1 Introduction and Motivation

Proof. In the following argument we use two standard and fundamental results from Banach space theory (cf., Dunford-Schwartz (1958), II.3.10, or the author, Rao (2004), Section IV.6), namely that there exist enough continuous linear functionals x∗ on X, i.e., x∗ ∈ X∗ , distinguishing points of X, [this is the HahnBanach theorem] and the uniform boundedness principle. Thus let x∗ ∈ X∗ and consider the scalar measure νx∗ = x∗ ◦µ : A → R(C). Then by Proposition 1.2.3 above, |νx∗ |(S) < ∞ for each x∗ , since |νx∗ |(S) = sup{|(x∗ ◦ µ)(A)| : A ∈ A}, and the mapping M : x∗ → |νx∗ |(S) is subadditive, so that M (x∗1 + x∗2 ) ≤ M (x∗1 ) + M (x∗2 ) ≤ (kx∗1 k + kx∗2 k)kµk(S), one can invoke the uniform boundedness result (cf., e.g., Rao (2004), IV, 6.5) to conclude supkx∗ k≤1 M (x∗ ) ≤ kµk(S) < ∞. The key step that the scalar measure x∗ ◦ µ is bounded for each x∗ ∈ X∗ , is a consequence of Proposition 1.2.3 above. This gives (8) and (9). We establish the last part with a counterexample, given immediately below, for a vector measure µ. 2 Example 1. The following well-known simple example, shows that, since the semi-variation of a vector measure is always finite in a Banach space, by (8) above, it is strictly weaker than the variation itself in such spaces. Thus let {An , n ≥ 1} be a disjoint sequence of subsets of a finite interval [a, b], −∞ < a 0 where λ is the Lebesgue measure on [a, b]. Consider µ(An ) = χAn so that µ : B([a, b]) → B([a, b]), where B([a, b]) is the σ-algebra of Borel sets of the interval and B([a, b]) is the Banach space of bounded Borel functions on [a, b] under the uniform norm. Then for any n, ∞ X

k=1

kµ(Ak )k =

n X

k=1

kχAk k + kχ∪∞ k = n + 1. k=n+1 Ak

Hence |µ|([a, b]) = ∞ so that µ does not have finite variation. But one can compute the semi-variation as: n o n X ai χAi : |ai | ≤ 1, ai ∈ C kµk([a, b]) = sup i=1

n o n X χAi k : Ai ∈ B([a, b]), disjoint ≤ sup k i=1

≤ sup{kχSni=1 Ai k : Ai as above} ≤ 1.

Thus the semi-variation of µ is finite but its variation is infinite. For a detailed study of vector measures, one can refer to Dinculeanu (1967) and Uhl and Diestel (1977). The following property is also of interest.

1.2 Basic Structures

9

Proposition 1.2.5 If µ : A → X is a vector measure, as above and |µ|(·), its variation-measure, then it is σ-additive but the semi-variation is only σsubadditive. Proof. That both the variation and semi-variation are σ-subadditive is routine, although not entirely trivial, and it will be left to the reader. The fact that |µ|(·) is superadditive (and hence σ-additive) is obtained as follows. S Let An ∈ A be disjoint and suppose, for non-triviality, that |µ|( n An ) < ∞, implying |µ|(An ) < ∞, n ≥ 1. By (3), given εk > 0, one can find Akm ∈ A, such that Akm ⊂ Ak and disjoint in the index m such that εk <

mk X

m=1

and

ℓ X

k=1

εk <

kµ(Akm )k ≤ kµ(Ak )k,

mk ℓ X X

k=1 m=1

kµ(Akm )k ≤ |µ|

Sℓ

k=1

Ak .

This gives on letting ℓ → ∞ and taking suprema on Akm ⊂ σ-super-additivity, since εk > 0 is arbitrary: ∞ X

k=1

|µ|(Ak ) ≤ |µ|

S∞

k=1

S

k≥1

Ak ∈ A the

Ak .

It follows, with the σ-sub-additivity, that |µ|(·) is σ-additive. 2. In showing that µ : A → X is bounded in Proposition 1.2.4, Gould (1965) observed that one may avoid an application of the uniform boundedness by adapting an indirect proof, as in Proposition 1.2.3. A sketch is as follows. Thus if µ is not bounded, kµ(E)k is arbitrarily large for some E ∈ A. So we let N ≥ 2kµ(E)k, and it is possible to find disjoint E1 , E2 ∈ A, E = E1 ∪ E2 such that at least one has a large value kµ(Ei )k, i = 1, 2. Suppose E1 has that property, i.e., kµ(E1 )k ≥ N . Consequently with the triangle inequality and additivity of µ, it follows that kµ(E2 )k = kµ(E) − µ(E1 )k ≥ kµ(E1 )k − kµ(E)k > N −

N N = . 2 2

So one of µ(Ei ) is unbounded. Now proceed as in Proposition 1.2.3 and find kµ(Ek )k > k, Ek ↓ . But x∗ ◦ µ is a signed measure, and by Proposition 1.2.3 it is bounded for each x∗ ∈ X∗ . Then by the preceding selection of Ek ∈ A, lim supk, kx∗ k≤1 |(x∗ ◦ µ)(Ek )| = ∞ contradicting the result of Proposition 1.2.3. Thus µ is bounded in X. The above result also holds for random measures in Xp = Lp (P ), if p ≥ 1 on a probability space (Ω, Σ, P ), p ≥ 1. But for 0 ≤ p < 1, it is a known fact that X∗p = {0} for (Ω, Σ, P ) as the Lebesgue unit interval. Thus the preceding results depending on the non-triviality of X∗p (i.e., the Hahn-Banach property)

10

1 Introduction and Motivation

cannot be used in a general study. But the random measures with values in Xp , 0 ≤ p < 1 are also important for applications. In this case, one assumes the σ-additivity of Z : A → Y, just a complete linear metric space in its (metric) topology so that the Hahn-Banach property may not be invoked, and one seeks some easily verifiable conditions for various concrete spaces Y such as Xp above. Thus we discuss several results on the subject that will be useful for applications at appropriate places in later studies.

1.3 Additivity Properties of Vector Valued Measures Let (S, A) be a measurable space and X a real S or complex topological vector space. If An ∈ A is a disjoint sequence with n An ∈ A, which is assumed if A is only a ring but is automatic for σ-rings, then µ will be called a vector valued measure if [ X µ µ(An ) (1) An = n

n

provided the right side sum converges in the topology of X. In our work the topology of interest is one which is invariant under translation and for which addition as well as scalar multiplication are continuous operations. X is called a Fr´echet space when it is a complete metric space. Thus d(x, y) = d(x − y, 0) and the distance function d(·, ·) will be denoted by d(·) itself for simplicity [d(x − y, 0) = d(x − y)] somewhat ambiguously. The Fr´echet ‘norm’ d(·) = k · k does not have the homogeneity property kaxk = |a|kxk for all scalars ‘a’s and unlike in the case of Banach spaces, X∗ , the space of continuous linear functionals (also called an adjoint space) need not have enough points (may have only ‘0’ element!). The crucial example is Lp (P ), 0 ≤ p < ∞ which is important in applications. Here the spaces for 0 ≤ p < 1 need not have the Hahn-Banach property of points being distinguished by the ‘adjoint space’. For vector measures µ with values in a Banach space X, the fact that µ is defined on a σ-ring or a σ-algebra makes very little difference as the following result due to I. Kluv´anek, which is not but should be well-known, shows. Proposition 1.3.1 Let (S, A) be a measurable space with A as a σ-ring of subsets of a (nonempty) point set S, and µ : A → X be σ-additive in a Banach space X. Then there exists an A0 ∈ A such that µ(A(A0 )) ⊂ X and µ vanishes outside A0 , where A(A0 ) = {A ∩ A0 : A ∈ A} is a σ-algebra (the ‘trace’ of A on A0 ). Moreover µ is bounded on A(A0 ), hence on A. [Thus for σ-additive measures, valued in a Banach space (or the space of scalars), the theory of integration will be on familiar lines.] Proof. By Proposition 1.2.4 above, the σ-additivity of µ : A → X, a Banach space, implies its boundedness in the norm of X. The point now is to find an A0 ∈ A on which µ “lives”. For this it is enough to show that {A ∈ A : µ(A) 6= 0} is at most countable. If D is the set of all such (countable) collections,

1.3 Additivity Properties of Vector Valued Measures

11

partially ordered by inclusion, then every finite subcollection of D has an upper bound in D, namely their union which is clearly countable. Then by Zorn’s S lemma, D has a maximal element A0 ⊂ A. Take A0 = {A : A ∈ A0 } ∈ A, ˜ 6= 0, then since the latter is a σ-ring. If there is an A˜ ∈ A, A˜ ∩ A = ∅ and µ(A) ˜ the enlarged collection {A} ∪ A0 ∈ D and contradicts the maximality of A0 . Thus it suffices to establish that the collection {A ∈ A : µ(A) 6= 0} is countable to complete the proof. For this, let {An ∈ A} be a disjoint collection (An ∩Am = ∅ for n 6= m) such that µ(An ) 6= 0. If ε > 0 is given, then {An : kµ(An )k > ε} is asserted to be finite when the An are disjoint. Indeed, if this is not true, consider an infinite (disjoint) such collection An and let B = ∪n An ∈ A. Using the fact that X is a Banach space, one has for each x∗ ∈ X∗ , with the σ-additivity of x∗ ◦ µ, the signed measure, X (x∗ ◦ µ)(B) = (x∗ ◦ µ)(An ) ∈ R (or C) (2) n

and the series converges (unconditionally) implying that limn→∞ (x∗ ◦ µ)(An ) = 0 uniformly in kx∗ k ≤ 1 since X∗ is also a Banach space. By the uniform boundedness theorem, this implies that limn kµ(An )k = 0, contradicting the choice of An (kµ(An )k > ε). Therefore for each m ≥ 1, the collection Am = 1 } must be finite, say has Nm members. Then {A ∈ A : kZ(A)k ≥ m A0 =

∞ [

n=1

An [=

∞ [

{A : kµ(A)k ≥

m=1

1 }] ⊂ A m

is countable implying that the set {A ∈ A : µ(A) 6= 0} is countable. Thus this collection with the largest element A0 is a desired “support” set in the (trace) σ-algebra A(A0 ) of the proposition. The boundedness of µ on A (or A0 ) was already established in Proposition 1.2.4 above using the fact that X is a Banach space. 2 Remarks: 1. It should be noted that the above argument depends on the fact that the range X of µ is a Banach space and that the signed measures are restricted to taking values in R or C. Also the fact that A is a σ-ring is crucial. If A is only a δ-ring, then the assertion does not hold as easy counterexamples show. Thus if µ : A → Lp (P ), 0 ≤ p < 1, these considerations do not extend and an alternative approach with different conditions will be necessary. 2. It is also important to note that in a Banach space the norm and weak σadditivities are equivalent by the classical Orlicz-Pettis theorem. The following vector metric space is frequently used as range of random measures: Proposition 1.3.2 Let (Ω, Σ, P ) be a probability space and Lp (P ) be a space of scalar random variables on Ω, 0 ≤ p < 1. Then it is a complete metric space when, as usual, equivalence classes are considered, under the distance functional k · kp : f 7→ kf kp where

12

1 Introduction and Motivation

kf kp =

(R

RΩ

|f |p dP,

|f | Ω 1+|f |

dP,

0
(3)

and the metric topology of k·k0 is equivalent to that of convergence in probability. Proof. Here we treat the case of p = 0, leaving 0 < p < 1 as an exercise. It 1 R should be remarked that if kf k′p = Ω |f |p dP p , then it is homogeneous (i.e., kaf k′p = |a| kf k′ ) but it does not satisfy the triangle inequality so that k · k′p will not define a metric. On the other hand k · kp of (3), 0 < p < 1 satisfies the triangle inequality but it is not homogeneous. The case of p = 0 must be considered differently as follows. First observe that from classical analysis (cf., e.g., Hardy-Littlewood-P´ olya (1934), p.130) a concave increasing function ϕ : R+ → R, ϕ(0) = 0, is representable as: Z x ϕ(x) = ϕ′ (t) dt, x > 0 (4) 0

where ϕ′ (·) is non-increasing, left-continuous and ϕ′ (t) ≥ 0. Hence for x, y > 0, Z x Z x+y ′ ϕ(x + y) = ϕ (t) dt + ϕ′ (t) dt 0 x Z y = ϕ(x) + ϕ′ (x + t) dt Z0 y ϕ′ (t)dt = ϕ(x) + ϕ(y). (5) ≤ ϕ(x) + 0

Taking ϕ(x) =

|x| 1+|x|

here, it follows that Z ϕ(f + g) dP kf + gk0 = Z ZΩ ϕ(g) dP = kf k0 + kgk0 , ϕ(f ) dP + ≤ Ω

(6)

Ω

and hence from d(f, g) = kf − gk0 = d(f − g, 0), k · k0 is an invariant metric. From the fact that fn → 0 in measure implies for each ε > 0 Z Z Z |f | |fn | d(fn , 0) = dP + dP ≤ dP Ω 1 + |fn | [|fn |<ε] 1 + |fn | [|fn |≥ε] ≤ P [|fn | ≥ ε] +

ε |x| , since ϕ(x) = ↑ 1, as |x| ↑ ∞. 1+ε 1 + |x|

Thus limn→∞ d(fn , 0) ≤ ε and fn → 0 in the metric d(·, ·) of L0 (P ). Conversely if d(fn , 0) → 0, then for any ε > 0 Z ε 1+ε dP P [|fn | ≥ ε] = ε 1 + ε [|fn |≥ε] Z 1+ε |x| |fn | ≤ dP, since ↑. ε 1 + |f | 1 + |x| n [|fn |≥ε]

1.3 Additivity Properties of Vector Valued Measures

13

1+ε lim d(fn , 0) = 0, and so fn → 0 in measure. Hence lim P |fn | ≥ ε ≤ n→∞ ε n→∞ Thus both are equivalent, and this also easily implies (the details are left to the reader) the completeness of L0 (P ). 2 If fn ∈ Lp (P ), p > 0, then kfn kp → 0 implies fn → 0 in measure, since for ε > 0, Z Z 1 1 P [|fn | ≥ ε] = p εp dP ≤ p |fn |p dP ε [|fn |≥ε] ε [|fn |≥ε] 1 ≤ p kfn kp → 0, as n → ∞. ε R Since Ω |fn |p dP is not necessarily finite, the converse implication is clearly false. In all cases one has kfn kp → 0 ⇒ fn → 0 in measure (P ), for 0 ≤ p < ∞. With these properties established, we can restate the σ-additivity of the mapping Z : A → Lp (P ), p ≥ 0, as follows. Definition 1.3.3 An additive vector-valued mapping Z : A → Lp (P ), is σadditive on the ring A of subsets of S, if (i) p ≥ 1 the conceptSis as in Definition 1.2.1(b) and if (ii) 0 ≤ p < 1, then for disjoint An ∈ A with n An ∈ A, [ X Z( An ) = Z(An ), (7) n

n

the series converging in probability measure for p = 0, and (also equivalently) in the metric k · kp of (3) for 0 ≤ p < 1.

Since Lp (P ), 0 ≤ p ≤ ∞, has a metric topology, (7) implies that the series there is unconditionally convergent in the topology determined by k · kp , p ≥ 0. This will be understood in all the following analysis without further explanation. Integration relative to a vector-valued measure of a scalar (or a vector) function is a nontrivial extension of that with a scalar measure. In order to use some classical ideas, it will be desirable to find a positive measure that regulates the vector case. We detail the argument as to how such a controlling (positive) measure can be found, which unfortunately is not entirely simple. Some readers may postpone reading the following proof for a later time. However, this is the most general result known in the Banach space context, and it plays a key role in the above stated integration theory to be developed where most of our vector measures have infinite (Vitali) variation and that the usual methods cannot be directly used. [Here-after S ∈ A will be assumed. Cf., Proposition 3.1 above.] Theorem 1.3.4 Let (S, A) be a measurable space and µ : A → X, (a Banach space), be σ-additive. Then there exists a measure λ : A → R+ which controls µ in the sense that for each ε > 0 there is a δ > 0 such that for A ∈ A, λ(A) < ε ⇒ kµ(A)k < δ, i.e., µ is λ-continuous. The result is extendible if S is locally compact, A is a δ-ring of bounded Borel sets of S, but then the controlling ¯ + will only be locally finite (or finite on compact [or bounded] sets of λ: A → R A or ‘strictly localizable’, a condition still more general than σ-finiteness).

14

1 Introduction and Motivation

Proof. We present a novel argument following Gould (1965) which is different from that in Dunford and Schwartz (1958) who jointly established this key result with R. G. Bartle in 1955. The following is given in a series of steps for convenience, but is also nonconstructive being an indirect one. 1. First observe that µ is λ-continuous, is equivalent to an apparently weaker condition that µ vanishes on λ-null sets. Indeed let µ vanish on λ : A → R+ , null sets. The equivalence is classical if µ is a signed measure, although the proof there is also not entirely simple (cf., e.g., Rao (2004), p.199 and p.268). To streamline the argument, we first show that limλ(A)→0 kµ(A)k = 0 which implies that the semi-variation satisfies limλ( A)→0 kµk(A) = 0. Indeed for each x∗ ∈ X∗ we have: n o n X ai µ(Ai )k : |ai | ≤ 1, ai ∈ R, Ai ∈ A(A), disjoint kµk(A) = sup k i=1

n o n X ai (x∗ ◦ µ)(Ai ) : ai , Ai as above = sup sup kx∗ k≤1

≤ sup sup kx∗ k≤1

i=1

n nX i=1

o |x∗ ◦ µ|(Ai ); Ai ∈ A(A), disjoint

≤ 4 sup |x∗ ◦ µ|(A),

(8)

kx∗ k≤1

since variation = semi-variation for scalar measures, = 4kµ(A)k ≤ 4 sup{kµ(B)k : B ∈ A(A)} = 4˜ µ(A), (say).

(9)

We now establish the stronger statement that limλ(A)→0 µ ˜(A) = 0 is true. This is again accomplished with an indirect argument. If the assertion is false, then there exist a δ > 0 and a sequence An ∈ A, n ≥ 1, An ↓ ∅ (so λ(An ) → 0) such that µ ˜(An ) > δ for all n. We follow the format of Proposition 1.2.3 and construct a sequence Bn ↓ from the An collection that gives us the desired contradiction. To get to µ from µ ˜, its definition allows one to find A ∈ A “close to A1 ”, such that kµ(A ∩ A1 )k > δ (because of ‘sup’ involved), and since the An ↓ ∅ so also A ∩ An ↓ ∅. Then choose nk such that kA ∩ Ank k < 2δ and let B1 = A ∩ A1 − A ∩ Ank ∈ A so that kµ(B1 )k = kµ(A ∩ A1 ) − µ(A ∩ Ank )k, since A1 ⊃ Ank and µ is additive δ δ ≥ kµ(A ∩ A1 )k − kµ(A ∩ Ank )k > δ − = . 2 2 Using induction, as in the proof of the scalar case earlier, take Bk = A ∩ (Ank − Ank−1 ). Then Bk ⊂ B1 ∩ Ank − B1 ∩ Ank−1 ⊂ Ank − Ank−1 , and kµ(Bk )k > 2δ . Since the Ank − Ank−1 are disjoint the Bk sequence is disjoint as well, and hence [ X µ µ(Bk ) ∈ X, by the σ-additivity of µ, Bk = k

k

1.3 Additivity Properties of Vector Valued Measures

15

and the series converge, weakly, but also strongly by the classical OrliczPettis theorem, (cf., e.g., Rao (2004), IV 6.6. on p.253 for a detailed sketch). Hence limk→∞ kµ(Bk )k = 0 because of the (strong) unconditional convergence. Since by the choice of Bk (as kµ(Bk )k > δ2 all k) we have a contradiction. This property can be extended if Bk ↓ B; Bk , B ∈ A, so that Bk − B ↓ ∅. Thus we may choose n0 such that µ ˜(Bn − B) < ε for n ≥ n0 . But µ ˜(·) is easily seen to be σ-subadditive and one has from Bn ↓ B, µ ˜(B) ≤ µ ˜(Bn ) = µ ˜((Bn − B) ∪ B) ≤ µ ˜(B) + µ ˜(Bn − B) < µ ˜(B) + ε. ˜n ∈ This implies that limn µ ˜(Bn ) = µ ˜(B) = µ ˜(limn Bn ), Bn ↓ B. If B ˜ ˜ ˜ ˜ A, Bn ↑ B(∈ A), so B − Bn ↓ ∅, and for any ε > 0, there is n ˜ 0 such that ˜−B ˜n ) < ε. Again by subadditivity, n ≥ n˜0 ⇒ µ ˜(B ˜ =µ ˜ −B ˜n ) ∪ B ˜n ) ≤ µ ˜ −B ˜n ) + µ ˜n ) ≤ µ ˜n ) + ε < µ ˜ + ε. µ ˜(B) ˜((B ˜(B ˜(B ˜(B ˜(B) Since ε > 0 is arbitrary, we have for any monotone sequence Cn ∈ A, Cn → C, that limn µ ˜(Cn ) = µ ˜(limn Cn ). 2. We now establish that µ ˜ is λ-continuous, i.e., for ε > 0, there is a δ > 0 such that A ∈ A, λ(A) < δ ⇒ kµ(A)k < ε must hold. If this is false, then for an ε > 0, there is a δ > 0 such that for some sequence An ∈ A, µ ˜(An ) > δ when S ε λ(An ) < n+1 . Here An need not be monotone. Let Bn = k≥n Ak ∈ A so 2 that Bn ↓ ∅ and µ ˜(Bn ) ≥ µ ˜(An ) > δ and for the positive measure λ, by its σ-subadditivity λ(Bn ) ≤

X

k≥n

λ(Ak ) < ε

X 1 = ε. 2k

k≥1

Thus 0 = limn λ(Bn ) = λ(limn Bn ) but that µ ˜ (hence µ) does not vanish on λ-null sets, contradicting the supposition. So µ must be λ-continuous. This argument is originally due to B. J. Pettis. 3. The preceding result admits an extension for an arbitrary bounded collection of vector measures {µτ , τ ∈ I} on A into X which are σ-additive simultaneously, i.e., the σ-additivity of µτ is uniform in τ in that [ X µτ (Cn ) ∈ X, Cn ∈ A, disjoint (10) Cn = µτ n

n

uniformly in τ. This may be reduced to the preceding step by considering a “vector of vector measures”: µ = {µτ , τ ∈ I} : A → XI = ⊗I X, the tensor product of X with itself “I times” (just as Rn is identified with N n I i=1 Ri , Ri = R) with any norm derived from X that makes X also a Banach space on the same field of scalars, e.g., if x(= (xi ∈ X, i ∈ I)) let P 1 |kx|kp = ( i kxi kp ) p , p ≥ 1, and this is an ℓpX -norm. Since ℓpX ⊂ ℓp∞ the weakest one here is to let |kx|k = |kx|k∞ = supτ kxτ k < ∞, and it is known (and easily verified) that {XI , |k · |k} is a Banach space. Now µ : A → XI

16

1 Introduction and Motivation

is σ-additive because of the uniform σ-additivity of µτ . Hence by step 2 above, if each µτ is λ-continuous uniformly in τ , then µ = {µτ , τ ∈ I} is uniformly λ-continuous. In fact we can make a stronger statement that λ(A) ≤ supτ µ ˜τ (A), in addition to µ being λ-continuous, as follows. 4. The argument is again indirect. We observe that the desired continuity of µ = {µτ , τ ∈ I} is governed by a finite subset J of I, i.e., for each ε > 0, there is a finite J(= Jε ) ⊂ I such that |µj |(A) = 0, j ∈ J ⇒ |µi |(A) < δ for all i ∈ I. If this is not the case, there exists an ε > 0 and an infinite sequence {µτn , n ≥ 1} and a sequence An ∈ A suchSthat |µτi |(Ai ) = 0, i = 1, . . . , n, but |µτn+1 |(An ) ≥ ε, n ≥ 1. If Bn = k≥n Ak , Ak ↓ ∅, then |µτn+1 |(Bn ) ≥ ε, n ≥ 1 and limn µτi (Bn ) = µτi (limn Bn ) = 0, by the (uniform) σ-additivity of µτi established in the preceding step. This is impossible since |µτi (Bn )| ≥ ε. Thus there can be no such ε > 0, and hence there is a finite set Jε ⊂ I as originally asserted. Take ε = k1 and let Jk ⊂ I be the corresponding finite subset of I such that |µj |(A) = 0 for j ∈ Jk and Pk |µi | |µi (A)| < ε for all i ∈ I. Let λk = i=1 2−i . Then λk : A → R+ 1 + |µi |(S) is a (finite) measure satisfying λk (A) = 0 ⇒ |µi |(A) < k1 , for all i ∈ I, and P P |µi |(A) ≤ 1. If we now let λ = n≥1 2−n λn (·), then λk (A) = ki=1 2−i 1 + |µi |(S) λ is a finite σ-additive function such that µ = {µτ , τ ∈ I} is λ-continuous by step 3 above, i.e., µτ is uniformly λ-continuous, τ ∈ I. 5. We now apply the above result for each µ : A → X, by considering the collection {x∗ ◦ µ, x∗ ∈ X∗ (= I)}, kx∗ k ≤ 1, which is a bounded set of scalar measures since |(x∗ ◦ µ)(A)| ≤ kx∗ k kµ(A)k ≤ µ ˜(A), A ∈ A for each kx∗ k ≤ 1. Hence x∗ ◦ µ(= µi ) is uniformly σ-additive with µ ˜ having the necessary monotonicity properties, irrespective of x∗ . There is thus a λ : A → R+ that controls µ by the preceding step, and the main part of the theorem is established. 6. It remains to extend the result for δ-rings D, say, instead of the σ-algebra A used thus far. Let S be locally compact and A ⊂ S, a bounded Borel set. Here D is the δ-ring of all bounded Borel sets (by hypothesis) and A ∈ D, and µ : D → X is a vector measure. Let µA = µ|D(A) where D(A) is the trace of D on A, which is a σ-algebra. By step 5 there is a λA : D(A) → R+ such that µA is λA -continuous. If A, B ∈ D, then λA and λB can be assumed to agree on A ∩ B, so that λA |D(B) = λB |D(A) = λAB on D(A ∩ B). Next ¯ + which is locally by a gluing process one can define a measure λ : D → R finite and µ is λ-continuous although λ will only be a ‘strictly localizable’ measure. If S can be covered by a countable collection of such An (or S is countably compact), then λ will be σ-finite. In general λ is only localizable relative to which the vector measure µ : D → X is absolutely continuous. In the σ-compact case, λ being σ-finite, it can be replaced by an equivalent finite measure there, so one can assume λ to be a finite positive controlling measure in that space which suffices for the work. 2

1.3 Additivity Properties of Vector Valued Measures

17

Remarks: 1. A gluing process in a locally compact space, considered in the last step of the above proof is similar to that described in Hewitt and Ross (1963, Sec. 11.45 on p. 133) and a more general case is discussed in Schwartz (1973, p.44). The result extends to δ-rings from σ-algebras used in the theorem. In fact, Dinculeanu(2000, Theorem 1.3.12, pp. 54-55) gives this extension for general (vector) measure spaces (S, S, µ) in which S is a δ-ring such that S = ∪∞ n=1 Sn where Sn ∈ S, and on occasion we may use this extension with a brief reference to the work above. This process need not be elaborated further here. 2. Another proof of this important result, using the Stone representation of Boolean algebras will be outlined in Exercise 4 below. The key aspect of controlling the regularity properties of a vector measure by a fixed positive measure which is important for developing an integration process relative to such measures, is not simple in contrast to that of signed measures. Also it should be noted that the range space X be a Banach space is essential for the above argument. In case X is a Fr´echet space such as Lp (P ), 0 ≤ p < 1, alternative methods with related assumptions have to be found, since the adjoint space of X can be trivial and the basic step 1 of the above proof breaks down. Some comments on the above work are in order. Specializing X to be a Hilbert space one can use its geometry effectively and in that case the random measure has special and interesting features. So Chapter 2 will be devoted to an extended analysis of the corresponding structure and various applications. One of the main reasons to search for a controlling measure is that the vector valued set functions need not have even locally finite variation (the Brownian or Weiner measure being a prime example) and the usual Lebesgue-Stieltjes type integration cannot be employed here with simple modifications. The necessary tools and results are contained in Chapters 3 to 5. Also it becomes clear in these applications that one has to focus on the δ-rings and then seek extensions. Chapter 6 illustrates this point. The distinction with the familiar LebesgueStieltjes type integration relative to a scalar integrator when the integrand is possibly a vector function and the present work with vector-valued integrators (scalar or vector integrands) will be prominent. These questions will be treated in the rest of this volume with greater specificity, and many specializations as well as applications are shown to justify our viewpoint. Hereafter σ-additive Banach space-valued set functions will be called vector measures, and those ranging in vector spaces based on some probability measure, such as Lp (P ), p ≥ 0, referred to as random measures. In the latter cases, metric convergence is used for σ-additivity and specializations as well as refinements resulting from the underlying probability measure will be considered. In the former one has the availability of weak convergence that can be used for the σ-additivity of vector measures because of the fact that this is equivalent to the (stronger) metric or norm convergence, due to the classical Orlicz-Pettis theorem. But this will not be true for non-locally convex spaces (such as the Lp (P ), 0 ≤ p < 1). Related integration of the latter will be considered in later chapters which is somewhat motivated also by Dobrakov’s view-point and his

18

1 Introduction and Motivation

publications on the subject. The fact that the range vector space has probabilistic structure allows a deeper study of measures in Lp (P ), p ≥ k ≥ 2, leading to some interesting applications using the theory of bimeasures and multi (or poly) measures. This is also discussed later on and used in integral representations of random process with respect to such (random) measures in the following Chapters. On the other hand if 0 ≤ p < 2, then the analysis splits into two parts, namely, when 1 ≤ p ≤ 2 and 0 ≤ p < 1 where martingale and stable measures enter in these cases respectively. Some of these are also discussed in later Chapters. The last part treats the subject based on (and influenced by) Brownian motion integrals and the related Itˆ o analysis. This is an outline and a preview of some aspects of the work to be detailed in the following pages.

1.4 Complements and Exercises 1. Let (Ω, Σ, P ) be a probability space and Lp (P ) be the usual Lebesgue function space on it. If f ∈ Lp (P ), define (as in the text) (R |f |p dP, 0
2.(a)

(b) (c)

3.

Show that dp (·) is a metric functional and {Lp (P ), p > 0} is a complete metric space, dp (af ) = |a| dp (f ) if p ≥ 1 but not for 0 < p < 1 for all scalars a. Here completeness means if dp (fn − fm ) → 0 as n, m → ∞, then there is an f ∈ Lp (P ) such that dp (fn − f ) → 0 as n → ∞. The result also holds for p = ∞ and p = 0, if we set d∞ (f ) = inf{k > 0 : P [|f | > k] = 0} R |f | and d0 (f ) = Ω 1+|f | dP as discussed in Proposition 1.3.2 above. p Let (S, A) be a measurableSspace, and PZ : A → L (P ), 0 < p < ∞ be a random measure S where Z( n An ) = n Z(An ) for any disjoint sequence An ∈ A with n An ∈ A, the series converging in the metric dp (·) of the range. Show that this implies that the series also converges in probability. In the above, with 0 ≤ p < ∞, suppose Z(·) takes only positive values, i.e., Z : A → (Lp (P ))+ . Then show that the convergence of the series is also a.e. [This part is for readers who are familiar with a standard course on Probability theory.] In part (b) above suppose Z : A → Lp (P ), not necessarily restricted to the positive part, has independent values on disjoint sets i.e., Z(Ai ) and Z(Bj ) are independent if Ai ∩ Bj = ∅, 1 ≤ i 6= j ≤ n, n ≥ 1. Show that the series again converges pointwise a.e. [Remark : This was a classical theorem due to P. L´evy which states that for a series of independent random variables convergences in probability and pointwise a.e are equivalent. See e.g., Chung (1964, p.120) or Lo`eve (1963, p.249).] This problem discusses the range space L∞ (P ) which was omitted so far. Again let Z : A → (L∞ (P ))+ be σ-additive positive random measure as in 2(b) above. Using the order relation (f, g ∈ L∞ (P )), f ≤ g to mean

1.4 Complements and Exercises

19

f (ω) ≤ g(ω) a.e. relative to S which the space is a vector lattice), if An ∈ A is a disjoint sequence [with n An ∈ A], Z(·) is σ-additive whenever m [ X Z( An ) = lubm Z(Ak ), Z(∅) = 0, Z(A) ≥ 0, ∀A ∈ A. n

k=1

Z(·) will be termed a (positive) random measure. Much of the Lebesgue type integration can be developed after observing for An ∈ A, An ↓ ∅, Z(Ak ) < ∞ for some k ≥ 1, implies glbn Z(An ) = 0. [The order topology employed here is weaker than the norm topology, and so a separate development is needed.] Verify that the following outline is sufficient. (i) Observe that L∞ (P ) can be replaced by C(S1 ), the space of continuous functions on a compact space S1 whose topology is extremally disconnected, i.e., is determined by the set of clopen (= closed and open) sets. This is a deep and classical theorem of M. H. Stone, and we use ∞ it here as a consequence of which ) and C(S1 ) are isometrically PnL (P n n n and orderPisomorphic. If fn = a χ i=1 i Ai , Ai ∈ A, disjoint and if n n n ℓ(fn )R = a Z(A ), then for each f ↓ 0, glb n ℓ(fn ) = 0 and dei=1 i Ri Pnn fine S f dZ = lubn S fn dZ = lubn i=1 ani Z(Ani ) and for each f ≥ 0 such that 0 ≤ fn ↑ f exist. Verify that many of the standard Lebesgue limit theorems extend. Thus our random measures Z(·) use only order topology and not the metric of L∞ (P ). (ii) The analogs of Lebesgue monotone and dominated convergence theorems hold in this context. [Follow the papers of Wright (1969a, 1969b) which also develop a Radon-Nikod´ ym theorem for a class called “ample” measures.] As yet the positivity of Z(·) is crucial, but an analog of the Jordan decomposition for general real random measures in order topology does not seem to have been established, and will be interesting to have a version of it for applications. In other words, an analog of Theorem 1.3.4 in the order topology should be considered. 4. This sketches an alternative approach to Theorem 1.3.4 if µ : A → X, is a Banach space, and a controlling measure λ dominating µ is desired. Our argument is based on the deep representation theorem of Kakutani’s (1941) abstract (L)-space concretely as L1 (T, T, ν) on some (T, T, ν), a finite measure space. Here again the Stone representation enters, as in the preceding problem. The set of measures {x∗ ◦ µ, x∗ ∈ X∗ } is of real bounded functions on (S, A). Observe that this set can be embedded in a (real) space of signed measures V on A which is a vector lattice since v1 , v2 ∈ V ⇒ v1 ± v2 ∈ V, max(v1 , v2 ), min(v1 , v2 ) ∈ V and V becomes an (L−) space. Indeed, define v = inf(v1 , v2 ) as v(A) = inf{v1 (A1 ) + v2 (A2 ) : A = A1 ∪ A2 , Ai ∈ A, disjoint}. Verify that v ∈ V. This involves some computation (cf, e.g., Rao (2004) p.197–198). Let V1 = sp{x∗ ◦ µ, x∗ ∈ X∗ } the abstract (L−) space containing an identity. Then by Kakutani’s representation, we find it (T, T, ν) as

20

1 Introduction and Motivation

described above and V1 is isometrically isomorphic to L1 (T, T, ν). If τ is this R ∗ ˜ with isomorphism, then each (x ◦ µ) : A 7→ A˜ gx∗ dν where A = τ −1 (A), A˜ as the Baire set of T , a totally disconnected compact Hausdorff space. It ˜ then λ : A → R+ is a measure that dominates is seen that if λ(A) = ν(A), µ as in Theorem 1.3.4. (Cf. Dunford-Schwartz (1958), Lemma IV. 9.10.) Verify that λ(·) given above is a regulator or controller of µ(·). [It is clear that the details are not trivial and shows that the domination question is a deep result, and yet in both approaches we have the existence of λ, but actual construction is not given, i.e., the procedure is not computational, as happens in many aspects of measure theory.] 5. We present a simplification of Theorem 1.3.4 for δ-rings A if it satisfies the condition S = ∪∞ n=1 Sn , Sn ∈ A, n ≥ 1. This will be sufficient for many applications and S is now an abstract set. Then A(Sn ) is a σ-algebra for each n. Suppose that µ : A → X, a Banach space, is σ-additive. Let λn : A(Sn ) → R+ be a (controlling) measure given by the theorem itself. Next define ∞ X λn (· ∩ Sn ) . λ(·) = 2−n 1 + λn (Sn ) n=1

Verify that λ : A → R+ is a (finite) measure that controls µ. [Here the gluing process is not explicitly employed, although it is a simple specialization of the former.] 6. Fr´echet function spaces, more general than Lp (P ), 0 ≤ p < 1 arise in applications of random measures, as will be seen later. Here we indicate such a space, motivated by the work of Urbanik (1968) and also Woyczy´ nski(1970). It has extensions to generalized Orlicz (or Musielak-Orlicz) spaces of interest in our work to be considered in more detail. The following is a beginning which assumes a knowledge of standard Probability Theory. Let (I, B) be the Borel unit interval, i.e., I = [0, 1], B = the class of its Borel sets, and let Z : B → L0 (P ) be a random measure as defined in Proposition 1.3.2. Suppose that Z(A) is infinitely divisible and so by the famous L´evy-Khintchine representation formula the Fourier transform (or characteristic function) of Z(A), denoted ϕZ(A) : t 7→ E(eitZ(A) ), t ∈ R is given uniquely as: Z itx 1 + x2 ) dGA (x)}, ϕZ(A) (t) = exp{−tγA + (eitx − 1 − 1 + x2 x2 R for a pair {γA , GA } where γA ∈ R and GA (−∞) = 0, GA (·) ↑≤ GA (∞) < ∞. Show that with the independent values of Z on disjoint sets, γ(·) : B → R is a signed measure and G(·) (x) : B → R+ is a bounded measure for each x ∈ R where we need to use the fact that Z is σ-additive in the Fr´echet metric of L0 (P ). Conclude that µ(·) = |γ|(·) + G(·) (∞) : B → R+ is a controlling measure for Z, which complements that of Theorem 1.3.4. 7. Using the ideas of Exercise 3(i) above, and the availability of a controlling measure as in the preceding problem, we can define an integral relative to the random measure Z(·) just considered. For a simple func-

Bibliographical Notes

Pkn

21

R

tion fn = i=1 ain χAin , Ain ∈ B, ain ∈ R define as usual A fn dZ = Pkn i=1 ain Z(Ain ∩ A), A ∈ B, and verify that this is well-defined and does not depend on the representation of fn . Then for any B-measurable f : IR → R, since there exist sequences fn → f pointwise, consider the set { A fn dZ, n ≥ 1} ⊂ RL0 (P ). If this Ris Cauchy in the metric of L0 (P ) for each A ∈ B, then let A f dZ = limn A fn dZ, A ∈ B. Show using the procedure of Dunford-Schwartz (1958, IV.10.8) that the D-S integral is welldefined, does not depend on the sequence {fn , n ≥ 1} using the fact that there is a controlling measure µ of Z established in the preceding problem, (assuming the random measure Z here satisfies the conditions there). If L is the space of all the thus defined Z-integrable functions f ∈ L0 (P ) verify that the former is a closed subspace of the latter under its Fr´echet norm. [Observe that the infinite divisibility of Z(·) can be dropped if Z : B → Lp (P ), p ≥ 1, since then by Theorem 1.3.4 there is already a controlling measure for it and the rest of the proof applies. Thus the result holds in this case with the same argument. Moreover the metric function now becomes a norm in L which can beR given by kχA k′ = E(|Z(A)|), or more generally for f ∈ L set kf k′ = E(| I f dZ|). These measures will be analyzed in detail later.]

Bibliographical Notes The preceding work motivates studies of various types of measures prompted by applications. Given a measure space (S, A, λ), the indefinite integrals µ : A 7→ R f dλ, A ∈ A, real or vector functions f : S → X generate vector measures, A and proceeding a step further, integration of a scalar or a suitable vector function g for µ will be of interest. To include these cases, we proceed for integration relative to a vector (or random) measure µ that may or may not be representable as an integral with respect to a positive (or scalar) measure λ. Our study includes both types. Actually the class of vector measures as indefinite integrals is another part of our subject, treated in some detail in Dinculeanu (1967), Diestel-Uhl (1977) as well as Dunford-Schwartz (1958) and Hille-Phillips (1957). For our more general study, the regulating (or controlling) measure λ dµ be defined in some sense. In for each µ should exist without demanding that dλ fact for the latter it is necessary that the variation |µ|(·) be at least locally finite, and this cannot be expected for random measures as the well-known Brownian motion (or Wiener) measure demonstrates. However, these have controlling or regulating measures. This is why we devoted considerable space for Proposition 1.2.3 and Theorem 1.3.4 to emphasize the importance of the work. The proof of this theorem is borrowed from Gould (1965) which is direct but not much simpler than that of Bartle-Dunford-Schwartz (1955). The alternative procedure indicated in Exercise 4 above is also nontrivial as it depends on the Stone isomorphism theorem. It is of interest to observe that for random measures probability theory enters and sharpens the results. Problem 2(c) indicates how the deep theorem of

22

1 Introduction and Motivation

P. L´evy on the equivalence of convergence a.e. and in probability for independent random variables is needed, since the measure takes values in L0 (P ) and need have no moments. There are several proofs of this result, such as using martingale convergence, Kolmogorov’s three series theorem, or the direct method of L´evy’s. Also our applications imply that the primary focus should be based on δ-rings possibly with extensions also to σ-rings. The only work with δ-rings as a primary construct seems to be the recent volume of Dinculeanu’s (2000), even though he has focused on vector integration for its own sake, and important applications of the whole subject abound. We study many of these aspects largely complementing the important work in Dinculeanu’s book, and include other parts of the theory of random measures in Lp (P )-spaces for 0 ≤ p < 1, as well as L which is a subspace of L0 (P ) introduced in Exercise 7. These are Fr´echet but not Banach spaces. [Here and elsewhere in this book, Fr´echet spaces are complete linear metric spaces that need not be locally convex, as in Dunford and Schwartz (1958). In Bourbaki the concept of a Fr´echet space includes the local convexity property which excludes spaces such as Lp (P ), 0 ≤ p < 1 of interest in our applications, and thus this restriction is not assumed without explicit mention.] In this respect the measures Z(·) will be restricted to have independent values on disjoint sets which itself leads to new applications even when p ≥ 1, and the spaces L will be identified with some generalized Orlicz spaces when Z is further restricted, as will be detailed later from Chapter 3 on. Thus the comparative analysis of random and vector measures demands a detailed study and it is considered in what follows.

2 Second Order Random Measures and Representations

A first view of second order random measures, their special structures and some applications are considered in this chapter. The work will indicate how the Hilbertian geometry available here motivates new directions of the subject when the range is not necessarily a Hilbert space. The translation invariance (or “stationarity”) of these measures defined on spaces with a group structure, and integral representations of such set functions are treated. Natural directions by which the theory leads to bimeasures and generalized random fields as well as analysis based on an application of the Bochner-Schwartz theorem are included. The resulting work is substantial with numerous consequences and applications.

2.1 Introduction There are different ways of introducing random measures, leading to the same eventual result. It will be of interest to indicate these approaches and then concentrate on one of them which will be of particular utility here. This is also operationally straight forward and connects with the vector measure approach to be employed. Recall that we started in the preceding chapter with a pair (S, S), a measurable space, and an Lp (P ), the usual Lebesgue space of pth order (p ≥ 0) (real) random variables on a probability triple (Ω, Σ, P ), with Z : S → Lp (P ) as a σ-additive (vector valued) function, called a random measure. Thus for any family {Ai ∈ S, i ∈ I}, the collection {Z(Ai ), i ∈ I} ⊂ Lp (P ) is a stochastic process indexed by the Ai , and its finite dimensional distributions for i1 , . . . , in ∈ I, zi ∈ R, are thus given by P [Z(Aij ) < zj , j = 1, . . . , n] = FZ(Ai1 ),...,Z(Ain ) (z1 , . . . , zn ),

(1)

which satisfy the classical Kolmogorov consistency conditions and are valued in the vector space Lp (P ). The theory of random measures can be developed from this point of view.

24

2 Second Order Random Measures and

Representations

For another approach, let (Ω, Σ, P ) be a probability triple and (S, S), measurable space. Consider the mapping Z(A, ·) : Ω → C, A ∈ S, so that Z( , ·) : ω 7→ C(S, S), ω ∈ Ω, is valued in the vector space C(S, S) of countably additive scalar measures on S which is a Banach space under the (total) variation norm. Thus Z(·, ω)(∈ C(S, S)) is σ-additive and Z(, ·) is a measurable mapping relative to (Σ, M) where M is a σ-algebra on the space of measures containing all open sets in its norm topology. Again one can establish all properties as before taking Z(·, ) as a vector valued mapping on Ω. Here one has to reestablish the work until the methodology becomes familiar and the results routine. This point of view is of interest when applications to point process is primary, and the procedure is detailed in Jagers (1974), (see also Harris (1971)). The approach in these papers has little relation with vector measure theory as in Dunford-Schwartz (1958) or Dinculeanu (2000), our treatment is essentially functional analytic, and hence the former will not be discussed in this book. There is yet another approach by considering Z : S× Ω → R as a generalized Stieltjes measure with Z(·, ω) satisfying a “conditional or kernel type” equation: Z ˜ Z(A, ω)dP (ω) = Z(A) (2) S

which was developed by L.C. Young (1970,1974), by treating it as a problem in an extended real analysis. The resulting stochastic integrals are of two types: (i) deterministic integrands with a stochastic integrator (or random measure), and (ii) a general integration with both a stochastic integral and a stochastic integrator. This leads to the Wiener and Itˆ o type integrals respectively. Thus Young develops an extension of the classical Riemann-Stieltjes type integral relative to a ‘measure’ that has finite quadratic variation, extending the well-known case of finite (Vitali) variation. Then his second extension is designed to include the martingale integrators which do not necessarily have finite variations but have increments that are ‘nearly orthogonal’ and are termed near-martingales or nigh-martingales (and informally called by him also as “nightingales”). The whole treatment is given, with details, when the ‘process’ Z(t, ω) treated as a function of two variables, with −∞ < a ≤ t ≤ b < ∞ and ω(∈ Ω) as an abstract parameter so that t 7→ Z(t, ·) is a Banach space (usually an L2 -space here) valued “Stieltjes measure”. A complete development using the ε-δ arguments of real analysis is provided for the case of functions having only (locally) finite quadratic variation, without any probabilistic intervention. The treatment is conducive for readers with little or no probabilistic preparation. Unfortunately this does not provide much motivation for the type of conditions needed for the familiar stochastic analyses. However, the development is essentially complete as it depends on two moduli of continuity, denoted χ and ϕ in terms of which p √ one defines the parameters ρ : u 7→ uχ(u), u > 0 and ϕ = ϕ1 ϕ2 where √ ϕ2 = 2 χ, and ϕ1 is then obtained. For integrators with orthogonal increments ρ = identity and ϕ = 0. His treatment gives somewhat more general stochastic integrals than those usually defined for martingales (of the Itˆ o-Doob types). Here Lipschitz conditions with moduli of continuity as well as certain inequali-

2.2 Structures of Second Order Random Measures

25

ties using conditional probability measures are employed. We see some of these in a different context later, using other methods. Since the natural probabilistic motivation is essentially different in the above procedures and what follows we merely point out the existence of this type of alternative work which implies that with some effort each result obtained by either of the approaches indicated above may be derived by the other two. Consequently, we proceed almost always with our method discussed above. Here we call attention of readers to the possibility of other procedures elucidating a particular view. Later it will be seen that all these are special cases of Bochner’s V-boundedness concept (slightly generalized). It may be observed that L.C.Young’s treatment of the problem is based on the classical Stieltjes methods and for his results null set exceptions play no significant role, whereas for stochastic analysis considered here this freedom is important both in its arguments and details. We thus use ‘random measure’ just as discussed in the recent work by the author (cf., Rao (2009)).

2.2 Structures of Second Order Random Measures Let B0 (Rn ) be the class of all bounded Borel sets of Rn which is a δ-ring. If Z : B0 (Rn ) → L2 (P )(= X) is a random measure always taken σ-additive in the norm (or,what is equivalent, to weak)-topology of the Banach space X, then β : B0 (Rn )×B0 (Rn ) → C defined by β(A, B) = (Z(A), Z(B))X , using the inner product notation, is a bimeasure. This means β has the property that β(·, B) and β(A, ·) are signed measures for each fixed A and B in B0 (Rn ). However, β does not in general define a (scalar) measure on the product ring B0 (Rn ) × B0 (Rn ), which is a new problem. We first give some examples to understand and analyze this nontrivial difficulty, and then present some (sufficient) conditions for the bimeasure β to define a (scalar) measure in the product space. It is convenient to recall the definition of a scalar (measurable) function relative to a vector measureP such as Z : A → X on a σ-ring A of a space S into n a Banach space X. If fn = i=1 ai χAi , ai ∈ R or C, Ai ∈ A, disjoint, it is a simple function, and define as usual Z

fn dZ =

A

n X i=1

ai Z(Ai ∩ A) ∈ X, A ∈ A.

(1)

It is a standard computation to see that the integral is well-defined and does not depend on the representation of fn . If f : S → R or C, A-measurable, there existRa sequence of A-simple functions fn → f pointwise, and |fn | ↑ |f |. Consider { A fn dZ : n ≥ 1} ⊂ X. If this sequence is Cauchy in X for each A ∈ A with limit αA (∈ X), then we set Z Z αA = f dZ = lim fn dZ, A ∈ A. (2) A

n→∞

A

26

2 Second Order Random Measures and

Representations

This integral is called the Dunford-Schwartz (or D–S) integral and is uniquely defined in that it does not depend on the sequence {fn , n ≥ 1} approximating f. This key fact uses the existence of a controlling (or regulating) measure λ for Z, established in Theorem 1.3.4. The detail is deemed standard and is found, R for f dZ instance, in Dunford-Schwartz((1958), Sec.IV.10). It is seen that f → 7 A R is additive for each A ∈ A and µf : A 7→ A f dZ is a vector measure for each f for which the integral is defined. In particular, the set of Z-integrable functions forms a vector space over R or C, and contains all bounded measurable functions. For convenience this property is stated as: Definition 2.2.1 If RZ : A → X is a vector measure into a Banach space, as above, then T : f 7→ S f dZ is called the Dunford-Schwartz (or D-S) integral. [It is proved there that T is linear and both the bounded and dominated convergence theorems hold for this integral.] The D-S integral will be used in examples and applications of vector measures in what follows. Our first illustration is in showing that a bimeasure need not define a (scalar) measure on the product space. Later on conditions will be given for a bimeasure to define such a scalar measure. Example 2. If a bimeasure β determines a (scalar) measure then by Proposition 1.2.3, it must have finite variation on the σ-ring generated by B0 (Rn )× B0 (Rn ), namely B(Rn ×Rn ). This will now be shown to be false in general. For simplicity let Rn (n = 1) be replaced by T = (0, 2π), so Z : B(T ) → L2 (P ) is σ-additive,and suppose it is even orthogonally valued, in the sense that Z(A) ⊥ Z(B) if A∩B = ∅. If V : L2 (P ) → L2 (P ) is a bounded linear operator, then from our work, in Chapter 1, Z˜ = V ◦ Z : B(T ) → L20 (P ) is also a vector measure but need not ˜ ˜ have orthogonal values. So G : (A, B) 7→ (Z(A), Z(B)) defines G as a bimeasure on B(T )×B(T ) → C. It is asserted that G does not have finite (Vitali) variation for a suitable V. To see this, consider the vector integral given by Z einλ dZ(λ) ∈ L2 (P ), n ∈ Z. (3) fn = T

Using the inner product notation and some simple properties of the D-S integral, one gets Z Z ′ (fn , fm ) = einλ dZ(λ), einλ dZ(λ′ ) T Z TZ i(nλ−mλ′ ) = e (Z(dλ), Z(dλ′ )). (4) T

T

Now Z(A) and Z(B) are orthogonal for disjoint A, B so that (Z(dλ), Z(dλ′ )) = δλλ′ ν(dλ) where ν(A ∩ B) = (Z(A), Z(B)) defines a measure, valued in R+ . Thus the integral in (4) reduces to that supported by the diagonal of T × T , and hence Z amn = (fm , fn ) = ei(m−n)λ dν(λ), m, n ∈ Z. (5) T

2.2 Structures of Second Order Random Measures

27

˜ ˜ However Z˜ = V ◦Z : A → L2 (P ) is not orthogonally valued and (Z(A), Z(B)) = G(A, B) defines just a bimeasure on B(T ) × B(T ). In this case, using the fact that the D–S integral and a bounded linear operator commute (this is a classical result due to Hille, see, e.g., Dunford-Schwartz (1958), IV,10.8(f)), one obtains from (3) Z V fn =

T

einλ d(V ◦ Z)(λ), n ∈ Z.

Then a computation of (5) in this case gives Z Z ′ ei(mλ−nλ ) G(dλ, dλ′ ), m, n ∈ Z. (V fm , V fn ) = T

(6)

T

To see that G does not concentrate on the diagonal of T × T for even simpler operators V , consider the closed subspace X1 determined by {fn , n ≥ 0} ⊂ X = L2 (P ) and let V : X → X1 be an orthogonal projection which thus annihilates fn , n < 0, so that V fn = fn if n ≥ 0, and = 0 for n < 0. Then (4)–(6) give ( (fn , fn ), n = m ≥ 0 amn = (fm , fn ) = (V fm , V fn ) = (7) 0, if m, n < 0. This implies that {fn , n ∈ Z} is an orthogonal sequence and by (3) it is uniformly bounded since kZk(T ) < ∞, the semivariation of Z. When Z(·) has orthogonal values, it can be verified that the fn given by (3) are not only orthogonal, but have the same norm, independent of n so that, by multiplying with a suitable constant, they can be taken to be orthonormal. It then follows that amn = δm,n α in (7), α > 0,for m, n ≥ 0 and = 0 for n < 0 or m < 0. Now in (6) if G defines a signed measure on T × T , then it must have finite variation and by an important generalization, due to S.Bochner, of a theorem of F. and M. Riesz, G must be absolutely continuous relative to the (planar) Lebesgue ∂2G (x, y) exists a.e. Hence (6) and (7) measure λ × λ, so that G′ (x, y) = ∂λ(x)∂λ(y) give Z Z ′ ei(mλ−nλ ) G′ (λ, λ′ )dλdλ′ , m, n ∈ Z, (8) 0 < α = am,n δm,n = T

T

and this implies by the Riemann-Lebesgue lemma (of Real Analysis) that amn → 0. The contradiction implied in (8) shows that G(·, ·) cannot determine a (signed ) measure in T × T and that the bimeasure G is not of bounded (Vitali) variation as well. One may feel that the situation noted in the above example is avoided if the bimeasure takes values in R+ or if Z(·) itself is a nonnegative vector measure. The following example (based on Berg et al (1984)) shows that the situation can still be bad. Example 3. Let (I, A, λ) be the Lebesgue unit internal, i.e., I = [0, 1], A = Lebesgue σ-algebra of I, and λ = Lebesgue measure. Let A1 , A2 ⊂ I be a pair

28

2 Second Order Random Measures and

Representations

of disjoint nonmeasurable sets each having outer measure = 1, i.e., λ∗ (A1 ) = λ∗ (A2 ) = 1. With the axiom of choice it is proved that such sets exist, shown in Real Analysis courses. Consider the trace σ-algebras A(Ai ) = {B ∈ A; Ai ∩ B ∈ A}, i = 1, 2. Define for each Ci ∈ A(Ai ) sets Bi ∈ A such that Ci = Ai ∩ Bi , i = 1, 2. Actually it is a consequence of a classical theorem due to F. Bernstein stating that the sets Ai of the type exist that meet every uncountable subset of the line and are nonmeasurable. This is proved as Theorems 5.3 and 5.4 in Oxtoby (1971). With this set up, define β(C1 , C2 ) = λ(B1 ∩ B2 ) ≥ 0.

(9)

Although B1 , B2 in A are not unique, the measure of B1 ∩B2 is uniquely defined, and hence β(·, ·) is a positive well-defined bimeasure. It is asserted that β of (9) cannot determine a measure β˜ on A(A1 ) ⊗ A(A2 ). Indeed if β determines a measure β˜ then for any decreasing sequence of sets ˜ n ) ↓ 0. However consider Cn ∈ A(A1 ) ⊗ A(A2 ) with Cn ↓ ∅ one must have β(C the sets Cn defined as product linear sets: k k+1 k+1 k n ∩ × A . (10) ≤ y < Cn = (x, y) : A1 ∩ x : n ≤ x < 2 2 2n 2n 2n ˜ n ) ≥ 1 for all n ≥ 1, since by Bernstein’s Then Cn ∈ A(A1 ) × A(A2 ) and β(C theorem (noted above) any subset of Ai is either λ-null or is of the first category ˜ n) 9 0 hence has λ-measure zero. Thus β˜ does not exist as a measure since β(C although Cn ↓ ∅, and so β(·, ·) does not determine a measure. A natural desire is to exclude such sets and impose some ‘regularity conditions’ on β assuming a topology on Si which is Hausdorff but not necessarily locally compact. It suffices to consider the real valued case. Thus if −∞ < β(A, B) < ∞ for all compact sets A, B contained in the Hausdorff spaces S1 and S2 respectively then β is regular from above (below) if it can be approximated from above (below) in the following sense: Given A ∈ B0 (S1 ) and B ∈ B0 (S2 ) and ε > 0, there exist open sets O1 ⊃ A1 , O2 ⊃ B1 (compact sets C1 ⊂ A and C2 ⊂ B) such that for each A2 ∈ B0 (S1 ), B2 ∈ B0 (S2 ) satisfying A ⊂ A2 ⊂ A1 , B ⊂ B2 ⊂ B1 (respectively C2 ∈ B0 (S1 ), C˜2 ∈ B0 (S2 ) satisfying C1 ⊂ C2 ⊂ A and C2 ⊂ C˜2 ⊂ B) for which one has |β(A, B) − β(A2 , B2 )| < ε. (|β(A, B) − β(C2 , C˜2 )| < ε.)

(11)

Then β is termed a regular bimeasure (also called a Radon bimeasure) if it is simultaneously regular from above and below. With this concept the following positive result can be obtained. Theorem 2.2.2 Let S be a locally compact Hausdorff space and B0 (S) be its δ-ring of bounded Borel sets. (Recall that a set A ⊂ S is bounded if the closure S¯ is compact.) Let Z : B0 (S) → L2 (P ) be a vector measure and β(A, B) = ¯ E(Z(A)Z(B)) = (Z(A), Z(B)), a (pos. def.) bimeasure on B0 (S) × B0 (S). If

2.2 Structures of Second Order Random Measures

29

moreover Z(A) ≥ 0 for all A ∈ B0 (S), then β determines a measure β˜ on the product space so that ˜ × B), A, B ∈ B0 (S). β(A, B) = β(A

(12)

Sketch of proof . A brief outline of the argument will be given, and the details take much more space. Also the result is not really essential for our work. Since L2 (P ) is a complete space, by a key result of Dinculeanu and Kluv´anek (1967) which holds not only for L2 (P ) but for all Banach spaces, Z(·) is Baire regular on B0 (S) and has an extension to a regular Borel measure. Then using the (sesqui-linearity) property of the inner product and the above noted regularity one has |β(A, B) − β(A0 , B0 )| = |(Z(A) − Z(A0 ), Z(B)) + (Z(A0 ), Z(B) − Z(B0 ))|

≤ kZ(A) − Z(A0 )kkZ(B)k + kZ(A0 )kkZ(B) − Z(B0 )k,

and the right side can be made small by choosing A0 ⊂ A, B0 ⊂ B suitably to satisfy (11) and infer that β is regular. But this is not always sufficient ˜ However, if β is also nonnegative, for the desired conclusion about finding β. which holds when Z(·) takes nonnegative values in L2 (P ), then we get β to be a regular nonnegative bimeasure. Thus one can use the same argument as in Berg et al ((1984), p.24) and deduce that there is a unique regular Borel or Radon measure β˜ satisfying (12). This concludes the sketch. 2 Remark 1. It should be observed that the measure Z(·) of Example 2 above satisfies all the conditions of this theorem except that it is not nonnegative. Since a scalar nonnegative bimeasure has finite Vital variation on Rn × Rn , the above example shows that a regular bimeasure need not determine a signed measure. The nonnegative hypothesis is satisfied for random measures appearing in the theory of point process (to be defined rigorously later) but not for all second order functions. However, it is not the nonnegativity of the (regular) bimeasure β that is crucial for its determination of a σ-additive scalar measure β˜ on the product space, but the property of (local) finite Vitali variation which is the key condition. Because of our intended applications to several problems that may not have the last property, our work has to proceed with the weaker concept of the Fr´echet variation. [The representing Z of a ‘weakly harmonizable field’ already presents this difficulty.] With a view to include in this theory some applications centering around classes of such “harmonizable processes”, we discuss an extension of the classical integration covering that relative to bimeasures which do not induce a measure on the product space. The necessary results were developed by M.Morse and W.Transue (1956 and earlier), and a relevant part (slightly restricted to be applicable for our case) will be recalled here and termed the strict MorseTransue (or sMT) integral which is somewhat weaker than the usual concept but stronger than Riemann’s so that the dominated convergence statement is still valid.

30

2 Second Order Random Measures and

Representations

Definition 2.2.3 If (Ωi , Σi ), i = 1, 2 are measurable spaces and (Ω, Σ) is their product (so Ω = Ω1 × Ω2 and Σ = Σ1 ⊗ Σ2 ), fi : Ωi → C (measurable relative to Σi , i = 1, 2) are given then the pair (f1 , f2 ) is said to be strictly β-integrable where β is a bimeasure on Σ1 × Σ2 , provided the following two conditions hold: (a) f1 is β(·, B)-integrable (L-S)for each B ∈RΣ2 and f2 is β(A, ·)-integrable (LS) for each A ∈ Σ1 such that β˜1 (F, A) = F f2 (ω2 )β(A, dω2 ) is σ-additive in R A ∈ Σ1 for each F ∈ Σ2 and β˜2 (E, B) = E f1 (ω1 )β(dω1 , B) is σ-additive in B ∈ Σ2 for each E ∈ Σ1 ; [L-S for Leb.-Stieltjes] (b) f1 is β˜1 (F, ·)-integrable (L-S), f2 is β˜2 (·, E)-integrable (L-S) and Z Z ˜ f2 (ω2 )β˜2 (E, dω2 ), E ∈ Σ1 , F ∈ Σ2 . (13) f1 (ω1 )β1 (dω1 , F ) = E

F

The common value in (13) is denoted by

R R E

F

(f1 , f2 )dβ.

If in the above, (13) is supposed to hold for just one pair, namely E = Ω1 and F = Ω2 , then the more general (but weaker) integral is the original MTformulation. However, the more stringent condition of (13) holding for all pairs (E, F ) will be needed for our applications. This is still much weaker than the original Lebesgue-Stieltjes definition used when the bimeasure β is demanded to be of finite Vitali variation. [See later Example 9.1.4 on the failure of (13) if (a) above is weakened.] We first present a few properties of the strict MT-integral and connect it with the vector or random integral of scalar functions relative to a random measure, valued in L2 (P ). It should be noted that the concept of “strict MT-integral” only restricts the class of β-integrable functions but not the bimeasures. Consequently the Jordan decomposition of β as β1 − β2 + i(β3 − β4 ) is not necessarily valid where βj are nonnegative bimeasures (hence of finite Vitali variations), and thus the corresponding L-S theory does not necessarily extend to the strict MT-class. [However such a decomposition with βj “positive definite” can exist. Chapter 9 has more advanced treatment of these and of multimeasures.] Recall that if Z : B0 (R) → L2 (P ) is a random measure and β(A, B) = (Z(A), Z(B)), A, B ∈ B0 (R), then β is a bimeasure on B0 (R) × B0 (R) → C, induced by Z. The respective integrals of β(·, ·) and Z(·) are related as follows: Proposition 2.2.4 Let (Si , Bi )2i=1 be a pair of measurable spaces and Zi : Bi → L2 (P ), i = 1, 2 be random measures on them where Bi ’s are δ− or σ-rings. If β : B1 ×B2 → C is a bimeasure defined as β(A, B) = E(Z1 (A)Z¯2 (B)), (A, B) ∈ B1 × B2 , the fi : Si → C being Bi -measurable and Zi -integrable (Dunford– Schwartz sense), then the pair (f1 , f2 ) is β-integrable in the (strict) MT-sense and one has: Z Z Z Z (f1 (x), f2 (y))β(dx, dy) = E f1 dZ1 f2 dZ2 , A ∈ B1 , B ∈ B2 . A

B

A

B

If (S1 , B1 ) = (S2 , B2 ), and Z1 = Z2 then β(·, ·) is also positive definite.

(14)

2.2 Structures of Second Order Random Measures

31

Proof. First observe that if Bi is a σ-algebra then fi is Bi -measurable signifies, as usual, that the set f −1 (A) ∈ Bi for each Borel set A ⊂ C so that by the structure theorem measurable functions there exists a sequence of simple Pmof in anij χAnij , Anij ∈ Bi disjoint i = 1, 2 and fin → fi pointwise functions fin = j=1 with |fin | ≤ |fi |, as n → ∞. If Bi is a δ-ring, then measurability of fi is understood as the limit of such a sequence fin , of simple functions. With this understanding the argument can be quickly given as follows. Thus for (A, B) ∈ (B1 , B2 ) one has Z m in X an1 β(Anij ∩ A, B), by definition, fin (ω1 )β(dω1 , ·) (B) = A

=

j=1 m in X j=1

= E

an1 E(Z1 (Anij ∩ A)Z2 (B))

Z

fin dZ1

A

→E

Z

f1 dZ1

A

Z

Z

dZ2

B

B

dZ2 , as n → ∞, by(2).

Consequently the following holds: Z Z f1 (ω1 )β(dω1 , ·) (B) = E f1 (ω1 )dZ1 (ω1 )Z2 (B) . A

(15)

A

Fix A ∈ B1 and so the left side as a function of B ∈ B2 is σ-additive denoted β˜1 (A, ·), and one has, for a simple function f2n , the following: Z m 2n X ¯ ˜ f2n (ω2 )β1 (A, dω2 ) = a ¯n2j β˜1 (A, An2j ∩ B) B

=

j=1 m 2n X

a ¯n2j

j=1

=

m 2n X

a ¯n2j

j=1

Z

A

lim

k→∞



= lim E  k→∞

f1 (ω1 )β(dω1 , ·) (An2j ∩ B) m 1k X

ak1l β(Ak1l , An2j

l=1

m 1k 2n m X X j=1 l=1

!

∩ B)

ak1l a ¯n2j Z1 (Ak1l )Z¯2 (An2j ∩ B) ,

using a property of the D-S integral, Z Z = E f1 (u1 )dZ1 (u1 ) f2n (u2 )dZ2 (u2 ) B A Z Z →E f1 (u1 )dZ1 (u1 ) f2 (u2 )dZ2 (u2 ) , A



B

as n → ∞ and fi is Zi integrable (cf.(2)).

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2 Second Order Random Measures and

Representations

Thus f2 is β˜1 (A, ·) integrable, and for each B ∈ B2 Z Z Z f¯2 (u2 )β˜1 (A1 , du2 ) = E f1 (u1 )dZ1 (u1 ) f2 (u2 )dZ2 (u2 ) . B

(16)

B

A

Similarly one gets for all (A, B) ∈ B1 × B2 Z Z ˜ f¯2 (u2 )β˜1 (A, du2 ) f1 (u1 )β2 (du1 , B) = B A Z Z = (f1 (u1 ))(f2 (u2 ))β(du1 , du2 ), A

B

so that (f1 , f2 ) is β-integrable in the strict MT-sense. If (S1 , B1 ) = (S2 , B2 ) and Z1 = Z2 (= Z), then for all ai ∈ C, one has n X n X

ai a ¯j β(Ai , Aj ) =

n X n X

i=1 j=1 n X

i=1 j=1

= E(|

i=1

¯ j )) ai a ¯j E(Z(Ai )Z(A ai Z(Ai )|2 ) ≥ 0.

Hence the bimeasure β is also positive definite. 2 It should be noted at this point that even if β(·, ·) is induced by a stochastic measure Z and (f1 , f2 ) is β-integrable in the MT-sense, fi is not necessarily Z-integrable as seen from Exercise 2 below. However, if the β-integral is in the strict MT-sense then the problem disappears. Thus given a positive definite bimeasure β, we can consider a vector measure with values in a Hilbert space and then find a random measure on some space that induces the original β. This dichotomy is of special significance for our analysis. Let (S, B) be a measurable space and β : B × B → C be a bimeasure so that β(·, B) and β(A, ·), A, B ∈ B, are both elements of ca(S, B), the space of countably additive scalar set functions on B, which is a Banach space under the (total) variation norm. Then the complex linear span H0 = sp{β(A, ·), A ∈ B} ⊂ ca(S, B), is a subspace, and since by definition β(A, ·) = β(·, A), it is also closed under complex conjugation. When β is positive definite, we can introduce a (stronger) norm in H0 and complete it to become Pn a Hilbert space H that is of interest for our representation. Thus if f = i=1 ai β(Ai , ·), g = Pn b β(B , ·), (same n is used by setting some a or b to be zero), then i j i j j=1 f, g ∈ H0 and the positive definiteness of β and its “complex symmetry” implies (f, g) =

n X n X

ai¯bj β(Ai , Bj )

(17)

i=1 j=1

is a sesqui-linear mapping on H0 , and (f, f ) ≥ 0 with equality iff ai = 0 all i by the strict positive definiteness, which we are assuming for convenience). It follows that (17) defines an inner product on H0 , and its completion, denoted

2.2 Structures of Second Order Random Measures

33

Hβ , is a Hilbert space, contained in ca(S, B). It has the remarkable reproducing Pkn property, namely if fn (A) = i=1 ani β(Ani , A) ∈ C(fn ∈ H0 ), then (fn , β(A, ·)) =

kn X

ani β(Ani , A) = fn (A),

i=1

which is a desirable relation. Since a bimeasure β has the finite Fr´echet variation on a σ- algebra (cf. Exercise 1(b)) one has for f ∈ H0 that f (A) = (f, β(A, ·)), and using the CBS (= Cauchy-Buniakowski-Schwartz) inequality, p |f (A)| ≤ kf kkβ(A, ·)k ≤ kf k β(A, A) ≤ kf k · kβk(A) < ∞,

A ∈ B.

The space Hβ is termed Aronszajn or Reproducing Kernel Hilbert Space (denoted RKHS), the former name is after its author Aronszajn (1950) for ˜ = β(, ·) : A 7→ β(A, ·)(∈ H) is the vector meahis fundamental work. If β(·) ˜ be the space of β-integrable ˜ sure determined by β let L1 (β) scalar functions 1 f : S → C in the D-S sense, and similarly L (Z) be the space defined for 2 2 RZ : RB → L (P ), with β determined by Z. If L (β) = {f : Ω → C, (f, f ) = ¯ S S f (s)f (t)β(ds, dt) < ∞}, where β is positive definite and the integral is in the strict MT-sense then the following useful relation can be established. Theorem 2.2.5 Let (S, B) be a measurable space, β : B × B → C be a positive definite bimeasure with β˜ as the associated Hβ -valued vector measure. Then ˜ = L2 (β), using the strict MT-integral for the latter. Moreover there exists L1 (β) a probability space (Ω, Σ, P ) and a random measure Z : B → L2 (P ) inducing ˜ = L1 (Z) = L2 (β), and β is positive definite. the bimeasure β, so that L1 (β) Proof. First let β be induced by the random measure, Z, andSconsider f = P n n 1 ˜ 1 i=1 Ai (∈ B) , i=1 ai χAi , Ai ∈ B, disjoint. Then f ∈ L (β) ∩ L (Z). If A = then

2

Z

2 n

X

˜ i )

f dβ˜ β(A a =

i

A Hβ i=1

=

Hβ

n X n X i=1 j=1

= =

n X n X

i=1 j=1 n X n X

˜ ai a ¯j (β(Ai , ·), β(Aj , ·)), by definition of β,

ai a ¯j β(Ai , Aj ), (reproducing property of Hβ ), ¯ j )) ai a ¯j E(Z(Ai )Z(A

i=1 j=1

 2  n X ai Z(Ai )  = E = E  i=1

Z 2 ! f dZ . A

(18)

34

2 Second Order Random Measures and

R

Representations

R

Thus τ : S f χA dβ˜ 7→ S f χA dZ ∈ L2 (P ), A ∈ B, is an isometry between the simple, and then all elements of Hβ and L2 (P ), since by definition of the D-S integral, f is β˜ integrable [Z-integrable]R iff there is a sequence of simple ˜ n ≥ 1} is Cauchy in Hβ functions fn → f pointwise, |fnR| ≤ |f | and { A fn dβ, and similarly for L1 (Z), i.e., { A fn dZ, n ≥ 1} is Cauchy in L1 (Z). It then ˜ = L1 (Z), must hold. follows that L1 (β) Next suppose that a positive definite bimeasure β is given with the associated β˜ and the space Hβ ⊂ ca(S, B). We now recall the classical fact that an abstract Hilbert space is isometrically embeddable into an L2 (Ω, Σ, P ) on some probability space, starting with a complete orthonormal system {ϕα , α ∈ I} and associating a Gaussian probability space with each ϕα and then using the Fubini-Jessen theorem to produce a product probability space (Ω, Σ, P ) in terms of which one can embed Hβ isometrically. This construction is due to P. Cartier, and is detailed in many places (cf., e,g., Rao(1995) p.482) and so will not be reproduced. Thus there exists a Z : B → L2 (P ), which induces β. Then upon using the strict MT-integration of β, one has L2 (β) = L1 (Z), and the desired equivalence is obtained. 2 Remark 2. If the general MT-integral is employed then the resulting L2 (β) is larger and one only has the proper inclusion L1 (β) ⊂ L2 (β). This shows the need for strict MT-integrals in our analysis. Moreover, the above theorem is also needed in establishing integral representations of certain second order random functions. For later applications the following form of the dominated convergence criterion for strict β-integrals will be established. Theorem 2.2.6 Let (Si , Bi ), i = 1, 2 be measurable spaces, fin : Si → C, n = 1, 2, . . . be measurable scalar functions and β : B1 × B2 → C be a bimeasure of finite Fr´echet variation (which is automatic if the Bi , i = 1, 2 are σ-algebras). Let h1 , h2 be β-integrable in the strict MT-sense, and let |fin | ≤ |hi |, i = 1, 2. If fin → fi as n → ∞ then the fin as well as the limits fi , i = 1, 2 are β-integrable in the same sense, and moreover for each A ∈ B1 , and B ∈ B2 , Z Z Z Z lim (f1m (s1 ), f2m (s2 ))β(ds1 , ds2 ) = (f1 (s1 ), f2 (s2 ))β(ds1 , ds2 ) m,n→∞

A

B

A

B

(19) holds, the order of the limits being immaterial. In particular if the Bi are σalgebras (not δ-rings), then hi = αi > 0 a constant is integrable and the resulting bounded convergence statement is also valid.

Proof. Since (h1 , h2 ) is β-integrable, we note (cf. Definition 2.2.3) R that (f1n , f2n ) and (f1 , f2 ) are integrable in the same sense. If β˜1B : D 7→ B f2 (s2 )β(D, ds2 ) which is a scalar measure one has for the strict MT-integrals of β

2.2 Structures of Second Order Random Measures

Z Z A

(f1 , f2 )dβ =

B

Z

Z

A

35

f1 (s1 )β˜1B (ds1 )

lim f1n (s1 )β˜1B (ds1 ), as fin → fi , a.e. as n → ∞, Z Z f2 (s2 )β˜A (ds2 ) fin (s1 )β˜B (ds1 ) = lim = lim =

A n

n

1

A

Z

n

B

2n

A f1n (s1 )β(ds1 , E) where β˜2n : E 7→ A Z A f2m (s2 )β˜2n (ds2 ), = lim lim n

m

B

by the classical D-S theory, Z Z (f1n , f2m )dβ, since = lim lim n

m

A

B

(fin , f2m ) is strictly MT-integrable for β.

It is clear that m and n can be interchanged here, giving (19). The last statement is immediate since β has finite Fr´echet variation on a σ-algebra. 2 For comparison, we now quote a general form of the above result for the more inclusive MT-integrals, due to Morse and Transue (1956) themselves, so that the reader may have a better idea of the subject under consideration. Theorem 2.2.7 (Morse-Transue) Let (Ωi , Σi ), i = 1, 2 be measurable spaces with Ωi as locally compact Hausdorff and Σi as its Borel σ-algebra. Let β : Σ1 × Σ2 → C be a bimeasure, Cc (Ωi ) be the space of scalar continuous functions with compact supports, and the corresponding bilinear form B : Cc (Ω1 )×Cc (Ω2 ) → C defined by the MT-integral Z Z B(f, g) = (f, g)(ω1 , ω2 )β(dω1 , dω2 ), fi ∈ Cc (Ωi ). (20) Ω2

Ω1

Suppose B(·, ·) is extended by setting (for p, q lower semi-continuous) B∗ (p, q) = sup{|B(f1 , f2 )| : |f1 | ≤ |p|, |f2 | ≤ |q|, fi ∈ Ci (Ωi )} and then for Borel functions u, v as B ∗ (u, v) = inf{B∗ (p, q) : p ≥ u ≥ 0, q ≥ v ≥ 0}. Suppose now for fi ∈ Cc (Ωi ) one has (a)

B ∗ (u, f2 ) < ∞, and (b)

B ∗ (f1 , v) < ∞.

Then β u (·) = β(u, ·) and βv (·) = β(·, v) are σ-additive and if further B (|u|, |v|) < ∞, then (f1 , f2 ) is β integrable (in the general MT-sense), and each bounded Borel pair (u, v) is β-integrable. ∗

36

2 Second Order Random Measures and

Representations

Moreover, the following approximation holds: if (f1 , f2 ) is β-integrable and lower semi-continuous, then for each ε > 0, there exist gi ∈ Cc (Ωi ), i = 1, 2, |gi | ≤ |fi |, i = 1, 2, and |B(f1 , f2 ) − B(g1 , g2 )| < ε.

(21)

A comparison of these two theorems shows that, although (20) and (21) can be applied for larger classes, the results here are not sharp enough for our applications, and the strict concept introduced earlier will be needed. We present some examples (as exercises) to clarify the differences which show the need for our specialization. Also the pair (f1 , f2 ) used in the definition of the βintegrals Rcan Rbe extended to f : Ω1 × Ω2 → C, and consider the integrals of the form Ω1 Ω2 f (ω1 , ω2 )β(dω1 , dω2 ) generalizing the Fubini-Tonelli type theory. However, here one encounters severe problems. For instance the dominated convergence theorem of the type given by Theorem 2.2.6 fails, as illustrated in Exercise 3(b) below. On the other hand, vector (or Banach space valued) functions and bimeasures can also be defined and studied. (See Section 9.1.) The following simple result motivates much of the later work in this chapter and the next ones, giving a relation between the bimeasure theory, random measures, and a general class of stochastic processes indexed by very general sets. Theorem 2.2.8 Let (Ω, Σ, P ) be a probability space L2 (P ) be a space of square integrable random variables on Ω and K ⊂ L2 (P ) be a separable subspace (closed linear manifold). If {Xt , t ∈ T } is an enumeration of K, then there exists a (possibly different) measurable space (S, S, µ), a class of µ-measurable functions g(t, ·) : S → C, g(t, ·) ∈ L2 (µ), and a random measure Z : S → L2 (P ) such that Z g(t, s)dZ(s), t ∈ T, (22) Xt = S

with (Z(A), Z(B)) = β(A, B) = µ(A ∩ B), A, B ∈ S. Here the bimeasure β is determined by the measure µ. In case (Ω, Σ, P ) is itself separable, then every element of L2 (P ) admits a representation as (22). Proof. Let α be the dimension of K which denotes the number of linearly independent (or basis) elements. By the separability of K, α ≤ ℵ0 . Since K is then isomorphic to Cα (or Rα ) and hence is isomorphic to L2 (µ) on a measure space (S, S, µ) which follows from the classical result that two Hilbert spaces of the same dimension are isomorphic (cf., e.g., Na˘ımark (1959), p.95). Since K and hence L2 (µ) is separable one can choose (S, S, µ) as the Lebesgue (unit) interval, so that K and L2 (µ) are isomorphic. [Here one may use Cartier’s remark to construct L2 (µ) as observed in the proof of Theorem 5 above.] Now let {ϕn , n ≥ 1} and {fn , n ≥ 1} be arbitrarily chosen complete orthonormal sets in K and L2 (µ) respectively. Such sets clearly exist and have the same cardinality (cf., Na˘ımark, loc. cit, p.93). The mapping τ : fn 7→ ϕn

2.2 Structures of Second Order Random Measures

37

can be extended linearly onto the spaces which sets up an isomorphism between K and L2 (µ), extended linearly so that it preserves the inner products. Then each element Xt ∈ K can be expanded in a series as: Xt = =

∞ X

n=1 ∞ X

an (t)ϕn , an (t) = (Xn , ϕn )L2 (P ) = E(Xn ϕ¯n ), an (t)τ (fn )

n=1

=τ

∞ X

an (t)fn

n=1

!

, by the linearity and continuity of τ.

(23)

If A ∈ S, define Z : A 7→ τ (χA ) and observe that Z : S → L2 (P ) is σ-additive and starting with step functions and extending to all of L2 (µ), one gets for τ : L2 (µ) → L2 (P ) the representation: Z f (u)dZ(u), f ∈ L2 (µ), (24) τ (f ) = S

2

as a D-S integral so that L (µ) ⊂ L1 (Z). Let g : (t, u) 7→ Then by (23), g(t, ·) ∈ L2 (µ), t ∈ T and Z g(t, u)dZ(u), t ∈ T. Xt =

P∞

n=1

an (t)fn (u). (25)

S

¯ Since E(Z(A)Z(B)) = (τ (χA ), τ (χB ))K = (χA , χB )L2 (µ) = µ(A∩B), it follows that (Z(A), Z(B)) = 0 if A ∩ B = ∅ implying that Z(·) is orthogonally valued. Thus (22) obtains, and the last statement is clear. 2 Note that the only serious assumption on the “process” {Xt , t ∈ T } is that it spans (linearly) a separable subspace of L2 (P ). If T is a topological space 2 and Xt : Ω → C is a scalar random collection and K = sp{X ¯ t , t ∈ T } ⊂ L (P ) is separable, then the representation (22) gives an integral form of the second moment function r(·, ·) : Z Z ¯ t ) = (Xs , Xt ) = r(s, t) = E(Xs X g(t, v)dZ(v) g(s, u)dZ(u), S S Z g(s, u)¯ g(s, u) dµ(u), s, t ∈ T. (26) = S

1

In the case that kXs − Xt k2 = (E(|Xs − Xt |2 )) 2 → 0 as s → t in the topology of T is also true, then r : T × T → C is a continuous function and leads to other considerations for classes of Xt -processes based on the behavior of their (second moment) functions r(·, ·). This specialization and the consequent behavior of the random measure Z : S → L2 (P ) will be studied. In this sense, the above elementary representation of a process is motivational, and additional restrictions on the second order process without any separability assumptions on L2 (P ) or K, dictated by serious applications will lead to a deeper analysis. These and other properties of random measures are also considered later.

38

2 Second Order Random Measures and

Representations

2.3 Shift Invariant Second Order Random Measures In order to define the shift or translation operation for random measures, one should have a group property for the space (S, S, µ) on which Z : S → L2 (P ) acts. With a view to include both the spaces Rn and Zn , the Euclidean n-space and its lattice of integer points, we take S = G as a locally compact abelian (or LCA) group, S = B0 (G), the δ-ring of bounded Borel sets and µ as some regular measure. Then a mapping Z : B0 (G) → L2 (P ) is shift (or translation) invariant if (i)E(Z(A)) = E(Z(τx A)), A ∈ B0 (G), where τx A = x + A = {x + y; y ∈ A}, (‘+′ being the group operation in the LCA group G), (ii) Z(·) is σ-additive on B0 (G), and (iii) E(Z(τx A)Z(τx B)) = (Z(τx A), Z(τx B)) = (Z(A), Z(B)), A, B ∈ B0 (G). Here τx : G → G is called the shift or translation operator. Thus the mapping β : (A, B) 7→ (Z(A), Z(B)) defined on B0 (G) × B0 (G) → C is then a shift invariant bimeasure. In general β will not define a measure but shift invariance is a sufficient condition. The group structure of G is crucial. ¯ + is called tempered if there Definition 2.3.1 A real measure ν : B0 (Rn ) → R R dν(x) is a number k ≥ 0 such that Rn (1+|x|2 )k < ∞ where |x| denotes the Euclidean length of x ∈ Rn . Similarly a positive definite bimeasure β (i.e., Pn a ¯j β(Ai , Aj ) ≥ 0, n ≥ 1, ai ∈ C, Ai ∈ B0 (Rn )) is tempered if it i,j=1 i a is of bounded Vitali variation on each relatively compact rectangle in Rn × Rn , whence of σ-finite Vitali variation, and for a k ≥ 0, one has Z Z β(dx, dy) (1) k < ∞. Rn Rn [(1 + |x|2 )(1 + |y|2 )] 2

It is seen immediately that a tempered measure is finite on each relatively compact set. As an example, one sees that a Lebesgue-Stieltjes measure generated by an increasing real function on Rn is tempered. We present the following comprehensive result, with two demonstrations, using different arguments. Theorem 2.3.2 Let Z : B0 (Rn ) → L2 (P ) be a random measure with β as its induced bimeasure on B0 (Rn ) × B0 (Rn ) and B : B(Rn ) × B(Rn ) → C be the associated bilinear functional on the bounded Borel function space through the strict MT-integral as eq.(20) of the preceding section (cf. Theorem 9 there). If Z(·) is shift invariant then the following integral representations for β, B and Z(·) hold. ¯ + such that (i) There exists a unique tempered measure ν : B0 (Rn ) → R Z fˆ(u)ˆ B(f, g) = g(u)dν(u), f, g ∈ B(Rn ), (2) Rn

and hence the bimeasure β admits a representation Z ¯ˆD )(u) dν(u), C, D ∈ B0 (Rn ), β(C, D) = (χ ˆC χ Rn

(3)

2.3 Shift Invariant Second Order Random Measures

39

where fˆ, gˆ, χ ˆC and χ ˆD are the Fourier transforms on Rn , so that β determines a signed measure on Rn × Rn . Moreover Z χA dλ(u), A ∈ B0 (Rn ) (4) M (A) = E(Z(A)) = a Rn

for a constant a ∈ C satisfying ν({0}) ≥ |a|2 , λ being the Lebesgue measure. (ii) The random measure Z(·) is uniquely representable as Z ˜ χ ˆA (u)dZ(u), (5) Z(A) = Rn

where Z˜ has orthogonal values although Z is not assumed to have such values, ˜ × D) defining a signed measure on B0 (R2n ). (Z(C), Z(D)) = β(C, D) = β(C 2 If limλ(An )→0 kZ(An )k2 = 0, then ν << λ where λ is as before the Lebesgue measure. If Z(·) is continuous at ‘0′ then E(Z(A)) = 0 for all A ∈ B0 (Rn ). First Proof. If Z : B0 (Rn ) → L2 (P ) is a random measure with β as its induced bimeasure, consider the associated bilinear form B : Cc (Rn ) × Cc (Rn ) → C and then extend it to B(Rn ) × B(Rn ) using the strict MT-integration and Theorem 2.8, so that Z Z B(f, g) = f (x)g(y)β(dx, dy), f, g ∈ B(Rn ) × B(Rn ). Rn

Rn

The shift invariance of Z implies the same for β and what is decisive, this induces in turn that B(·, ·) is also translation (or shift) invariant which is positive definite and hermitian. The last property follows from a similar fact regarding β. Since the space of infinitely differentiable functions K, vanishing outside compact sets, is a subspace of B(Rn ), with the (stronger) Schwartz topology used in generalized functions, it results that B : K × K → C is continuous in this topology. Consequently by a form of the classical Bochner-Schwartz theorem (cf. Gel’fand and Vilenkin (1964), Theorem 6 on p.169)) that B(·, ·) admits the representation (2) for a positive tempered measure ν : B0 (Rn ) → R+ and then it extends to B0 (Rn ). Then one obtains the uniqueness from the Plancherel theorem easily. Now for a compact C ⊂ Rn there exist fn ∈ K with supports in C and fn → χC pointwise and boundedly, with a similar statement for gn → χD . Thus B(fn , gn ) → β(C, D) and fˆn → χ ˆA , gˆn → χ ˆB pointwise as well as boundedly. Substituting these into (2) and applying the bounded convergence theorems, namely Theorem 2.6 on the left and the standard Lebesgue theorem on the right, one obtains (3) since compact subsets of Rn generate the δ-ring B0 (Rn ). (Although, because of the application of the Bochner-Schwartz ˜ × D) which defines a σ additive signed measure and theorem, β(C, D) = β(C so the MT-integration need not be invoked, for the general case of bimeasures, Theorem 2.6 is needed. For such a generalization see the next result.) If M (A) = E(Z(A)), A ∈ B0 (Rn ), then M (·) is a regular signed measure that is shift or translation invariant on Rn . A classical result (Jordan

40

2 Second Order Random Measures and

Representations

decomposition) then applied to M = M + − M − where M + = max(M, 0) and M − = min(M, 0), implies that M + = αλ, M − = α′ λ for a pair of fixed numbers α, α′ ≥ 0, λ being the Lebesgue measure, so that M = (α − α′ )λ = aλ (say). In case M takes complex values, considering its real and imaginary parts, we get the same result with ‘a’ as a complex number. This implies (4). The relation between |a| and ν({0}) is established as follows. Since β is a positive definite bimeasure, β(A, A) ≥ 0 for each A ∈ B0 (Rn ). Hence taking a symmetric relatively compact A, λ(A) = c > 0, one has λ(A) 2 E(Z(A)) 2 ≤ β(A, A) , by the CBS inequality, = |a | = a c c c2 Z 1 = 2 |χ ˆA |2 (u) dν(u) c Rn Z δ0u dν(u), as c → ∞ → 2

Rn

= ν({0}).

(6)

For (ii), since the given shift invariant random measure Z(·) need not R be orthogonally valued, we associate a generalized random field F : f 7→ Rn f dZ, through the D-S integral for f ∈ Cc (S) and extend it to B(S), the space of bounded Borel functions on S, retaining the same (shift) invariance property from Z(·). Next applying (2) one obtains the representation for the positive definite bilinear form B(·, ·) as : Z ¯ fˆ1 (u)fˆ2 (u)dν(u), f1 , f2 ∈ B(S), (f1 , f2 ) = B(f1 , f2 ) = Rn

¯ + . Now consider the linear span H0 = for a tempered measure ν : B0 (Rn ) → R 2 sp{F (f ) : f ∈ Cc (S)} ⊂ L (P ), and let H be its completion in L2 (P ). It follows that (F (f1 ), F (f2 ))L2 (P ) = B(f1 , f2 ) = (f1 , f2 )L2 (ν) (7) is a semi-inner product. The mapping τ : f1 7→ F (f1 ) is an isometry of B(S) into H induced by (7). It may be uniquely extended from L2 (ν) onto H. If ˜ A ∈ B0 (S), then χA ∈ L2 (ν) and let Z(A) = τ (χA ) so that Z˜ : B0 (Rn ) → H is ˜ ˜ a random measure such that (Z(A), Z(B)) = ν(A ∩ B), since its σ-additivity is easily verified. It is now almost immediate that Z ˜ A ∈ B0 (Rn ), χ ˆA dZ, Z(A) = F (χA ) = Rn

giving (5) in which Z˜ has orthogonal values. Finally the hypothesis of nonatomicity of Z(·) at the origin gives 0 = ν({0}) ≥ |a|2 so that a = 0 and since M (A) = aλ(A) = 0, A ∈ B0 (Rn ) by (4). Thus E(Z(A)) = 0 for all A ∈ B0 (Rn ), by shift invariance. 2

2.3 Shift Invariant Second Order Random Measures

41

Second Proof. We outline an alternative argument which does not use the Bochner-Schwartz theorem in obtaining (2) and (3), essentially due to Thornett (1979), and the analysis is also influenced by the work of Yaglom’s (1958). Thus the crucial thing is to establish (2) or equivalently we assert that for all A, B ∈ B0 (Rn ), the following relation holds: Z Z Z Z ¯ ¯ y A)) dx dy. E(Z(τx+t B)Z(τy B) )dx dy = E(Z(τx+t A)Z(τ (8) A

A

B

B

To see this, we approximate the sets A and B by those of the form Cmj = {x = (x1 , . . . , xn ) ∈ Rn :

pmjk − 3m

1 2

< xk ≤

pmjk + 12 , k = 1, . . . , n}, 3m j = 0, 1, . . . ,

for suitable integers pmjk , and let xnj be the ‘midpoint’ of Cmj . Then χCmj (x) = χCmr (x − xmj + xmr ), x ∈ Rn , for some r. With this choice (as in Gel’fand-Vilenkin (1964), pp.165–169) one verifies that (8) holds when A and B there are finite unions of sets Cmj or equivalently, X anj χCmj + χAn , χA = j

where λ(An ) is arbitrarily small. Similarly for B. Thus verifying (8), one ob¯ serves that β(τy A, A) = E(Z(τy A)Z(A)) is positive definite as a function of y and is continuous. Then by the classical Bochner theorem (as in the BochnerSchwartz abbreviated Pn argument) one has a unique bounded measure νA such that [with (t, y) = j=1 tj yj ] Z ei(t,y) dνA (t). β(τy A, A) = Rn

Substituting this into (8) one has for all A, B ∈ B0 (Rn ), Z Z ei(t,y) |χ ˆA (t)|2 dνB (t) = ei(t,y) |χ ˆB (t)|2 dν(t). Rn

(9)

Rn

But then the collection {νA , A ∈ B0 (Rn )} determines a unique tempered measure ν as: Z dνA (y) , C ∈ B0 (Rn ), λ(A) > 0. (10) ν(C) = 2 | χ ˆ | A C Thus (9) and (10) imply (take y = 0) Z β(A, A) = |χ ˆA |2 dν, A ∈ B0 (Rn ). Rn

By the polarization identity of the inner product, this gives

42

2 Second Order Random Measures and

β(A, B) =

Z

ˆB )(u)dν(u), (χ ˆA χ

Rn

Representations

A, B ∈ B0 (Rn ).

(11)

[This method, however, does not give the result that β(·, ·) also extends to a signed measure on B0 (R2n ).] The rest of the argument is as in the first proof.2 Remark 2.1 It is of interest to observe that the translation invariant random ˜ measure Z(·) in L2 (P ) determines an orthogonally valued Z(·) by the integral representation (5) or its function theoretical equivalence (2) which is a type of Plancherel’s relation. Although the positive definite bimeasure β(·, ·) is translation invariant, and determines a ν, by (2), which may not be translation invariant, this relation motivates a study of measures having this special connection and its extension to Fourier analysis. [See the Remark after Cor. 2.3.4 below.] The preceding result has the following generalization leading to a corresponding representation of bilinear forms relative to tempered bimeasures (cf. Definition 1 above). This in turn gives an extension of the classical Plancherel’s isometric relation when the random measure is both translation invariant and orthogonally valued opening up an interesting avenue of Fourier transforms of certain unbounded measures. Theorem 2.3.3 Let Z : B0 (Rn ) → L2 (P ) be a random measure with its posi¯ tive definite bimeasure β : (A, B) 7→ E(Z(A)Z(B)), A, B ∈ B0 (Rn ). If Z Z B : (f, g) 7→ f (x)¯ g (y)β(dx, dy), f, g ∈ Cc (Rn ) (12) Rn

Rn

is the induced positive definite continuous bilinear form (by the strict MTintegral), then there exists a positive definite tempered bimeasure µ : B0 (Rn ) × B0 (Rn ) → C such that the following (strict MT-sense) integral representation obtains: Z Z B(f, g) = g(v)µ(du, dv), f, g ∈ Cc (Rn ). (13) fˆ(u)ˆ Rn

Rn

Moreover, the bimeasures β and µ are related by the equation: Z Z β(C, D) = χ ˆC (u)χ ˆD (ν)µ(du, dv), C, D ∈ B0 (Rn ). Rn

(14)

Rn

Further, the random measure Z admits the following representation: Z ˜ Z(A) = χ ˆA (u)dZ(u), A ∈ B0 (Rn )

(15)

Rn

where Z and Z˜ have the associated bimeasures β and µ respectively. Conversely, each positive definite bimeasure β : B0 (Rn ) × B0 (Rn ) → C admits a representation (14) for a unique tempered bimeasure µ and a random measure Z : B0 (Rn ) → L2 (P ) given by the (D-S) integral as (15).

2.3 Shift Invariant Second Order Random Measures

43

Proof. The argument uses some nontrivial results on generalized functions, as in the first proof above. Thus let β(·, ·) be the covariance bimeasure of Z : B0 (Rn ) → L2 (P ), the given random measure, and let B(·, ·) be the bilinear form induced by β on Cc (Rn ) × Cc (Rn ) using the strict MT-integration. Using the distribution theory and the kernel theorem of Schwartz, one notes that a Hermitian positive definite bilinear form on Cc (Rn ) can be expressed by a positive definite linear functional F1 on K2 , the Schwartz space of 2n variables on R2n , with compact supports (of infinitely differentiable functions of compact ¯ ≥ 0, ϕ ∈ K2 and supports) so that (F1 , ϕϕ) B(f, g) = (F1 , f g¯) = F1 (f g¯),

f, g ∈ Cc (Rn ),

(16)

where {f g¯ : f, g ∈ Cc (Rn )} spans a dense subspace of K2 . We then use the generalized Riesz representation theorem due to Morse and Transue given as Theorem 2.9 above to get Z Z F1 (φ) = φ(x, y)µ(dx, dy), φ ∈ K2 , (17) Rn

Rn

for a unique bimeasure µ as described in the statement (cf., e.g. Gel’fand and Vilenkin (1964), pp.158–161, particularly Lemmas 1–3 there), and µ is tempered and unique. In fact F1 is also defined on the space of rapidly decreasing functions containing K2 as a dense subspace in its topology. There are several details to be filled in, but they are spelled out in the above reference. This implies the representation Z ˆ v)µ(du, dv). F1 (φ) = φ(u, (18) R2n

ˆ v) = fˆ(u)g¯ˆ(v) one obtains (13) from Taking φ(x, y) = f (x)g(y) so that φ(u, (18). Extending these representations to B(R2n ), and then setting f (x) = χA (x), g(y) = χB (y), A, B ∈ B0 (Rn ), the relation (14) follows from (13). It is now a standard procedure to associate a generalized random functional F : Cc (Rn ) → L2 (P ), as in Proposition 2.6, and then find a random measure Z : B0 (Rn ) → L2 (P ), where we can assume the existence of a probability space (Ω, Σ, P ) rich enough to carry both the representations, since we can enlarge the space by adjunction if necessary and this is a standard (cf., e.g., Doob (1953), p.71, or Rao (1984), p.38) procedure. The random functional F then admits a representation relative to a random measure Z Z ˜ fˆ(u)dZ(u), f ∈ Cc (Rn ), (19) F (f ) = Rn

which implies (15) after extending this to B(Rn ), and setting F (χA ) which coincides with Z(A), A ∈ B0 (Rn ). Further detail seems unnecessary. The preceding arguments can be reversed for the converse implications with just the (strict) MT-integration (and without Schwartz distributions) and the verification can be left to the reader. 2

44

2 Second Order Random Measures and

Representations

If the given Z : B0 (Rn ) → L2 (P ) is also translation invariant, and orthogonally valued, then µ(A) = kZ(A)k22 = E(|Z(A)|2 ), A ∈ B0 (Rn ), and then in (15), Z˜ is also orthogonally valued by Theorem 2 already (cf.(5)). These two conditions imply the following statement. Corollary 2.3.4 Let Z : B0 (Rn ) → L2 (P ) be a translation invariant orthogonally valued random measure. Then µ : A 7→ E(|Z(A)|2 ) and the related tempered measure ν given by (2) satisfy the Plancherel type relation: Z Z g(u) dν(u), f, g ∈ Cc (Rn ). (20) fˆ(u)ˆ f (x)¯ g (x) dµ(x) = Rn

Rn

In particular, one has µ(A) =

R

Rn

|χ ˆA |2 dν, A ∈ B0 (Rn ).

Remark 2.2 If µ is Lebesgue measure, then ν = √12π µ in the classical Plancherel’s theorem (cf. Goldberg (1961), p.48), and the relation (20) leads to a study of Fourier transforms of unbounded measures. Such an investigation, without the random measure background or motivation, was considered by Argabright and Gil de Lamadrid (1974). Using this as a basis, one can generalize the corresponding work for both random and vector measures. (See Exercise 5 for further motivation.) The full details are not available at this time. So far we considered the invariant (and other) properties of random measures Z : B0 (Rn ) → L2 (P ), and briefly noted that if Z(·) is translation invariant so that its (signed) mean measure as well as the covariance bimeasure are shift or translation invariant, then its integral representations are obtained. If M (A) = E(Z(A)), A ∈ B0 (Rn ) and Z ∗ (·) = Z(A) − M (·), then Z ∗ (·) has zero mean measure and is shift invariant. Hence the preceding analysis applies to Z ∗ (·) whatever the signed measure M (·) is (i.e. not necessarily shift invariant). But this raises the question of determining the measures M (·) that qualify to be means of random measures Z(·) since clearly not every signed measure can be the mean of a random measure. This problem is not obvious, and we present its solution here. To appreciate the real problem, let Z : B0 (Rn ) → L2 (P ) be a random measure and M : B0 (Rn ) → C be a signed measure. In order that M (A) = E(Z(A)), A ∈ B0 (Rn ) it must also satisfy for Z0 = Z − M , that E(Z0 (A)) = 0 ¯ ¯ (B) = β(A, B) − M (A)M (B) and (Z0 (A), Z0 (B)) = E(Z(A), Z(B)) − M (A)M n for all A, B ∈ B0 (R ) be positive definite. Thus M (·) should obey this constraint and cannot be any signed measure on B0 (Rn ), while β(A, B) = ¯ E(Z(A)Z(B)) is always a positive definite bimeasure on B0 (Rn )×B0 (Rn ) which then has a representation (14), irrespective of M (·). In the following we use the notation: for Z : B0 (Rn ) → L2 (P ) a random measure with its bimeasure β, which induces a bilinear form B : Cc (Rn )×Cc (Rn ) → C admitting the integral representation (13) relative to a unique positive definite bimeasure µ : B0 (Rn ) × B0 (Rn ) → C. Let L2 (B) and L2 (µ) be the corresponding Hilbert spaces determined by the (semi-)inner products (f, g)B = B(f, g),

2.3 Shift Invariant Second Order Random Measures

R

45

R

ˆ(v)µ(du, dv), then by Theorem 3 above the mapping and (fˆ, gˆ)µ = Rn Rn fˆ(u)g¯ ˆ T : f 7→ f is an isometric isomorphism of the spaces L2 (B) and L2 (µ) relative to the inner products (·, ·)B and (·, ·)µ respectively when equivalence classes given by (f, f )B = 0 = (fˆ, fˆ)µ are identified. A solution to this problem is given by the following: Theorem 2.3.5 Let Z : B0 (Rn ) → L2 (P ) be a random measure with the associated bimeasure β, the bilinear form B and the representing positive definite tempered bimeasure µ all satisfying relations (12)–(14) above. Then a signed measure M : B0 (Rn ) → C is the mean of the random measure Z(·) iff M (·) admits a strict MT-integral representation: Z Z g (v)µ(du, dv) = (χA , g)B , A ∈ B0 (Rn ) (21) M (A) = χ ˆA (u)ˆ Rn

Rn

for a unique g ∈ L2 (B) satisfying (g, g)B = (ˆ g , gˆ)µ ≤ 1. Proof. Let Z(·) be a random measure into L2 (P ), with M (·) as its (signed) mean measure. Then by Theorem 3 above, its covariance bimeasure β is (positive definite and) tempered. Let ℓ : L2 (β) → C be defined as Z ˜ ℓ(f ) = E fˆ(u)dZ(u) , f ∈ L2 (β) Rn

˜ is determined by the representation of Z(·), given by (15). where Z(·) Then ℓ : L2 (β) → C is evidently linear. Consider Z 2 2 ˆ ˜ |ℓ(f )| = E f (u) dZ(u) n R Z 2 ! ˜ , by Jensen’s inequality ≤ E fˆ(u) dZ(u) Rn Z Z = fˆ(u)fˆ(v)µ(du, dv), by (14) and (15) Rn

Rn

= (fˆ, fˆ)µ = (f, f )B < ∞.

(22)

It follows that |ℓ(f )| ≤ kf k2,B and hence kℓk ≤ 1, ℓ ∈ (L2 (B))∗ = L2 (B), by identification of the adjoint space of the Hilbert space with itself. On the other hand by the Riesz representation theorem, there exists a unique g ∈ L2 (B) such that kℓk = kgk2,B = kˆ g k2,µ and hence Z Z M (A) = ℓ(χA ) = χ ˆA (u)g¯ˆ(v)µ(du, dv) = (χA , g)B (23) Rn

Rn

using the isometry. This establishes (21). For the converse, let M (·) be given by (21) and β be the bimeasure determined by Z(·). Then M (·) will be the mean measure of Z(·) iff β˜ : (A, B) 7→

46

2 Second Order Random Measures and

Representations

¯ (B) is positive definite. Indeed if β˜ has the stated property β(A, B) − M (A)M ˜ ¯ (·) is also positive definite and then by the clasthen β(·, ·) = β(·, ·) + M (·)M sical Kolmogorov existence theorem there is a random measure (can be taken ˜ Hence Gaussian distributed) Z˜ with mean zero and covariance bimeasure as β. ˜ we only have to verify that, under the given condition, β is positive definite. This is confirmed with the following computation. P Let Ai ∈ B0 (Rn ), ai ∈ C, i = 1, . . . , n and take f = ni=1 ai χAi ∈ L2 (B). Then one has : Z Z X n n X ˜ i , Aj )ai a ¯ˆA (v)µ(du, dv) β(A ¯j = ai a ¯j χ ˆAi (u)χ j i,j=1

Rn

Rn i,j=1

−

n X

¯ (Aj )ai a M (Ai )M ¯j

i,j=1

2 n X ¯ˆ ˆ = ai M (Ai ) f (u)f (v)µ(du, dv) − Rn Rn i=1 2 n X Z Z ¯ ai χ ˆAi (u)gˆ(v)µ(du, dv) , by (23), = (f, f )B − n n R R i=1 Z Z 2 = (f, f )B − fˆ(u)g¯ˆ(v)µ(du, dv) n n Z

Z

R

R

≥ 0,

since the second term in the penultimate display is at most (fˆ, fˆ)µ · (ˆ g , gˆ)µ ≤ (fˆ, fˆ)µ = (f, f )B by the CBS-inequality and condition (20) using the fact that gˆ is in the unit ball of L2 (µ). This establishes the converse assertion. 2 It is natural to look for conditions on the random measure so that its mean (signed) measure is representable in (21) for a measure µ instead of a bimeasure as given. Corollary 4 suggests a way to settle this question. The answer in fact is as follows: Proposition 2.3.6 Let Z : B0 (Rn ) → L2 (P ) be a random measure which is translation or shift invariant and has orthogonal values so that the equation µ(A) = E(|Z(A)|2 ) defines µ as a measure, (and often such a µ : B0 (R) → R+ is called the spectral measure of Z(·)). Then the signed measure M : B0 (R) → C is the mean of Z(·) in the sense that M (A) = E(Z(A)), A ∈ B0 (R), iff there is a unique g ∈ L2 (µ) ∩ L1 (µ) whose Fourier transform gˆ satisfies (ˆ g , gˆ)ν ≤ 1 where µ and ν are connected by the equation (20). In this case one has the representation for M as Z χ ˆA (u)g¯ˆ(u)dν(u), A ∈ B0 (Rn ) (24) M (A) = Rn

with kgk2,µ = kˆ gk2,ν ≤ 1 and µ, ν are connected by

2.3 Shift Invariant Second Order Random Measures

µ(A) =

Z

Rn

|χ ˆA |2 dν,

A ∈ B0 (Rn ).

47

(25)

The proof of the result is immediate from Theorems 3 and 5. This will be left as an exercise. The proposition extends a result due to Balakrishnan (1959) who characterized possible mean functions of second order stationary processes, and the above one is the counterpart for random measures. Remark 2.3 It may be observed that, from (25), when M (·) is also desired to be translation invariant then ν is equivalent to the Lebesgue measure, and dλ ˆ for A ∈ B0 (Rn ). In case the spectral measure of Z(·) is also ˆA g¯ dν |A = χ nonatomic or continuous at the origin of Rn , then by Theorem 2 above, M = 0 must hold so that g = 0 a.e.[µ](or gˆ = 0, a.e.[ν]). Thus Z(·) must be centered. This analysis gives a general description of second order (real or complex) random measures in terms of their mean and covariance bimeasures. The (adˆn ∼ ditive) group property of Rn and its dual R = Rn (identified) allowed us to study a translation invariant random measure Z : B0 (Rn ) → L2 (P ) for which the pair (β, µ) are related by (14) and when Z(·) also has orthogonal values, the pair of (positive) measures (µ, ν) on B0 (Rn ) given by (25), is analogous to the Plancherel pair in the familiar (classical) harmonic analysis sense. Thus a study of random measures of various types in L2 (P ) leads us into an interesting nontrivial extension of the classical theory into many new directions. We indicate how this may be accomplished. Observe that (20) can be expressed also as: Z Z Z \ fˆ(u)ˆ (f ∗ g¯)(u)dν(u), (26) g (u)dν(u) = f (x)g(x)dµ(x) = Rn

Rn

Rn

where f ∗ g¯ is the convolution of f and g¯. If the measure ν on the right can be replaced by µ ˜ so that (26) becomes Z Z (f ∗ g¯)(x)d˜ µ(x) = fˆ(u)g¯ˆ(u)dν(u), f, g ∈ Cc (Rn ), (27) Rn

Rn

such a measure µ ˜ is termed positive definite, since (25) is nonnegative when R µ(x) ≥ 0 for all f ∈ Cc (Rn ), and ν on the right side of f = g, i.e., Rn (f ∗ f¯)(x)d˜ (27) is called the (Plancherel type) transform measure of µ ˜, by Thornett (1979). The following exact statement connecting these two terms is due to him. Proposition 2.3.7 A measure µ ˜ : B0 (RnR) → R+ is positive definite iff there is n + ˆA |2 dν < ∞, A R∈ B0 (Rn ) and (b) a unique ν : B0 (R ) → R such that (a) Rn |χ n ˆAm |2 dν = 0 for all An ↓, An ∈ B0 (R ) with λ(An ) ↓ 0 implies limm→∞ Rn |χ where ν is also a transformed measure. [However, not all measures satisfying the conditions (a) and (b) need be transformed ones.] Proof. If µ ˜ is positive definite, then the bimeasure

48

2 Second Order Random Measures and

β(A, B) =

Z

Rn

(χA ∗ χ ¯B )(x)d˜ µ(x),

Representations

A, B ∈ B0 (Rn )

is positive definite and by the converse part of Theorem 2.3.3 it is associated with a random measure that is translation invariant. Hence by Corollary 2.3.4 there is a ν satisfying (20) so that (a) and (b) hold, i.e. is a transform of µ ˜. Conversely if ν satisfies (a) and (b), then µ ˜ defined by the equation (27) is clearly positive definite. 2 Since each bimeasure β induced by a shift invariant random measure determines a measure ν, as in Theorem 2(i), (see also (20) above) which plays an important role in this analysis one may consider a few more of its properties for a better appreciation. We include one more result on this topic here. The following additional terminology will help the description. A measure ν : B0 (Rn ) → R+ is called translation bounded if supx ν(τx A) < ∞ for each A ∈ B0 (Rn ) where τx is the shift operator for x ∈ Rn . Further ν is said to be uniformly bounded if supx ν(τx Sh ) < ∞ for a symmetric fixed width cube Sh at each point u ∈ Rn , where Sh = {u ∈ Rn : − h ≤ ui < h, i = 1, . . . , n, u = (u1 , . . . , un )}. An interesting characterization of these boundedness concepts is given by: Proposition 2.3.8 For a measure ν : B0 (Rn ) → R+ , the following four statements are equivalent: (i) (ii) (iii) (iv)

ν is translation bounded. ν is uniformly bounded. χ ˆA ∈ L2 (ν), for all A ∈ B0 (R). There exists a fixed A0 ∈ B0 (Rn ), λ(A0 ) > 0, such that Z |χ ˆA0 (u − x)|2 dν(u) < ∞. sup x

Rn

Moreover, the measure ν is a transform of a shift invariant random measure Z(·) iff the induced bimeasure β(·, ·) of Z(·) is of locally bounded Vitali variation in the sense that for each A ∈ B0 (Rn ), there exist sets Ak ∈ B0 (Rn ) ∩ A, disjoint, satisfying the condition sup m

m X m X

j=1 k=1

|β(Aj , Ak )| < ∞.

We omit a proof of this proposition, as the result is not used in our later developments. The last part is already considered in Theorem 2.3.2(i) above, and the details of the result are given in Thornett (1979). It is also detailed with simplifications in Daley and Vere-Jones (1984), Sec.(11.1) and Lin (1969), as well as Robertson and Thornett (1984). Instead we consider some special properties of these measures utilizing the group structure of the underlying measure space more fully which indicates further extensions when Rn is replaced by suitable LCA groups G.

2.4 A Specialization of Random Measures Invariant

on Subgroups

49

2.4 A Specialization of Random Measures Invariant on Subgroups We first describe an instance where a random measure is always nonnegative and then include some special analysis in such cases. Let X : Rn → L2 (P ) be a measurable mapping called a random field. The collection A(a, ω) = {t ∈ Rn : X(t, ω) ≥ a} is the set of its indexing points of X at which the field X(t) is above the level a(≥ 0), and then A(a, ·) is often called the level set of the field over a. The mapping Za : C → #C ∩ A(a, ·) is also a type of random measure (under the standard measurability assumptions) of the set C ∈ B0 (Rn ). The collection {Za (C) : C ∈ B0 (Rn )} is often called a point process which is nonnegative and takes values in L2 (P ), i.e, Za (C) ∈ L2 (P ). Thus nonnegative additive random set functions Z(·) : B0 (Rn ) → Lp (P ) can be studied for p ≥ 0. Here we analyze the case that p = 2, and discuss 0 < p < 2 later. It was already noted in Theorems 2.4 and 3.2 that, for shift invariant random measures, the induced bimeasures determine a (scalar) measure on B0 (R2n ). They are also hermitian symmetric and inherit the positive definiteness property. Moreover we have for C, D ∈ B0 (Rn ), Z Z ¯ˆD )(u)dν(u) ˜ (1) χC×D (s, t)dβ(s, t) = (χ ˆC χ R2n

Rn

where β˜ is a (scalar) measure determined by the bimeasure of β of Z(·). Using a detailed group structure of the space G = Rn , (1) will be analyzed. The underlying problem will be visible, if the abstract version of its structure is briefly described. This is done here, and further details may be found in Bourbaki (1963) with an elaboration in Reiter (1968). Now if G is any locally compact group and H ⊂ G is a closed subgroup, let G/H be the quotient space (this will be a group again only when H is also normal). If µ is a (complex) measure on G (i.e. on its Borel σ-algebra or even a σ-ring) then, with λ as a left invariant (or a Haar) measure on H, there exists a unique regular measure µ ˜ (complex if µ is such) satisfying Z Z Z f (x)dµ(x), f ∈ Cc (G), (2) f (xu)dλ(u) d˜ µ(x) ˙ = G/H

G

H

iff µ satisfies the relation Z Z −1 f (x)dµ(x), u ∈ H, f ∈ Cc (G). f (xu )dµ(x) = ∆H (u) G

(3)

G

Here x˙ = πH (x) ∈ G/H where πH : G → G/H is the quotient mapping and ∆(·) is the modular function. A proof of this result is in Reiter (1968, p.157) to which the reader is referred. When G is abelian, as here, ∆H (x) = 1, x ∈ H, and (3) holds automatically so that (2) can be applied, and then µ may be written symbolically as dµ = d˜ µ × dλ. We discuss the special case that G is separable σ-compact and µ is also (translation)invariant in a particular form.

50

2 Second Order Random Measures and

Representations

Suppose that G is a Lie group (an important example is Rn or nonsingular n × n matrices on Rn ) which can be represented as G = KH where K and H are subgroups satisfying K ∩ H = {e}, the identity so that K = G/H is subsumed. Then G can be identified with K × H/{e}, the correspondence being a homeomorphism. We apply this for G = Rn . Suppose that a special translation τx = τx1 e = τx1 · Ie : G = K × H → G is given by τx y = (τx1 y1 )u1 where y1 ∈ K and u1 ∈ H, τx1 is as usual (τx1 y1 = x1 y1 ) and Ie u1 = u1 so that µ(τx (A × B)) = µ(τx1 A × Ie B) = µ(A × B). [Group operation is denoted by juxtaposition of its elements, which may be for instance addition or multiplication.] In this case µ can be written as a product of an invariant (or left Haar) measure λ on K and a measure µ on H so that µ = λ × µ1 . This follows from the fact that µ(· × B) is a τx1 -invariant measure on the group K, for each B ∈ B0 (H), and since K is locally compact, by the essential uniqueness of the Haar measure it must equal c(B) · λ(·) where λ is the Haar measure on B0 (K) and c(B) is a positive constant for each B ∈ B0 (H). Thus µ(A × B) = c(B)λ(A) and since µ(A × ·) is σ-additive, it follows easily that c(·) is σ-additive so that we may call it µ1 and µ = λ × µ1 is an essentially unique representation. Expression (2), then reduces to Z Z Z f (x, u)dλ(u) dµ1 (x1 ), f ∈ Cc (G). (4) f (x)dµ(x) = H

G

K

[See Wijman (1990), particularly Sections 5.9 and 7.6 on this formulation.] Suppose now that G = Rn and K = {x ∈ Rn : x1 = x2 = · · · = xn } the diagonal subgroup. Let the translation (= shift) be τx1 (y) = (x1 +y1 , . . . , x1 +yn ) for x1 ∈ K and y = (y1 , . . . , yn ), so that G/K1 = H and τx1 moves every coset by x1 units called the diagonal shift. Then (4) becomes for a unique regular measure µ1 on H, and λ as Lebesgue measure on K1 , Z f (x1 , . . . , xn )dµ(x1 , . . . , xn ) G Z Z f (x1 , x1 + y1 , . . . , x1 + yn−1 )dµ1 (y1 , . . . , yn−1 ) dλ(x1 ). (5) = H

K

Using this specialization, we present a representation of the second moment measure under these invariance conditions, called moment stationarity by Daley and Vere-Jones for obvious reasons. The relation between µ and µ1 is that the spaces on which µ, µ1 act are such that the latter is on K, a subspace of G (which is G/H) and so it is also termed a “reduced” measure of µ. The formula (5) shows that if µ1 is shift invariant on G for “diagonal shifts” τx1 the µ is also invariant for τx1 . This shows that a random measure Z : B0 (Rn ) → L2 (P ) is translation invariant for τx1 and the resulting bimeasure, which determines a measure µ of (5), has the invariance property for its reduced measure µ1 . It can be stated as a summary of the preceding discussion as: Proposition 2.4.1 Let Z : B0 (Rn ) → L2 (P ) be a random measure which is translation invariant on the diagonal subgroup H ⊂ Rn and whose bimeasure

2.4 A Specialization of Random Measures Invariant

on Subgroups

51

determines a regular measure µ (in particular invariant for translations on Rn and also invariant for diagonal shifts). Then the reduced measure µ1 of µ by (5) is the shift invariant part of µ on the subspace K = G/H, a factor space identified as a subgroup of G. Several examples from point processes and further discussion on this topic are in Daley and Vere-Jones ((1984), Chapters 10 and 11). There is also an interesting application for positive (second order) random measures which determine positive bimeasures that are positive definite with their covariance bimeasures, whose mean measures satisfy Theorem 3.5. Thus they are different and are “positive-positive definite” bimeasures, which can then be analyzed. In this context it is useful to discuss further the random measures which have k-moments and satisfy the corresponding invariance condition. This again leads to nontrivial analysis bringing in some new ideas suggested by applications, and motivating a study of (vector) “polymeasures” considered in a later chapter. Thus the relation to be satisfied is: β(A1 , . . . , Ak ) = E

k Y

Z(Ai ) = β(τx A1 , . . . , τx Ak )

(6)

i∈λ

where τx A = A+x, x ∈ Rn , is the translation operator and assume the existence of the k th moment (k ≥ 2) so that β(A1 , A2 , . . . , Ak ) is a measure in each Ai ∈ B0 (Rn ) when others are held fixed. Thus Z : B0 (Rn ) → Lp (P ), for p ≥ k, and β, Z satisfy the corresponding regularity conditions extending the bimeasure situation. Then Z will be called k-stationary. However, as Theorem 3.5 implies, the k-stationarity does not imply the same for the lth order moments, 1 ≤ l < k, and that property has to be assumed explicitly. If one desires to use a strengthened concept called cumulant (also called semi-invariant) stationarity, it includes all the lower moments, but is still weaker than the strict stationarity which is the invariance of all finite dimensional distributions of the random process {Z(A), A ∈ B0 (Rn )}. The former was considered by Brillinger (1972) and others. We recall the concept precisely and present a key result to get some feeling for such an extension of the analysis. In the classical theory if a random variable X has n moments then its characteristic function given by ϕ(t) = E(eitX ), satisfies ϕ(t) = 1 +

n X

αk

k=1

(it) + o(|t|n ), k!

αk = E(X k ),

(7)

and hence ϕ(t) 6= 0 in some neighborhood of the origin 0 ∈ R. We can then define in that neighborhood ψ(t) = log ϕ(t) as a suitable (often the principal) branch of the complex function defined by log ϕ(t). Then using a Taylor series expansion of the latter in that neighborhood, one finds ψ(t) =

n X j=1

kj

(it)j + o(|t|n ), j!

(8)

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2 Second Order Random Measures and

Representations

where k1 = α1 , k2 = α2 − α21 = Var X, k3 = α3 − 3α1 α2 + 2α31 , . . . , and the constants kj are the cumulants of X, each kj being a polynomial in α1 , . . . , αj as one verifies after a simple computation with (8). To generalize this for the vector variable Y = (Y1 , . . . , Yr ) each Yi , 1 ≤ i ≤ r, having k-moments, one studies the function given by ϕ(t) = ϕ(t1 , . . . , tr ) = E(ei(t1 Y1 +···+tr Yr ) ) so that ϕ is continuously (partially) differentiable, and since ϕ(0) = 1, it is again nonvanishing in a neighborhood of 0 ∈ Rn so that ψ(t) = log ϕ(t) which is definable as a principal branch of the ‘log’ function in that neighborhood. Then one gets after a (nontrivial) computation for each vector j = (j1 , . . . , jr ) with |j| denoting the sum of the integers, ji ≥ 0 the following: ψ(t) =

X

0≤j1 +···+jr =|j|

kj

(itr )jr (it1 )j1 ··· . j1 ! jr !

(9)

The kj are called the j th order cumulants, and are given by   ! X Y Y kj = cum (Y1 , . . . , Yr ) = (−1)p−1 (p − 1)!E Yi  (10) Yi · · · E  i∈vi

i∈vp

where the sum extends to all partitions (v1 , . . . , vp ) of (1, . . . , r) and p = 1, . . . , r. This formidable expression has some useful properties and more extensive relations are in Leonov and Shiryaev (1959,1960). With these and related relations one can obtain the following: Proposition 2.4.2 Suppose (S, S, ν) is a measure space, Z : S0 → Lp (P ), 2 ≤ k ≤ p, a random measure and consider the multi or polymeasure: ! k Y Z(Ai ) (11) β(A1 , . . . , Ak ) = E i=1

˜ 1 × · · · × Ak ) = for Ai ∈ S0 ,the δ-ring of all ν-finite sets in S. Suppose that β(A β(A1 , . . . , Ak ) determines a scalar measure on the product space S k , for such k > 1. If Z(·) has the k th order translation invariance property defined by (6) for τx as in Proposition 1, then the (scalar) measure β˜ can be factorized as: dβ˜ = dβ˜1 dλ

(12)

˜ a (scalar) invariant measure on Rk−1 for τx as in Proposition 1, and where β,is λ is the Lebesgue measure on R. Moreover, the same property holds for the k th order cumulant measure of Z(·), denoted γ(·) which admits a decomposition as dγ = dγ1 dλ

(13)

where γ1 is a scalar measure on R(k−1)n that is similarly translation invariant. If also the γ has the property that

2.4 A Specialization of Random Measures Invariant

Z

Rk

Z

R(k−1)n

on Subgroups

|xi |d|γ1 |(x)dλ(x) < ∞, 1 ≤ i < k,

then the random measure Z(·) can be represented as: Z Z ˜ 1 , . . . , xk−1 )dλ(xk ) Z(A) = eit(x1 +···+xk−1 ) dZ(x R(k−1)n

53

(14)

(15)

A1

where A = R(k−1)n × A1 , the cylinder set with base A1 ∈ B0 (Rn ). Sketch of proof. We include a sketch of the argument to give an idea of the general theory of expanding random measures different from and sharper than the vector measure case. The formulas (12) and (13) are really specializations of (5) if G = Rk = Rk−1 × R. Then (15) follows on using the further property (14) when this condition is applied to the stationary (second order) case. It should be noted that the translation invariance of β˜ implies (12) may also be established directly in the present special case without invoking the general form (4) which is really a theorem due to A.Weil. This alternative method was employed by Brillinger (1972) and we shall briefly sketch it for the convenience of the reader. It is comparatively elementary. Since the scalar measure β is translation invariant, one can make a change of variables t1 , . . . , tk where ti ∈ R by translation of tk to (t1 −tk , . . . , tk−1 −tk , tk ), the Jacobian being unity, the new measure γ is invariant under the mapping t 7→ tk + t for all t ∈ R. Hence the transition from β to γ takes the translation to ui = ti − tk , i = 1, . . . , k − 1, uk = tk , so that γ((u1 , . . . , uk−1 , tk ) − γ(u1 , . . . , uk−1 , 0)

= γ(u1 , . . . , uk−1 , tk + t) − γ(u1 , . . . , uk−1 , t)

(16)

which is the key observation in going from β to γ. It is a functional equation in tk and t. In fact set γu˜ (tk ) = γ(u1 , . . . , uk−1 , tk ) and rewrite (16) as γu˜ (tk + t) = γu˜ (tk ) + γu˜ (t) − γu˜ (0), tk , t ∈ R.

(17)

This is analogous to the classical Cauchy functional equation. Setting t, tk to be rationals and observing γu˜ (·) is a function of locally bounded variation (hence continuous for (17)) one has (as in the classical Cauchy case) the solutions as: γu˜ (t) = Ku˜ t + Ku˜′ (0),

(18)

for any fixed u ˜ ∈ Rk−1 . Thus γu˜ (·) is continuous and γ(·) (t) is (locally) bounded and measurable. Hence the measure determined by γu˜ (t) is the same as that obtained for dKu˜ dt = γ1 (d˜ u)dλ(t) where λ(·) is the Lebesgue measure and (13) follows. Here γ1 (·) is invariant on the subgroup H. [It should be observed that this specialization for R, of Weil’s formula which is valid for any LCA group, cannot be simply extended to more general groups. In fact the result has been generalized by G.W. Mackey for the nonabelian separable (locally compact) case, and a further extension (removing the last separability restriction) by

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2 Second Order Random Measures and

Representations

F. Bruhat. The general proposition has numerous applications to harmonic analysis and also to probability theory, for martingale analysis, as well as to stationary random fields (cf., e.g., Rao (1981), Sec 5.5).] 2 The corresponding result for the k-stationary cumulants can be given using formula (10) and the above representation for moments j, 1 ≤ j ≤ k. In this work γ1 (·) depends on ti − tk , i = 1, . . . , k − 1 and is a complicated function of all the lower order moments of Z(·). An integral representation of the random measure Z(·) in the case that it has all moments (so its Laplace transform exists) was given by Brillinger (1972), (see also Daley (1971) and Vere-Jones’s addendum in that paper). When the polymeasure defines a scalar measure, one can extend Theorem 3.2 in principle, to polymeasures. The details of formula (15) may be obtained from Brillinger’s work noted above. Remark 2.4 Formulas (12) and (13) are particular instances of the general class of measures that admit disintegration. A deep analysis of this problem, in an abstract setting has been presented in the monograph on ‘Lifting Theory’ by the Ionescu Tulceas (1969) Chapter IX. Optimal conditions are given in order that, for a pair of Radon measures µ, ν on compact spaces S and T , ν(·) = R α (·)dµ(t) where αt (·) is a suitable real measure. Thus ν is disintegrated into T t parts αt (·) relative to µ. This general problem is related to the lifting theory, and has interest in many areas of abstract analysis. Thus we have to omit its further discussion here.

2.5 Complements and Exercises 1. (a) This example shows the differences between the MT- and strict MTintegration of Definition 2.5. Let (Ωi , Σi ), i = 1, 2 be identical copies if Ωi = Z, the integers, and Σi = P(Z) = 2Z , the power set. Define the mapping β : Σ1 × Σ2 → R+ by the equation, β({n}, {m}) = (|m| + |n|)−4 χ[(m,n) : m2 +n2 >0] . As a positive bimeasure, it defines a measure with finite (Vitali) variation. Show that f = g : Z → Z, where f (x) = x, and Z X X f (x)β(dx, B) = n(|m| + |n|)−4 , A ∈ Σ1 , B ∈ Σ2 , A

n∈A−{0} m∈B

and (f, g) is MT-integrable with value = 0. Deduce that if A = Z+ , f = f χA , (f = g), then it is not a strict MT-integrable function. (b) Verify however that a bimeasure on a σ-algebra B, (i.e. on B×B) always has finite Fr´echet variation. 2. Let Z : S → L2 (P ) be a random measure on a measurable space (S, S) ¯ and consider its induced bimeasure β : (A, B) → E(Z(A)Z(B)), A, B ∈ S.

2.5 Complements and Exercises

55

There exist β-integrable f (i.e.,(f, f ) is MT-integrable) which are not Zintegrable, using the D-S integration, as the following example implies. Let (Ω, Σ, P ) be the Lebesgue unit interval, Ω = [0, 1]. Let {ϕn , n ≥ 1} be the orthonormal sequence in L2 (P ), where ϕn (ω) = e2πinω P , ω ∈ Ω. Take (S, S) as S = N, (positive integers) and S = 2N . Let Z(A) = i∈A αi ϕi , A ∈ S, and αk = n−2 if k = 2n, but = −n−2 if k = (2n+1). Verify that Z(·) is a random measure and let β(·, ·) be its induced bimeasure. Let f (2x) = x2 = f (2x+1). R R Show that (f, f ) is β-integrable and S S (f, f )dβ = 0. However, if f were Z-integrable in the D-S sense, then f χA is also Z-integrable, A ∈ S, but note that if A = {2n : n ≥ 1}, then Z ∞ 2 X E f χA dZ = E(|ϕn |2 ) = ∞ S

n=1

so that f is not Z-integrable! (Thus the strict MT-definition as in Proposition 2.6 is useful here.) 3. (a) In the definition of the MT-integration, we used only f : Ω1 → C and f2 : Ω2 → C to define an integral of the map (f1 × f2 ); Ω1 × Ω2 → C. It is perhaps desirable to consider f : Ω1 × Ω2 → C which is not necessarily a product f = f1 f2 . Here we indicate the existence and some new difficulties. If f : Ω1 × Ω2 → C is a measurable function for Σ1 ⊗ Σ2 (Σi is a σ-algebra of Ωi ) and fnP : Ω1 ×P Ω2 → C is a sequence of simple (measurable) functions n n where fn = i=1 j=1 aij χEi ×Fj , Ei ∈ Σ1 , Fj ∈ Σ2 then for a (even a positive definite) bimeasure β : Σ1 × Σ2 → C define the MT-integral Cn (A, B) =

Z Z A

fn (ω1 , ω2 )β(dω1 , dω2 ) =

B

n X

i,j=1

aij β(Ei ∩ A, Fj ∩ B),

for A ∈ Σ1 , B ∈ Σ2 . If β has finite Vitali variation and {Cn (A, B), n ≥ 1} is Cauchy in C define the limit as Cf (A, B). Verify that Cf (A, B) is well defined for each A, B and does not depend on the fn sequence of the definition and Cf is also a bimeasure having finite Vitali variation. (b) When f and fn are not products such as f (ω1 , ω2 ) 6= f 1 (ω1 )f 2 (ω2 ) then with the above definition, Theorem 2.8 need not hold. Construct an example to verify this statement. [A somewhat involved construction of an example substantiating the above statement is in Chang and Rao ((1986), p.48), but a simpler example should be devised, and it is worth while for the readers to find one.] (c) Bimeasures can be generated in many ways. Let (Ωi , Σi ), i = 1, 2 be measurable spaces and λ : Σ2 → R+ be σ-additive. Suppose µ : Σ2 × Ω1 → R+ is a mapping such that µ(·, ω1 ) is a finite measure for each ω1 and µ(·, E2 ) is measurable (Σ1 ) for each E2 ∈ Σ2 . Consider Z ν(A, B) = µ(B, ω1 ) dλ(ω1 ), A ∈ Σ1 , B ∈ Σ2 . (∗) A

56

2 Second Order Random Measures and

Representations

Such an integral representation is also called the disintegration of ν(A, ·) into measures µ(·, ω1 ) relative to λ for each A (usually taken A = Ω1 ). Verify that ν(·, ·) is a positive bimeasure. Sometimes µ(·, ·) is also called R ˜ 2 ) and ν(A, B) = a measure kernel. If µ(B, ω1 ) = B f (ω2 , ω1 )dλ(ω R R ˜ A B f (ω2 , ω1 )dλ(ω2 )dλ(ω1 ), (∗) is an example of a bimeasure that is not generated by a product. All the above noted problems and new questions arise. If Ω1 , Ω2 are locally compact spaces and if A = Ω1 (fixed), then the corresponding questions have been investigated thoroughly in the Ionescu Tulcea (1969) volume, as noted before. Still our bimeasure problems are more general and present new challenges. The above (∗) is also closely related to conditional probability measures. 4. (a) A classical method of generating random measures from second order (random) functions, following the known procedure with Brownian motion, is as follows. Let {Xt , t ∈ [0, 1]} be the Brownian motion on (Ω, Σ, P ) so that it is a Gaussian process with mean zero and covariance r(s, t) = E(Xs Xt ) = min(s, t), 0 ≤ s, t ≤ 1. If we set Z([a, b]) = Xb − Xa , then Z(·) R1 is a random measure and f ∈ L2 ([0, 1], dt)is defined as T f = 0 f (t)dZ(t) = R1 f (t)dXt . Since the Xt -process is known not to have finite variation on any 0 non-degenerate subinterval of [0, 1], the integral cannot be understood in the Lebesgue-Stieltjes sense. Wiener was the first to have given a rigorous mathematical definition of this in 1923, but it is relatively complicated. Later in 1933, Paley-Wiener-Zygmund have given another method which is much simpler and equally satisfactory. Namely consider for f : [0, 1] → R a continuously differentiable function with derivative vanishing R 1 at t′ = 1, R1 and a formal integration by parts gives T f = 0 f (t)dXt = − 0 Xt f (t)dt, since X0 = 0 (f ′ = df dt ). It then is verified without difficulty that (taking the last integral in the vector or Bochner sense), kT f k2L2(0,1)

= =

Z Z Ω Z 1 0

=

Z

0

1

Z

0

Z

0

1 ′

Xt f (t)dt

0 1 Z

Ω

1

2

dP

Xs Xt dP f ′ (s)f ′ (t)ds dt

min(s, t)f ′ (s)f ′ (t)ds dt = kf k2L2(0,1) .

Thus T is our isometry on C01 ([0, 1]) into L2 (P ) where C01 ([0, 1]) is the space of continuously differentiable scalar functions vanishing at t = 1, which is dense in L2 ([0, 1]), and thus T extends to all of L2 ([0, 1]) uniquely. R1 Then by the classical Riesz representation theorem T f = 0 f (t)dZ(t)(= R1 f (t)dXt ), for all f ∈ L2 (0, 1], where Z : B([0, 1]) → L2 (P ) is a unique 0 (vector) random measure. Taking f = χ[a,b] ,one obtains Z([a, b]) = Xb − Xa as we started at the beginning. The reader is invited to fill in the details for the last integral.

2.5 Complements and Exercises

57

(b) This method has a beautiful extension to a vast class of processes with continuous rapidly decreasing covariance functions, utilizing a generalized Fourier transform introduced by S. Bochner (1959) and employed by T.Kawata (1965). We present it here to generate classes of second order 2 random measures. Let us call a k-operator √ of R → C denoted by Lk , k ≥ 0 (integer), as L0 = 0 and for k ≥ 1, (i = −1) define it by: Lk (x, y) =

k−1 X j=0

(−ixy)j , |x| ≤ 1, and = 0 if |x| > 1, y ∈ R. j!

Then a k-transform of an L20 (P )-continuous process {Xt , t ∈ R} with means zero is defined as: Z 1 e−its − Lk (s, t) Yk (t, ω) = √ Xs (ω) ds, ω ∈ Ω, (∗) (is)k 2π R when this integral is considered as a vector or Bochner integral. If k = 0, this is the usual Fourier transform, and the integral exists if the covariance function r(·, ·) of the Xt -process is rapidly decreasing, i.e., satisfies for k > 0, Z Z |r(s, t)| ds dt < ∞. (∗∗) k )(1 + |t|k ) (1 + |s| R R Show that (using results of the classical Fourier analysis) the Yk : Ω → C has almost all continuous paths and that the Xt and Yk (t)-process determine each other uniquely. [Verify this at least for k = 2 and Xt Brownian motion.] (c) With k = 2, for any mean continuous second order process {Vt , t ∈ R} and f : R → C twice continuously differentiable, show by a formal integradf tion by parts one has the expression, for T > 0 and u > 0, f ′ = ds , Z

0

T

Z

u

−u

f (x) d2 V (x)(ω) = ∆2T {f (−T )V (−T )(ω)} −2 +

Z

Z

0 T

T

∆2s {f ′ (−s)V (−s)(ω) ds du

0

Z

u

f ′ (s)V (s)(ω) ds,

(†)

−u

where we abbreviated for each ω ∈ Ω1 ∆2T {f (−T )V (−T )(ω)} = (f (T )V (T ) − 2f (0)V (0) + f (−T )V (−T ))(ω) ∆2s {f ′ (−s)V (−s)(ω)} = f ′ (s)V (s)(ω) − f ′ (−s)V (−s)(ω). The right side of (†) is well-defined as the (Bochner or) standard vector integral and the left side is defined by the right side, and makes dV as a random measure, as in (a) above.

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2 Second Order Random Measures and Representations

(d) Let {Xt , t ∈ R} ⊂ L2 (P ), with E(Xt ) = 0, be a mean continuous process with covariance r(·, ·) satisfying the temperedness condition (**). If the 2-transform of the Xt -process is denoted as {Yt , t ∈ T } as in (b) and (c) above, then, using a procedure of the familiar L´evy inversion formula, we have the integral representation of Xt with the random measure by Y : B(R) → L2 (P ) Z u Z T Xt = slimT →∞ eits d2 Y (s) du 0

−u

where “slim” stands for the L2 (P )-norm limit. [Hints: There is some algebraic simplification, and then the limit procedure uses delicate estimates of integrals with (∗∗). Consult Kawata (1965) for details.] 5. If Zi : B0 (R) → L2 (P ), i = 1, 2, are a pair of random measures, it is possible to define a convolution product to obtain a new random measure with the following procedure, using their induced bimeasures βi , assumed to have finite Vitali variations. For a pair of sets A, B employing the Lebesgueintegration which is permissible here, consider Z Z (β1 ∗ β2 )(A, B) = β1 (A − x, B − y)β2 (dx, dy), A, B ∈ B0 (R). R

R

Show that (β1 ∗ β2 )(·, ·) is a well-defined positive definite bimeasure on B0 (R) × B0 (R) and that there is a random measure Z : B0 (R) → L2 (P ) whose induced bimeasure is (β1 ∗ β2 )(·, ·). We define Z = Z1 ∗ Z2 as a new random measure. Show that Z is well-defined with values in L20 (P ) again. [Since the convolution of the sum of a pair of random variables, not necessarily independent, can have their joint characteristic functions just at each t ∈ R as the product of marginals or individual characteristic functions, the above concept through the bimeasures of the individual random measures is appropriate for the L2 (P )-valued Zi , i = 1, 2, and leads to a theory generalizing the corresponding convolution algebra of the classical harmonic analysis or the one valued even in an operator algebra.] (In connection with this problem, the reader may consult Graham and Schreiber (1984), and the analysis in Rao (1989). Using the ideas in both these sources, a study of convolution algebras of random measures, valued in L2 (P ), can be considered. This matter will be discussed further in Chapters 5 and 9.)

Bibliographical Notes Although random measures analyzed here, having values in an L2 (P )-space, are a subclass of vector measures (in Banach spaces, so that there exist controlling positive measures), the Hilbert space geometry enables a finer analysis of their structure. Typically the ‘randomness’ of measures is endowed in the range space

Bibliographical Notes

59

based on a probability triple (Ω, Σ, P ) which presents possibilities of new ideas that are not present in the abstract theory of vector measures. Analysis included in this and several other places that follow signifies this distinction vividly. The key concept associated with second order (vector) measures is an introduction of the concept of bimeasures and the resulting functionals. Thus random measure on locally compact groups through their expected values allows an interplay with the classical Fourier analysis and certain properties via generalized functions. A brief look into the Schwartz distribution theory is exhibited as a natural growth in Section 2.3, which connects bimeasure theory and bilinear functionals. There are close connections and applications here with vector analysis. Starting from the Vitali variation and its extension to vector valued bimeasures is a natural avenue. This has been explored in some detail by Dinculeanu and Muthiah (2000). It generalizes certain problems related to Lo`eve’s extension of (second order) stationarity to what are (currently) called strongly harmonizable classes. Another aspect of this method, with the classical Bochner k-transform of Fourier analysis can be used in going beyond Brownian (or Wiener-) measures, as was detailed by Kawata (1965). On the other hand, keeping the Fr´echet variation which is weaker than that of Vitali’s, Ylinen (1978) has presented some aspects of the (vector) bimeasure theory that is directly applicable to what are (nowadays) called weakly harmonizable families which are also second order classes. Our study in later chapters will consider both these points of view in analyzing several other aspects of random measures. It may be worth observing the distinction between the MT and the strict MT concepts which is somewhat analogous to that of Gel’fand-Dunford type and of Pettis integration in Banach spaces while Lebesgue and Bochner integrals have more similarities. An interplay of generalized functions also appears in Section 2.2 and in a sharper form in Section 2.3. Our treatment in part of Section 2.2 is influenced by Thornett’s (1979) analysis of shift invariant random measures on an LCA group as it shows clearly the connection between the classical Fourier analysis of unbounded measures since they appear in some integral representations of bimeasures with the strict MT-integration and their dominating (positive) measures. The underlying group structure is crucial here. Its analog with the work by Argabright and Gil de Lamadrid (1974) has interesting prospects for further analysis. Most of the textual treatment follows the recent work of the author’s (Rao 2006) on the subject. This generalization actually opens up new areas for further research. In particular the convolution of second order random measures to get a second order one of the same type (cf., Exercise 5.5) appears to have a good prospect of extending some analysis in Graham and Schreiber (1984) and others in this context. (See also Chapter 5.) Another interesting property of second order random measures included in Theorems 3.2 and 3.3 is worth pointing out. Namely, if Z : B0 (R) → L2 (P ) is a shift invariant random measure, then it admits a disintegration relative to a deterministic measure kernel g : (A, u) 7→ g(A, u) = χ ˆA (u), and an orthogonally valued random measure Z˜ of a very special form (cf. eq. (5) there). A similar

60

2 Second Order Random Measures and Representations

disintegration of a nonorthogonally valued second order random measure can be studied with a deterministic measure kernel and a nonorthogonally valued random measure having a less sharp property (cf., eq. (15) there). These particular type of results seem special for random measures and are worthy of future research. The cumulant stationarity is another area started by Leonov and Shiryaev (1959) and its extension by Brillinger (1972) for higher order (moment) stationarity. Here the analysis of random measures whose expectations connect the theory of multi or polymeasures of the different works of Voropolus (1967) and Dobrakov (1980–1999) will be of special interest. This will be discussed in the last chapter, but the present analysis in Section 2.4 is a prelude for that. The correlation study on quotient groups which is a specialized application of the classical theory presented by Weil-Mackey-Bruhat also is of great interest for the future random measure analysis. This is discussed here, and its important applications to point processes, treated in some detail with many examples, by Daley and Vere-Jones can form a motivation for such a study. We have also included a few complements as exercises with comments and hints for readers desiring to pursue further on these topics. An additional point of interest is an analysis of Fr´echet type algebras of convolutions indicated in Exercise 5.5 above on an LCA group. (See Hille-Phillips (1957), p. 77, and also Chapter 9.) The theory of this chapter forms a basis for much of the work in the following chapters. The analysis in the second order case has a special and concrete utility for both an independent study and a powerful motivation for moving forward. Further nonstationary extensions of this analysis for more general classes of random fields will be considered later in Chapter 4. Many of the other references related to the work presented here have been interspersed in the text and are therefore not repeated. With this preamble we proceed to the next chapter, concentrating on various random measures admitting controlling scalar measures in order to develop integration of random integrators with Bochner’s boundedness principle for various applications as well as theory. It should be emphasized that our treatment here and in the rest of the book is generally influenced by the vector measure theory of functional analysis (as contained in the title of the volume itself), and specific applications are incidental although not avoided. It is possible to develop a kind of parallel account where ‘random measures’ are regarded as measure kernels motivated by and devoted to specific applications such as point processes. However this possibility is not considered to any significant extent, and certainly not synonymous with ‘point processes’ as noted in the introduction of this chapter.

3 Random Measures Admitting Controls

It was shown in Chapter 1 that σ-additive random measures with values in Banach spaces in the weak or strong (= norm) topology always have controlling positive (and finite) measures on the underlying σ-rings (cf. Theorem 1.3.4). This result is valid for all vector (or Banach-valued) measures that need not be related to ‘randomness’ in any sense. On the other hand random measures Z : B → Lp (P ), p ≥ 0, do not have such a general result, especially if 0 ≤ p < 1. In fact if the basic (Ω, Σ, P ) is the Lebesgue unit interval then the adjoint space (Lp (P ))∗ = {0} and hence random measure analysis in Lp (P ), 0 < p < 1, must be considered from a different view that does not depend on the richness of the adjoint space. Here all the special properties of the underlying probability space must be utilized and the new concept of stochastic independence plays a key role along with their (probability) distributions and many of their properties must be brought into the analysis. Thus one may consider the collection {Z(A), A ∈ B0 (Rn )}(⊂ Lp (P )) as a stochastic process indexed by sets that bring in new ideas without relying only on the geometry of the range space. It will be shown for classes of measures Z : B0 (A) → Lp (P ), controlling measures µ : B0 (R) → R+ exist and integration of functions f : R → K (scalars) can be developed. This is necessary for integral representations of processes relative to such measures Z. In this chapter we consider mostly random measures valued in Lp (P ), 0 < p < ∞, and derive some of their properties. These are studied along the lines of Chapter 2 which was devoted largely to the case p = 2. Processes and their integral representations will be used for important applications later on. Special mention should be made about the translation invariance condition for distributions themselves, called ‘strict stationarity’. Even when a process has two moments, strict stationarity shows some deeper relations for the distribution family than those obtainable when a fixed (finite) number of moments are assumed to have invariance properties. It should be compared with the corresponding work of Chapter 2.

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3 Random Measures Admitting Controls

3.1 Structural Analysis As already noted in the preceding work, a random measure Z : B0 (Rn ) → Lp (P ), p ≥ 0, is vector valued and σ-additive (in its metric) for disjoint countable collections of sets whose union belongs to B0 (Rn ), the δ-ring of bounded Borel sets. Theorem 1.3.4 implies for p ≥ 1, the existence of a (finite) measure µ : B0 (Rn ) → R+ which dominates Z. In the case that p = 2, the geometry of the (range) space allowed us to consider the important class of measures Z(·) having orthogonal values on disjoint sets. We now analyze some families of those (vector valued) Z(·) that use the underlying probability structure and derive some properties implied by the condition that p 6= 2. Following the work in Chapter 2, we first consider the independence (in lieu of orthogonality) conditions of the random measure Z(·). Let us start with the case of a measurable space (S, S) and Z : S → Lp (P ), a random measure, where S is a δ-ring of subsets of the space S, Z(·) having independent values on disjoint sets. Then A, B ∈ S, A ⊂ B implies that Z(A) and Z(B − A) are independent random variables, since A, B − A ∈ S are disjoint. If p ≥ 1, then there exists a measure µ : S → R+ that dominates Z(·) as in Chapter 2. This independence property implies a sharper and more concrete form of µ. As a motivation, consider S = R, and S = B0 (R), the δ-ring of its bounded Borel sets. Suppose Z : S → L0 (P ) is also nonatomic so that S n n Z({a}) = 0 a.e. for every a ∈ R. Then for each n, A ∈ S and A = ki=1 Ai , disjoint union, Ani ∈ S, one has Z(A) =

kn X

Z(Ani ),

(1)

i=1

which is a sum of independent random variables for each A. By definition this property implies that Z(A) is infinitely divisible [in the generalized sense to begin with, and by an interesting theorem due to J.L. Doob, this is equivalent to (the ordinary) infinite divisibility]. Classically the infinite divisibility property assumes that all the Z(Ani ), i = 1, . . . , kn , in (1) should be identically distributed, and the Doob version states that the ‘generalized form’ extends or collapses essentially to the standard case, for each A ∈ B0 (R). Now more generally let us replace the nonatomicity condition by simply assuming that Z(A) is infinitely divisible which includes the Poisson case which is discrete but infinitely divisible. Such a random variable Z(A) has been the subject of investigations of mathematicians such as Kolmogorov, Khintchine, L´evy and many others. In fact they have shown that the characteristic function ϕZ(A) : t 7→ E(eitZ(A) ),

A ∈ B0 (R),

(2)

has a nicely determined form, although Z(A) need not have any moments and the Fourier transform with its uniqueness property is crucial. The final form, due to L´evy and Khintchine, is known to be given by the formula:

3.1 Structural Analysis

Z eixt − 1 − ϕZ(A) (t) = exp iγA t + R

itx 1 + x2

1 + x2 dG (x) , A x2

63

(3)

where for each A ∈ B0 (R), γA ∈ R and GA (·) is a bounded, left-continuous nondecreasing function with GA (−∞) = 0, and that (γA , GA (·)) uniquely determine (and determined by) ϕZ(A) (·). Now using the fact that Z(·) is additive with independent values on disjoint sets, it is deduced from the uniqueness of representation (3) that γ(·) : B0 (R) → R is a signed measure and G(·) (+∞) is a finite measure. Thus if µ(·) = |γ|(·) + G(·) (+∞) : B0 (R) → R+ , then µ is a σ-additive function that dominates Z(·) in the sense that µ(An ) → 0 as n → ∞ implies that Z(An ) → 0 in probability (even almost everywhere). Thus Z(·) is controlled by µ(·). Since no moments of Z(·) need to exist, the controlling measure µ is obtained for a subclass of vector valued set functions Z(·), namely those that are infinitely divisible, and Z(·) is σ-additive in the (Fr´echet) metric of Lp (P ), p ≥ 0. We now consider subclasses of the infinitely divisible Z(·) for a more detailed analysis where such random measures play a key role. Using the fact that the range space of Z(·) has an underlying probability structure, it is possible to obtain several important properties of random measures. It was already seen in Chapter 2 that, when p = 2, the Hilbert space geometry allows a detailed structural analysis of these measures and integrals of scalar functions are defined relative to them. A similar specialization for a large subclass of infinitely divisible random measures, termed the L´evy stable class will be of particular interest here where Z : B0 (R) → Lp (P ), 0 < p < 2, plays a key role. This will be needed for a proper understanding of the ensuing discussion and its application. We include a motivational example of considerable interest. Recall that a random variable X is termed stable if for any pair of independent identically distributed random variables X1 , X2 whose distribution is that of X, and each linear combination (a1 X1 + b1 ) + (a2 X2 + b2 ), with ai > 0, bi ∈ R, i = 1, 2, has the same distribution as aX + b for some a > 0, and b ∈ R (both a, b naturally depending on ai , bi ). This unmotivated concept is an abstraction of a property of the Gaussian or normal random variable. Indeed if X1 , X2 are independent and normal with means µ1 , µ2 variances X i − µi , i = 1, 2 are independent standard normal random σ12 , σ22 so that σi variables, N (0, 1), then Z = X1 + X2 has mean µ = µ1 + µ2 and variance Z −µ σ 2 = σ12 + σ22 and is again N (0, 1). Surprisingly G. P´ olya has shown in σ the 1920’s that this property of the independent variables (linear combinations) µi 1 > 0, bi = ∈ R) will have the same a1 X1 + b1 , a2 X2 + b2 (with ai = σi σi distribution as X, then there exist a > 0, b ∈ R, such that aX + b has the same distribution if and only if X1 , X2 are Gaussian distributed so that X is a “2-stable” random variable. This motivational example can explain the above concept of stability which turns out to be (a subclass of) infinitely divisible variables. Thus Z : B0 (R) → Lp (P ) is a stable measure if Z(A) is, for each

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3 Random Measures Admitting Controls

A ∈ B0 (R), a p-stable random variable and is nontrivial only when 0 < p ≤ 2 with p = 2 corresponding to the Gaussian case. The ensuing specialization of (3) is more demanding and is usually given in graduate probability text books. The final formula of (3) becomes the following for all A ∈ B0 (R) : (4) ϕZ(A) (t) = exp iγA t − cA |t|p 1 − iβA sgn t · h(t, p)

where γA ∈ R, |βA | ≤ 1, cA ≥ 0, 0 < p ≤ 2, and  pπ tan , if p 6= 1, 0 < p ≤ 2 2 h(t, p) = 2 − log |t|, if p = 1. π

The fact that Z(·) is σ-additive with independent values implies that γ(·) and c(·) are σ-additive and β(·) has the following property. For each An ∈ B0 (R), disjoint, ∪n An ∈ B0 (R), one has ∞ X

c(An ) β(An ) = c(∪n An ) β(∪n An ) .

(5)

n−1

(cf. Problem 1.) Here |β(An ) | ≤ 1, but β(·) is not necessarily σ-additive although (c ◦ β)(A) = cA βA , A ∈ B0 (R) is. Consequently (c ◦ β)(A) = 0 if cA = 0 which implies that µ defined as : µ(A) = |γ|(A) + c(A) , A ∈ B0 (R) is a σ-finite measure on the δ-ring B0 (R) and which dominates Z(·) in the sense that µ(A) = 0 implies P [Z(A) = 0] = 1, because of the uniqueness of the L´evy-Khintchine representation (4), i.e. (γ(·) , c(·) , β(·) ) uniquely determine Z(·) and conversely. It is clear that β(·) plays no role in obtaining the controlling measure µ, although it is an important parameter in the existence assertion of a p-stable random measure Z(·) and in the study of its distributional properties. When Z : B0 (R) → Lp (P ) is a p-stable measure, it is called symmetric if Z(A) and −Z(A) have the same distribution. This means ϕZ(A) (t) = E(eitZ(A) ) = E(e−itZ(A) ) = ϕ−Z(A) (t) = ϕZ(A) (t). Hence in (4), by the uniqueness, γ(·) = 0 = β(·), and cA ≥ 0 so that ϕZ(A) (t) = exp{−cA |t|p },

t ∈ R, 0 < p ≤ 2, A ∈ B0 (R).

(6)

In this case the controlling measure is µ(·) = c(·) . This discussion is summarized with some additions in the following form for reference. Theorem 3.1.1 Let Z : B0 (Rn ) → Lp (P ), p > 0 be a σ-additive function when R 1 the range space is endowed with the natural metric k · kp : f 7→ ( |f |p dP )1∧ p . Then there exists a controlling measure µ : B0 (Rn ) → R+ for Z(·) which is

3.1 Structural Analysis

65

finite for p ≥ 1, but for 0 < p < 1 and Z is p-stable with independent values on disjoint sets, µ exists (controlling Z(·)) but may be σ-finite. In both cases the stochastic integral for f ∈ Lp (µ) is defined Z T : f 7→ f (u) dZ(u), (7) Rn

and in case that Z(·) has independent values, the mapping T : Lp (Rn , µ) → Lp (P ) has the property that for f, g ∈ Lp (Rn , µ), f · g = 0, T f and T g are independent random variables. Moreover, νf : A → T (f χA ), A ∈ B0 (Rn ) is again a random measure in the sense that νf is σ-additive in norm for p ≥ 1, and defines a p-stable random measure in case 0 < p < 1. Proof. The first part on µ is just a summary of the preceding discussion and R the second half is established as follows. Observe that the integral Rn f dZ is well-defined because of the availability of a controlling measure µ for Z and the method given in Dunford-Schwartz (1958) is applicable. When ν is σ-finite on a σ-algebra S, it can be replaced by an equivalent controlling finite measure µ. [In fact, let (S, S, ν) be σ-finite and S = ∪∞ n=1 Sn , ν(Sn ) < ∞ so that if we define µ ˜ : S → R+ as µ ˜(A) =

∞ X 1 ν(Sn ∩ A) , n 1 + ν(S ) 2 n n=1

A∈S

then µ ˜ : S → R+ is the desired measure, and this procedure applies to B0 (Rn ) since Rn is σ-compact and ν is a Borel measure so that ν(A) < ∞ for A ∈ B0 (Rn ) for which A¯ is compact.] With this modification, the proof of Theorem IV.10.9 of Dunford-Schwartz (1958) applies without change and shows that T : Lp (µ) → Lp (P ) is well-defined and (7) holds. [The computations using the Vitali-Hahn-Saks theorem are nontrivial and the reader should go through these details to appreciate that property.] If f, g ∈ Lp (µ), f · g = 0, then let A = supp(f ), B = supp(g), the supports, so that A ∩ B = ∅, and f + g = f χA + gχB . The linearity of the integral implies Z Z g dZ. (8) f dZ + T (f + g) = T (f χA) + T (gχB ) = A

B

Taking f, g as simple functions, it follows that the independent values of Z(·) imply that the integrals in (8) are mutually independent, and the dominated convergence for these integrals is available which then implies that T f and T g are independently distributed for f, g with disjoint supports from Lp (µ). This R also shows that νf : A 7→ A f dZ ∈ Lp (µ) is a random measure as asserted. 2

Remark 3.1 The above result remains true if Rn is replaced by a locally compact space S with its δ-ring of bounded Borel sets and also admits an extension if (S, S S, ν) is a measure space with S0 = {A ∈ S : ν(A) < ∞}, a δ-ring, and ∞ S = n=1 An , An ∈ S0 . Then Z : S0 → Lp (P ), 0 < p ≤ 2, can be p-stable

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3 Random Measures Admitting Controls

or if p ≥ 1, and Lp (P ) is considered as a Banach space, with Z σ-additive. In both cases a controlling measure exists and the integral (7) is well-defined and νf : S0 → Lp (P ) is σ-additive without having independent values. Further properties will be outlined in exercises later. Remark 3.2 If Z(·) has independent values but not necessarily is of stable class, with values in Lp (P ), p ≥ 1, then also (7) is well-defined and T : Lp (µ) → Lp (P ) will also have independent values on functions of disjoint supports. This has important consequences, as generalized fields, and will be considered separately later. Thus the ‘stochastic integral’ (7) has special properties which will be analyzed in some detail. It is useful to highlight the distinctions between the stable and the much larger class of infinitely divisible random variables to have a better idea of these collections under consideration. If {Z(A1 ), . . . , Z(An )} is a set of pstable Pn (or symmetric stable) random variables, then every linear combination i=1 ai Z(Ai ), ai ∈ R (or C) is also a p-stable variable. On the other hand there exist random elements Z(A P1 ), . . . , Z(An ), not necessarily p-stable (p > 0), but every linear combination ni=1 ai Z(Ai ) is a p-stable random variable. If the Z(Ai ) are symmetric, then this difficulty disappears. An example will be outlined in Exercise 2 below. On the other hand if X = (X1 , . . . , Xn ) is a random vector of infinitely divisible elements it is an infinitely divisible vector provided its characteristic function is given as Pn

) = E(eiu·X ) Z iu · x 1 iu·x ′ ν(dx)} e −1− = exp{−γ · u − u · Γ u + 2 1+x·x Rn (9)

ϕX (u1 , . . . , un ) = E(ei

i=1

uj Xi

where γ = (γ1 , . . . , γn ) ∈ Rn , u = (u1 , . . . , un ), Γ = (Γik , 1 ≤ j, k ≤ n); γ, u are vectors and Γ is a symmetric n × n positive (semi)-definite matrix, with prime (′ ) denoting transposition, and where ν(·) is a measure on the Borel sets of Rn satisfying Z Z x · x dν(x) + dν(x) < ∞. (10) [x·x≤1]

[x·x≥1]

This representationP is due also to P. L´evy. From it one can verify that each linear combination nj=1 ai Xj is also infinitely divisible. However the converse is not necessarily true. This fact was illustrated by L´evy himself in the 1940’s by showing that for the “Wishart matrix” of a jointly multivariate normal vector that, using n = 2, each linear combination was shown to be infinitely divisible but the full matrix itself was not. If each component Xi is infinitely divisible and all are independent then this anomaly does not arise. But since each stable random vector is infinitely divisible, it follows that the infinitely divisible class is substantially larger than the p-stable class, 0 < p ≤ 2. Also an infinitely

3.1 Structural Analysis

67

divisible variable can have all moments finite, as in the case of normal, Poisson and gamma variables, but in the p-stable case if 0 < p < 2, the pth order moment does not exist. [p = 2 gives the Gaussian case, and this is the only stable variable with all moments finite.] Let us note a few other facts for symmetric p-stable collections, motivating further analysis. Namely, X = (X1 , .P . . , Xn ) is a p-stable symmetric vector if and only if every linear combination ni=1 ai Xi has Pnthat property. However, if the Xi , 1 ≤ i ≤ n are individually p-stable and i=1 ai Xi is p-stable for all ai ∈ R, 0 < p ≤ 2, does not ensure that the vector X is p-stable. The positive conclusion obtains if X is infinitely divisible in addition. This was shown by D.J. Marcus (1983). Thus we first restrict to the symmetric case and the general result being somewhat technical, will be postponed. Now the characteristic function ϕX of a symmetric p-stable X is given by : ϕX (t) = exp{−c|t|p },

t ∈ R, 0 < p ≤ 2,

and if X = (X1 , . . . , Xn ) is a vector of symmetric stable, not necessarily independent, variables then its characteristic function ϕX is representable as: ϕX (t1 , . . . , tn ) = exp

n

−

Z

Sn

n p X o tj λj dG(λ1 , . . . , λn )

(11)

j=1

where G is a unique (finite) measure on the unit sphere S n of Rn . This result is also due to P. L´evy. The following important consequence will be proved here as it illuminates the relationship between the symmetric p-stable random measures and the parameters in the L´evy representation. Theorem 3.1.2 Let Z : B0 (R) → Lp (P ), 0 < p ≤ 2 be a symmetric pstable random measure and for Ai ∈ B0 (R), i = 1, . . . , n, consider the set Z(A1 ), . . . , Z(An ) of its values so that ϕZ(Aj ) (t) = exp{−c(Aj ) |t|p }, where c : B0 (R) → R+ is the controlling measure of Z(·). [Thus the mapping 1 k · kp : Z(A) 7→ kZ(A)kp = [c(A)]1∧ p defines a Fr´echet metric in Lp (P ) equivalent to convergence in probability.] Moreover, if Z(·) is independently valued then k · kp is additive on Z(Ai ), Z(Aj ) when Ai ∩ Aj = ∅ and it has the monotonicity property kZ(A)kp ≤ kZ(B)kp where A ⊂ B. If in fact 1 < p ≤ 2, then for any A, B in B0 (R) one has the conditional expectation, often also called a linear regression equation, given as : E(Z(A)|Z(B)) = aZ(B),

(12)

for an a ∈ R determined only by A, B and the measure c(·), so that the regression of Z(A) on Z(B) is always linear.

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3 Random Measures Admitting Controls

Proof. The statement on the metric is an extension of Proposition 1.3.2, and the general case is an adaptation of an argument from Schilder (1970). Thus let A1 , A2 ∈ B0 (R) and since Z(A1 ) + Z(A2 ) ∈ Lp (P ), p > 1, (11) gives Z p |t1 λ1 + t2 λ2 | dG(λ1 , λ2 ) ϕ(Z(A1 ), Z(A2 )) (t1 , t2 ) = exp − S2

where S 2 is the unit sphere of R2 (i.e. λ21 + λ22 = 1), and the L´evy measure G is supported by it. Setting t1 = t2 = 1 and using the definition of k · kp , we get kZ(A1 ) + Z(A2 )kp = ≤

Z

S2

Z

S2

1∧ p1 |λ1 + λ2 | dG(λ1 , λ2 ) p

1∧ p1 Z + |λ1 | dG(λ1 , λ2 ) p

S2

1∧ p1 |λ2 | dG(λ1 , λ2 ) p

using the triangle inequality since G is a Stieltjes

measure (cf. e.g. Rao (2004) p.207), = kZ(A1 )kp + kZ(A2 )kp . Recall that for 1 ≤ p ≤ 2, k · kp is a norm functional while for 0 < p < 1 it is simply a (Fr´echet) metric, and is not homogeneous. In any case Lp (Rn , B0 (Rn ), c(·)) is a Fr´echet space, i.e., it is a complete [translation invariant] metric space and is a Banach space whenever p ≥ 1. Next suppose that Z(·) is independently valued on disjoint sets. Then ϕ(Z(A1 ), Z(A2 )) (t1 , t2 ) = ϕZ(A1 ) (t1 )ϕZ(A2 ) (t2 ) for all t1 , t2 ∈ R. Taking t1 = t2 = 1, one gets ϕ(Z(A1 ),

Z(A2 )) (1, 1)

1

1

= exp{−[c1 (A1 )1∧ p + c2 (A2 )1∧ p ]}.

(13)

Hence kZ(A1 ) + Z(A2 )kp = ln ϕ(Z(A1 ), Z(A2 )) (1, 1) = ln(ϕZ(A1 ) (1)ϕZ(A2 ) (1)) = kZ(A1 )kp + kZ(A2 )kp using (13) and the definition of k·kp . It is clear that kZ(An )kp → 0 is equivalent to c(An ) → 0 so that ϕZ(An ) (t) → 0, t ∈ R. But this is the same as Z(An ) → 0, in probability as n → ∞, and An ↓ ∅. For the last part, let 0 < p ≤ 2, Z(·) independently valued and A ⊂ B. Then by additivity and (13), Z(A) and Z(B − A) are independent, so that kZ(B)kp = kZ(A) + Z(B − A)kp = kZ(A)kp + kZ(B − A)kp ≥ kZ(A)kp . Now if 1 < p ≤ 2, so that E(Z(A)) ∈ R for all A ∈ B0 (R), then the conditional expectation E(Z(A)|Z(B)) is defined. We next use the identity E(E(X|Y )) = E(X) for any integrable (or positive) random variable X and any random variable Y , with X = Z(A1 ) and Y = Z(A2 ) for A1 , A2 ∈ B0 (R). The following computation holds:

3.1 Structural Analysis

69

E(Z(A1 )|Z(A2 )) = E(Z(A1 − A1 ∩ A2 ) + Z(A1 ∩ A2 )|Z(A2 )), by additivity of Z(·), = E(Z(A1 − A1 ∩ A2 )|Z(A2 )) + E(Z(A1 ∩ A2 )|Z(A2 )), by the linearity of conditional expectations, = E(Z(A1 − A1 ∩ A2 )) + E(Z(A1 ∩ A2 )|Z(A2 ), by independence of Z(A1 − A1 ∩ A2 ) and Z(A2 ),

= E(Z(A1 )) − E(Z(A1 ∩ A2 )) + E(Z(A1 ∩ A2 )|Z(A2 )), by additivity of Z(·) = g(A1 , A2 ), (say).

(14)

We now establish that the random variable g(A1 , A2 ) is a constant multiple of Z(A2 ) to complete the argument. Since by the symmetry hypothesis E(Z(A)) = 0 for all A ∈ B0 (R), we see that g(A1 , A2 ) = E(Z(A1 ∩ A2 )|Z(A2 )), and if Z(A2 ) = 0 with probability one then it is a constant independent of all Z(B), B ∈ B0 (R). Hence Z(A1 ∩ A2 ) will be independent of Z(A2 ) so that g(A1 , A2 ) = E(Z(A1 ∩ A2 )) = 0 whence (12) holds for any a ∈ R. After this consider P [Z(A2 ) 6= 0] > 0, and set B = A1 ∩ A2 ⊂ A1 . Then by the Doob-Dynkin lemma, since Z(A1 ) is integrable, E(Z(A1 )|Z(A2 )) = hA1 (Z(A2 )), (cf. e.g. Rao (1981), p.4) and we simplify hA1 (Z(A2 )) to show that it is a constant multiple of Z(A2 ), which depends on a simple but interesting trick. The σ-additivity of Z(·) implies that there is a measure c(·) ≥ 0 dominating Z(·),√and c(A2 ) > 0 since P (Z(A2 ) 6= 0) > 0, and Z ≪ c. Now consider, with i = −1 and 1 < p ≤ 2, the following calculation setting B = A1 ∩ A2 ⊂ Ai : E(Z(B)eitZ(A2 ) ) = E(Z(B)eitZ(B∪(A2 −B) )) = E(Z(B)eitZ(B)+itZ(A2 −B) ) = E(Z(B)eitZ(B) ) · E(eitZ(A2 −B) ), since A2 − B and B are = =

= = =

disjoint and Z(·) has independent values on these sets, 1 d itZ(B) E (e ) · E eitZ(A2 −B) i dt 1 d E(eitZ(B) ) · E(eitZ(A2 −B) ), since under the current i dt d and E(·) commute, hypothesis dt p 1 d −c(B)|t|p · e−c(A2 −B)|t| , by (5), e i dt p p 1 [(−c(B))e−c(B)|t| p|t|p−1 ][e−c(A2 −B)|t| i c(B) − [c(A2 )e−(c(B)+c(A2 −B))|t| p|t|p−1 ], since c(A2 ) > 0, ic(A2 )

70

3 Random Measures Admitting Controls

d −c(A2 )|t|p c(B) , since c(·) e ic(A2 ) dt c(B) d = · E(eitZ(A2 ) ) ic(A2 ) dt c(B) · E eitZ(A2 ) · Z(A2 ) = c(A2 ) c(A1 ∩ A2 ) E(eitZ(A2 ) · Z(A2 )). = c(A2 ) =

is additive,

(15)

Thus from (14) and (15), on using the conditioning identity E(X) = E(E(X|Y )), noted earlier, one has E(E(Z(B)eitZ(A2 ) |Z(A2 ))) = E(eitZ(A2 ) E(Z(B)|Z(A2 ))) = E eitZ(A2 ) hB (Z(A2 )) ,

by the Doob-Dynkin lemma,

(16)

and the left side, by the basic identity, is E(eitZ(A2 ) · Z(B)). Subtracting this, (16) becomes on using (15): E(eitZ(A2 ) [hB (Z(A2 )) −

c(B) · Z(A2 )]) = 0, t ∈ R. c(A2 )

(17)

Since [ ] is just a function of Z(A2 ), (17) shows that its Fourier transform vanishes for all t. Written differently this is just Z eitX dQX , t ∈ R, 0 = E(eitX (E(Y |X) − h(X)) = Ω

where QX is the signed measure on the σ-algebra generated by E(Y |X)− h(X), and determined by the variable X, so that QX has finite variation. Then the uniqueness theorem for Fourier transforms implies that QX = 0 identically and hence, in the original notation, we must have in (17) that hB (Z(A2 )) = c(B) e(A1 ∩A2 ) c(A2 ) Z(A2 ), with probability one. Taking a = e(A2 ) , since B = A1 ∩ A2 , this establishes (12), and the result follows. 2 Remark 3.3 It is interesting to observe that both the basic identity of conditioning, E(X) = E(E(X|Y )) for integrable X and arbitrary random variable Y , as well as the Doob-Dynkin lemma along with the symmetry of Z(·) have played significant parts in this theorem. Also if Z(·) is p-stable with p = 1, then it is Cauchy and its expectation does not exist so that much of the above computation does not hold. The case that E(|Z(A)|p ) < ∞, p > 0 is used in d (e−|t| ) = e−|t| sgn t, but the other computathe form in (15), for p = 1 as dt tions depend on the existence of E(|Z(A)|) < ∞ which effectively eliminates the Cauchy case.

3.1 Structural Analysis

71

A related problem of the last part is about the multiple regression. It is as follows. Let A, B1 , . . . , Bn be sets form B0 (Rn ) and observe that the σalgebra generated by Z(B1 ), . . . , Z(Bn ) is also given as σ(Z(Bi ), 1 ≤ i ≤ n) = σ(σ(Z(B1 )), σ(Z(B2 )), Z(B3 ), · · · , Z(Bn )), 1 ≤ i ≤ n which does not depend on the order of occurrence of Z(B1 ), · · · , Z(Bn ). Hence for instance E(Z(A)|Z(B1 ), Z(B2 )) = E(E(Z(A)|Z(B1 ))|Z(B2 )) = a1 E(E(Z(B1 )|Z(B2 )) = a1 a2 Z(B2 ),

by (12).

Repeating the procedure gives (for 1 < p ≤ 2), E(Z(A)|Z(B1 ), . . . , Z(Bn )) = (a1 a2 . . . an−1 )Z(Bn ). Since the order is immaterial, this equals = a′1 a′2 . . . a′n−1 Z(Bk ), for k = 1, . . . , n. Hence adding, averaging and noting that E(·|Z(Bi ), 1 ≤ i ≤ n) is a linear operator, it follows that there exist constants, akn , determined by the sets B1 , . . . , Bn , such that with probability one, E(Z(A)|Z(B1 ), . . . , Z(Bn )) =

n X

akn Z(Bk ),

(18)

k=1

so that this conditional equation is always a linear (multiple) regression, and the coefficients akn ∈ R are determined by B1 , . . . , Bn which are not unique in general. This result can be summarized as follows: Corollary 3.1.3 If Z : B0 (Rk ) → Lp (P ), 1 < p ≤ 2, is a symmetric p-stable random measure and A, B1 , . . . , Bn ∈ B0 (Rk ), then the conditional mean of Z(A) given Z(Bi ), 1 ≤ i ≤ n, is a linear equation represented by (18). We now discuss integration of deterministic functions relative to random measures, having controlling measures, valued in Fr´echet spaces such as Lp (P ), p > 0, without relying on properties special to Banach spaces. This complements the work of Chapter 2 where the range was a Hilbert space and its geometry played key roles. As seen above, a large class of random measures with control is the infinitely divisible family which includes the stable collections, as well as different types of ‘semi-stable’ ones. To explain this better, let us recall the latter concept. Let {Xn , n ≥ 1} be a sequence of independent random variables with a Pn S n − Bn → X in distribution common distribution and Sn = k=1 Xk . If An for some numbers Bn ∈ R and An > 0, then X must be infinitely divisible. an+1 → 1 as Suppose now that for a subsequence of integers an such that an S a − B an ˜ in distribution. The problem of characterizing X ˜ was n → ∞, n →X Aan ˜ is raised by Khintchine in 1936 and in the following year L´evy has shown that X

72

3 Random Measures Admitting Controls

an+1 → r, 0 < r < 1, the an corresponding variable is called r semi-stable which is again infinitely divisible. These are all distinct forms of stable classes, and thus the random measures Z : B0 (R) → Lp (P ), 0 < p ≤ 2 can belong to these subclasses, and being infinitely divisible have controlling measures. There are some other classes of random variables that are more general than the semi-stable class which are still infinitely divisible. A direct definition, by P.L´evy, is that a random variable X is semi-stable if its characteristic function ϕX : t 7→ E(eit X) is nonvanishing and satisfies the functional equation ϕ(t) = [ϕ(βt)]r for some r > 0 and 0 < β < 1, t ∈ R. It is called generalized semi-stable, by B. Ramachandran and C.R. Rao (1968), if ϕX satisfies the more inclusive (hence weaker) condition : also infinitely divisible, called a semi-stable one, and if

s Y

j=1

[ϕX (βj t)]rj =

s+k Y

j=s+1

[ϕX (βj t)]rj ,

t ∈ R, rj > 0,

(19)

and 0 < |βj | < 1 for 1 ≤ j ≤ s + k for any integer s ≥ 1. It is then verified that, from the positive definiteness of ϕX , if ϕX (a) = 0 for some a > 0, then ϕX (t) 6= 0, 0 ≤ t ≤ sa, and at the same time ϕX (t) = 0 for 0 ≤ t ≤ a giving a contradiction to the fact that ϕX (0) = 1. Thus ϕX (·) never vanishes so that one can choose the principal branch and define log ϕX . This implies for (19) a functional equation from which after a detailed computation one deduces that ϕX (·) is an infinitely divisible characteristic function. [See B. Ramachandran and C.R. Rao (1968) for details of this observation.] Consequently the infinitely divisible class is large enough to have many distinct subclasses. In all those cases the random measure Z : B0 (R) → L0 (P ) for which ϕZ(A) (·) satisfies (19) or other conditions such as r-semi-stable, 0 < r ≤ 1, willR have a controlling measure µ so that one can define an integral T : f 7→ Rn f (s) dZ(s), f ∈ L1 (Rn , µ), to study many of its properties. The integral is called the Dunford integral which coincides with that of Brownian motion (or Wiener’s integral) for Z([a, b]) = Xb − Xa , 0 < a < b < 1 and {Xt , 0 ≤ t ≤ 1} is Brownian motion where Z(·) thus defined extends to a random measure with independent values. [For details, see the author’s book (Rao (1995, p. 458).] It must be noted that if Z(·) does not have independent values, then the corresponding characteristic function need not be infinitely divisible even if it never vanishes. An example to this effect can be constructed from a result due to W.F. Kibble (1941) for a bivariate gamma distribution (given as an Exercise 6 below). The case of (19) for s = 1 is called a generalized semi-stable characteristic function, and the corresponding random variable or measure will be called by the same name. The preceding discussion with some additional properties will be given as follows [here and below (S, S) is a measurable space]: Theorem 3.1.4 Let Z : S → Lp (P ), p ≥ 0, be a random measure, and that it is generalized semi-stable, so is independently valued. Then Z(A) is infinitely

3.1 Structural Analysis

73

¯+ divisible for each A ∈ R S, and hence0 has a controlling measure µ : S → R . Also the integral T : f 7→ S f dZ, f ∈ L (S, µ) is defined and has independent values in that for bounded fi ∈ L0 (S, µ), i = 1, 2, with f1 · f2 = 0 the random variables T f1 , T f2 are defined and independently distributed. Proof. Most of the assertions have been detailed above. It is noted that, in the very definition of generalized stability of B. Ramachandran and C.R. Rao, the nonvanishing of ϕZ(·) was assumed. But it is seen to be automatic when other conditions hold. In fact, (19) implies with r = n. ϕ(at)r = ϕ(a1 t)r1 . . . ϕ(ak t)rk ,

rj , rk > 0, 0 < |aj | < 1.

so that ϕ(t)n = ϕ(a′1 t)r1 . . . ϕ(a′k t)rk , and hence |ϕ(t)| = Replacing t by

t a′k

k Y

j=1

for some |a′j | > 0, rj

|ϕ(a′j t)| n .

here gives (for some a′′j ) n k−1 Y ϕ t = |ϕ(t)| |ϕ(a′′j t)|rj ≤ |ϕ(t)|. ′ ak j=1

(20)

Let S ⊂ R be the set of points t at which ϕ(t) = 0. Then (20) implies a′k S ⊂ S, and also

1 S ⊂ S. a′k

Since ϕ(·) is continuous so that S is closed, we get that a′k = 1 and (20) implies 1

|ϕ(t)| ≤ |ϕ(t)| n . Now letting n → ∞, we get |ϕ(t)| = 0 or 1, ϕ being continuous, and ϕ(0) = 1, one must have ϕ(t) = 1 for all t. Thus ϕ(·) does not vanish and S = ∅. Hence log ϕ is well-defined as a principal value on all of R. So one can proceed with the argument of B. Ramachandran and C.R. Rao (1968), to conclude that ϕ is infinitely divisible. Then by the L´evy-Khintchine formula, it can be concluded that ϕZ(A) (t) satisfies (3) which implies the existence of a controlling measure µ : S → R+ for Z(·). p space, p ≥ 0, then the mapping T : f 7→ R If L (µ) is the standard Lebesgue 0 f (s)dZ(s) is well-defined on L (µ) into L0 (P ), and is independently valued S 0 for disjointly supported fi ∈ L (µ), i = 1, 2, as before. The other statements now follow as in the earlier work. 2 The preceding result admits an extension. A random field X : S → Lp (P ), for p > 0, such that FX (x) = P [X < x] has an associated mapping

74

3 Random Measures Admitting Controls

(TX f )(t) =

Z

f (t, s) dFX (s),

S

t∈T ⊂R

(21)

where f : R × S → C, called a kernel, satisfies the following restrictions: (i) f (·, ·) is jointly measurable (B(R) × S) and bounded, (ii) {f (t, ·), t ∈ T } ⊂ Lp (FX )} is a fundamental set. R Then the mapping ϕX : t 7→ S f (t, s) dFX (s), is also defined and has the following properties: [|f (t, ·)| ≤ Mt ≤ M < ∞, t ∈ T ] Z ||ϕX || = sup |ϕX (t)| ≤ sup |f (t, s)| dFX (s), t t S Z ≤ sup Mt dFX (s) = sup Mt = M < ∞. t

S

t

We normalize the set {f (t, ·), t ∈ T } so that M = 1 and f (0, ·) = 1. Thus (a) ϕX (·) is complex valued, (b) ϕX (0) = 1, and (c) |ϕX (t)| ≤ M = 1, but ϕX need not be positive definite. If f (t, s) = eits , where s ∈ S ⊂ R, then ϕX (·) is positive definite, but this is not implied by (21) in general. Consider a class of processes Xj and functions ϕX which has the following important property: For each n, there exist constants cn > 0, γn ∈ R such that ϕX (t)n = ϕX (cn t)eiγn t ,

n = 1, 2, . . .

(22)

If ϕX is the characteristic function of X, then this is the stability condition and the general case (22) is motivated by it, but it is not necessarily a characteristic function. A random variable X for which (22) holds will be termed weakly stable. This class includes both the generalized as well as semi-stable families and is of interest in considering a class of random measures that have controlling measures so that stochastic integrals can again be defined for them. A simple example satisfying the above conditions, and not positive definite, is 4 ϕX (t) = e−t , t ∈ R. It is desirable to characterize this class which was introduced abstractly by Bochner (1975). There are several details to establish, and so we devote the next section for it and show that they have the infinite divisibility property, and establish an analog of the L´evy-Khintchine representation to deduce the existence of a controlling measure and proceed with their integrals, generalizing the preceding work substantially. All of this analysis is used since the Kakutani-Klee (cf. p. 445) result corresponding to the BartleDunford-Schwartz is yet to be refined for Lp (P ), 0 ≤ p < 1 as range of Z(·).

3.2 Controls for Weakly Stable Random Measures The general problem of weak stability as defined above will be detailed without considering results from Fourier transforms but using only the basic ideas. Such

3.2 Controls for Weakly Stable Random Measures

75

a procedure was successfully formulated, perhaps for the first time, by Trotter (1959) to obtain a direct demonstration of the classical Lindeberg-Feller central limit theorem. It was soon extended by others, including the author [(Rao (1961)] and will be employed here in adapting it for weak stability of random variables, possibly without moments, and what is more, to obtain a characterization of the analog of the L´evy-Khintchine formula which is different from the former, and which is not deducible from it by some modification. First let us motivate the concept itself. If X is a random variable with distribution function FX , and C(R) denotes the space of uniformly continuous scalar bounded functions on R, equipped with the uniform norm || · || : f 7→ supx∈R |f (x)|, define Z (TX f )(y) = E(f (X + y)) = f (x + y) dFX (x), (1) R

so that TX : C(R) → C(R) is a linear contractive operator, since ||TX f || = supy |(TX f )(y)| ≤ ||f ||, |(TX f )(y1 ) − (TX f )(y2 )| ≤ supx |f (x + y1 ) − f (x + y2 )|, and by the uniform continuity of f, (TX f ) ∈ C(R) and ||TX || ≤ 1. It is clear that if X1 , X2 are independent random variables, then (1) implies that (TX1 +X2 f ) = TX1 (TX2 f ), since TX2 f ∈ C(R) again so that TX1 +X2 = TX1 TX2 = TX2 TX1 . It also follows from (1) that if Xn → X in distribution, then, for f ∈ C(R), ||TXn f − TX f || → 0 and the statement also holds if C(R) is replaced by C ∞ (R) ⊂ C(R), the space of infinitely differentiable functions. Using the classical Helly-Bray theorem it is seen that E(f (Xn )) → E(f (X)), for all f ∈ C p (R) ⊂ C(R), as n → ∞ where C p (R) is the space of p-times continuously differentiable functions in C(R), p ≥ 0. The preceding analysis admits the following abstraction which uses a clasA sical result due to A.D. Alexandroff and is stated for a motivation. Let TX be the same operator as in (1) defined for each A ∈ B(R) so that Z A (TX f )(y) = f (x + y) dFX (x), y ∈ R, (2) A

R A A A and set TX = TX of (1), so that TX = TX T A (= TX T A ) for independent 1 +X2 1 X2 2 X1 random variables X1 , X2 . The following is a version of Alexandroff’s result (cf. Dunford-Schwartz (1958) IV.9.15 for a proof).

Proposition 3.2.1 If FXn and FX are distribution functions of Xn and X then FXn (x) → FX (x) as n → ∞ at all continuity points of FX if and only A A if limn→∞ TX f = TX f for all f ∈ C(R) and every continuity interval A of n F, (A ⊂ R) (so the end points of A are also continuity points). This is an operator version of the Helly-Bray theorem, and the Alexandroff version holds if FX , FXn or merely F, Fn are replaced by regular bounded Borel measures on a normal topological space S (instead of R). The interest now is that if Fn is the distribution of the normalized sums of independent random variables such that Fn → F in the above sense, then F is a (standard) normal

76

3 Random Measures Admitting Controls

distribution if and only if the sequence satisfies the Lindeberg-Feller condition. This result, usually proved with characteristic functions, can be given an operator limit version. Trotter (1959) called it an ‘elementary’ operator-theoretical version which translates the classical form, and the author (cf. Rao (1961)) gave an extension of it so that the limit F is merely infinitely divisible. The latter will be formulated here as it gives a further motivation for obtaining a condition, not depending on Fourier transforms and thereby possibly omitting the positive definiteness condition. Recall thatPa random variable X is infinitely divisible if for each integer n kn > 1, X = ki=1 Xni , where Xni are independent random variables having the same distribution, and further conditions need be imposed to conclude that X is also Gaussian. The classical volume of Gnedenko and Kolmogorov (1954) is devoted to many aspects of this subject. The operator version is given here to motivate our work leading to the desired weak stability. Proposition 3.2.2 Let Fn , n = 1, 2, . . . and F be distribution functions of Xn A A and X, and let TX , TX be the corresponding operators introduced in (2). In n order that Fn → F and that F be Gaussian, it is sufficient that the following holds: A (3) = TYA1 · · · TYAk f, A ⊂ R, lim TYAn f = TX n→∞

A exists for f in a fundamental set of C(R) and for A ⊂ R and TX as product satisfying k X (4) kTYAi k ≥ k − ε i=1

where A = {x : |x| < ε} for each ε > 0, and k(≥ 2) depends on ε > 0. If (4) is replaced by the weaker condition kTYAi k > 1 − ε,

1≤i≤k

(5)

and the remaining hypothesis holds, then F will just be infinitely divisible. The details will be omitted here (they are given in the above reference) and the reader can take it as an exercise. The point of this procedure is that we need not stipulate positive definiteness in this work. Let us therefore recast condition (2) in the following (not necessarily linear) form. Thus let gX : R → C be defined as Z f f (t, s) dFX (s), t ∈ T, f ∈ T, (6) gX (t) = R

where T = {f (t, ·), t ∈ T } ⊂ C0 (R), the space of continuous functions vanishing at infinity, and separating points of R, normalized so that f (0, s) = 1 and f ||f (t, ·)|| ≤ 1. The class gX (·) is subject to some additional conditions motivated by the theory of Fourier transforms but without demanding the positive definiteness condition.

3.2 Controls for Weakly Stable Random Measures

77

f Definition 3.2.3 Let {gX (·), f ∈ T} be a family of functions defined by (6) relative to an arbitrarily fixed class {f (t, ·), t ∈ T ′ }, and the distribution FX f f f of X with the following properties: (i) gX (0) = 1, (ii) |gX (t)| ≤ gX (0) = 1, f f n ibn t and especially (iii) (gX (t)) = gX (cn t).e , n = 1, 2, . . . , cn > 0, bn ∈ R. Such f a gX (·) will be called a weakly stable function determined for a normalized fundamental set T = {f (t, ·), t ∈ R} ⊂ C0 (R).[T = T ′ of (6) is possible.]

Note that Condition (iii) is motivated by the stability postulate of charac4 f teristic functions. As remarked before, the elementary example gX (t) = e−t satisfies this concept for a suitable f (t, s), but is not a characteristic function f since otherwise, the second moment should vanish implying gX (t) must be identically a constant, which it is not. A somewhat sophisticated example, satisfying the conditions is the following where for simplicity g f (·) is written as g(·) : gα,β (t) = exp{iαt − ctα (1 ∓ βt)},

t > 0, 0 < α ≤ 2, α 6= 1, c > 0,

(7)

and take 1 + βt if t < 0. This is a characteristic function if and only if |β| < | tan απ 2 |. This fact is not obvious, but is verified by Dharmadhikari and Sreehari (1976) using some complex (not Fourier) analysis methods. In general if α 6= 1, |β| > | tan απ 2 | then (7) is not a characteristic function. Our aim now is to find a large class of random measures Z : B0 (R) → Lp (P ), p > 0, that have controlling measures. The existence of this result will be established below without using the Fourier-analytic methods, namely for weakly stable classes of Definition 3.2.3. These complementary classes are useful since as remarked already, there is no simple result on control that corresponds to the Bartle-Dunford-Schwartz theorem, the latter applying only to Banach space valued σ-additive set functions. We start with an example, showing a need for new methods. Let p(y; α, γ) be defined by : Z p(y; α, γ) = e−ity ψαβ (t) dt, y ∈ R, (8) R

where ψα,β (t) = exp[−|t|α e±

iπγ 2

t ∈ R, β = tan

],

πγ , |γ| < 1. 2

If in (8) we choose the signs so that ψα,γ (−t) = ψα,γ (t), and consider Z ∞ p˜(y; α, γ) = 2Re( e−ity ψα,γ (t) dt),

(9)

0

the function may be seen to qualify for the weak stability. [See Feller (1960), pp. 548-549 on this example.] We later consider a restricted class gX (·), called minimally positive definite if only the following condition holds : 3 X 3 X i=1 j=1

gX (ti − tj )zi z¯j ≥ 0,

zi ∈ C

(10)

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3 Random Measures Admitting Controls

for just t1 = 0, t2 = u, t3 = 2u, 0 < u < ∞. [Recall that gX will be “fully positive definite” if (10) is true for any n (not just 3) points ti ∈ R and zi ∈ C. This minimally positive definiteness is much weaker than the usual concept.] The weakly stable class (not the name) was introduced originally by Bochner (1975) who moreover characterized all such functions. Since we are utilizing this class for considering weakly stable random measures, it is desirable to present a characterization of this class and then use it in representing many random fields as integrals relative to such measures. Theorem 3.2.4 (a) A continuous mapping ϕ : R+ → C, extended to R by the equation ϕ(−t) = ϕ(t), t ≥ 0, is weakly stable if and only if the following representation holds : ϕ(t) = exp{−ψ(t)},

t ≥ 0,

(11) 1

where ψ : R+ → C is continuous, ψ(0) = 0, or cn = n p , n ≥ 1 in Definition 3.2.3, and ψ(t) = (A + iB)tp + iCt, 0 < p < 1 or 1 < p < ∞,

(12)

while for p = 1 ψ(t) = (A + iB log t)t + iCt,

(13)

with A ≥ 0, B, C ∈ R and A + iB 6= 0. The representation is unique if ψ(0) = 0 is also stipulated. (b) Further when ϕ(·) is minimally positive definite so that (10) also holds for ϕ, then in the representation of ψ(·) conditions (11)–(13) are restricted to satisfy (i) 0 < p ≤ 2, (ii) |B| ≤ Mp A (14) 1

where the constant Mp = (1 − 22−p ) 2 (1 − 21−p )−1 , 0 < p < 1, 1 < p ≤ 2, and M1 = (log 2)−1 . The bounds in (11)–(14) cannot be replaced by any smaller numbers. Proof. The basic outline of the argument is somewhat similar to the classical stable case, but there are very significant differences. Hence we include essentially the detailed methodology. Here part (a) is the main result needed for our random measures to obtain controlling set functions. It is presented in numbered paragraphs for convenience. Then part (b) will be sketched in the problem section with essentially all details. Step 1. It is first asserted that, as in the classical case, a weakly stable ϕ never vanishes. This is seen also in the generalized semi-stable result. In fact since ϕ(−t) = ϕ(t), 0 < t < ∞, let S = {t0 : ϕ(t0 ) = 0} ⊂ R+ . Then from the equation |ϕ(t)|n = |ϕ(cn t)| for a cn > 0, and all t > 0, we have |ϕ( ctn )|n = |ϕ(t)| and hence, as before, cn S ⊂ S ⊂ c1n S ⊂ S. Since ϕ(·) is continuous, S is closed so that S ⊂ c1n S ⊂ S and cn = 1 implying |ϕ(t)|n = |ϕ(t)|, n ≥ 1 which shows the fact that (since ϕ(0) = 1) ϕ never vanishes, as discussed in the earlier

3.2 Controls for Weakly Stable Random Measures

79

result. Then ψ(t) = log ϕ(t) is well-defined and even continuous. Now take the principal value of the logarithm, so that ϕ = e−ψ establishing (11). We next establish (12) and (13). Step 2. Also ψ(0) = 0, since ϕ(0) = 1, and ϕ(t)n = ϕ(cn t)e−iγn t and |ϕ(t)| ≤ 1 so that Re ψ(t) ≥ 0 as well as nψ(t) = ψ(cn t) + iγn t,

0 < t < ∞.

(15)

To establish (12) and (13), from (15), replacing t by ctn and dividing by n one gets, on multiplying through by m > 0, m γn γn imγm cm t m −i ·t= ψ t −i t− t. ψ(t) = mψ n cn n cn cn cn ncn It then follows that for all rationals r(=

m n)

rψ(t) = ψ(c(r)t) + iγ(r)t,

one has c(r) > 0, γ(r) ∈ R.

(16)

Step 3. We assert that c(r) > 0 and γ(r) ∈ R are unique in (16). Indeed if c′ , c′′ and γ ′ , γ ′′ satisfy (16) then the right side of (16) implies ψ(c′ t) + iγ ′ (t) = ψ(c′′ t) + iγ ′′ t,

t > 0, c′ > 0, c′′ > 0.

(17)

In case ψ(t) = ict (the degenerate case), then we must have c′ = c′′ as well as γ ′ = γ ′′ since t > 0 is arbitrary. If ψ(t) 6= ict, then (17) gives on letting ′′ ′ ′′ , the equation q = cc′ , r = γ c−γ ′ ψ(q m t) − ψ(q m+1 t) = iδq m t,

m = 0, 1, 2, . . .

Adding for m = 0, 1, . . . , n, and using the telescopic cancellation on the left we get for q 6= 1, 1 − qn ψ(t) = ψ(q n t) = iδ t. (18) 1−q

If 0 < c′′ < c′ so that 0 < q < 1, and letting n → ∞, we get from (18) ψ(t) = iδt ct (say) and as noted above uniqueness follows in this case. If ψ(·) is not 1−q = i˜ degenerate, i.e., not of the above type, we assert that {c(rn ), n ≥ 1} is bounded for a bounded rn -sequence. For otherwise there is a sequence 0 < |rn | < A < ∞ and limn c(rn ) = ∞. In this case, (16) gives rn t = ψ(t) + icn t, cn = . (19) rn ψ c(rn ) c(rn ) Letting n → ∞, since rn is bounded, c(rtn ) → 0 and the left side tends to zero so that cn → c0 (say), and ψ(t) = −ic0 t the excluded degenerate case. Thus c and γ are unique. Also c : R+ → R+ is continuous. In fact if ζn ∈ R+ , ζn → ζ0 , then there are rationals rn such that |rn − ζn | < n1 and |c(rn ) − c(ζn )| < n1 , by the above work. This means c(·) is continuous. Also if rn → ζ ∈ R, then

80

3 Random Measures Admitting Controls

c(rn ) → c˜0 ≥ 0. Indeed, by the above work, the sequences {c(rn ), n ≥ 1} and {γ(rn ), n ≥ 1} are bounded. Let rn ↓ 0. Hence c(rn ) → c(0+) and γ(rn′ ) → γ(0+) for some subsequence {rn′ , n ≥ 1} of {rn , n ≥ 1}, so that (16) implies (since ψ(0) = 0), ψ(c(0+)t) + iγ(0+)t = 0, t ≥ 0.

Thus if α = c(0+) > 0 then ψ(αt) = −ic0 t so that ψ(·) is linear. In case rn → ∞, then 0 < c(rn ) → a0 < ∞, so that (16) gives ψ(t) =

γ(rn ) 1 ψ(c(rn )t) + i t rn rn

which implies (rn → ∞ now), ψ(t) = iαt again, the first term going to 0. Thus in all cases, either ψ(t) = iat for some a ∈ R or c(r) = β for all 0 ≤ r < ∞. In either case the degenerate form of ψ(·) results. If ψ(·) is not degenerate, then c(·) is one-to-one. Let d(·) = c−1 (·) so that with d(β) = r we get from (16) the following relation: d(β)ψ(t) = ψ(tβ) + iδ(β)t,

δ(β) = γ(d(β)),

0 ≤ β < ∞.

(20)

Step 4. We now characterize ψ(·) as a solution of (20). Setting t = 1 in (17) one has iδ(β) = d(β)ψ(1) − ψ(β), by eliminating δ(β) in (20) and using this equation we get d(β)ψ(t) = ψ(tβ) + t(d(β)ψ(1) − ψ(β)). Now interchanging t and β here results in the relation: d(t)ψ(β) = ψ(tβ) + β(d(t)ψ(1) − ψ(t)), from which, eliminating ψ(tβ), one gets (ψ(t) − ψ(1)t)(d(β) − β) = (ψ(β) − ψ(1)β)(d(t) − t).

(21)

Since this equation is valid for all 0 < β, t < ∞, we have either ψ(t) = ψ(1)t or else ψ(t0 ) 6= ψ(1)t0 for some t0 . Since Re ψ(t) ≥ 0, the complex number ψ(1) = A + iC gives A ≥ 0 (C ∈ R), so that ψ(t) = (A + iB log t)t + iC with B = 0, and hence (12) holds in this case. If ψ(t0 ) 6= ψ(1)t0 , then (21) gives for t = t0 , ψ(β) − ψ(1)β (d(t0 ) − t0 ) ψ(t0 ) − ψ(1)t0 β[ψ(t0 ) − ψ(1)t0 ] + [ψ(β) − ψ(1)β](d(t0 ) − t0 ) = ψ(t0 ) − ψ(1)t0 β[ψ(t0 ) − ψ(1)t0 − ψ(1)(d(t0 ) − t0 )] d(t0 ) − t0 + = ψ(β) ψ(k0 ) − ψ(1)t0 ψ(t0 ) − ψ(1)t0 d(t0 ) − t0 = ψ(β) · D + β[1 − ψ(1)D], where D = . (22) ψ(t0 ) − ψ(1)t0

d(β) = β +

3.2 Controls for Weakly Stable Random Measures

81

This equation holds for all 0 < β < ∞. If D = 0, then d(β) = β for all β, and gives for (20) that ψ(tβ) = βψ(t) − iδ(β)t,

for all t, β.

If t = 1 this implies ψ(β) = βψ(1) − iδ(β), 0 < β < ∞ so that (20) becomes ψ(tβ) = βψ(t) − iδ(β) = βψ(t) − βtψ(1) + tψ(β), Setting ρ(t) =

ψ(t) t

t, β > 0.

− ψ(1), this equation reduces to the functional equation ρ(tβ) = ρ(t) + ρ(β).

(23)

Its well-known solution is ρ(t) = (R + iβ) log t, t > 0, so that ψ(·) is given by ψ(t) = R + (iβ)t log t + (A + iC)t,

t > 0.

(24)

Since 0 ≤ Re ψ(t) = Rt log t + At can hold only if R = 0, we get in this case ψ(t) = (A + iB log t)t + iCt, which is (13). Step 5. If D 6= 0, then substituting (22) with (20) results in the relation: Dψ(β)ψ(t) + (1 − Dψ(1))βψ(t) = ψ(tβ) + iδ(β)t,

t, β > 0.

(25)

To eliminate iδ(β) we consider t = 1 so that Dψ(β)ψ(1) + (1 − Dψ(1))βψ(1) = ψ(β) + iδ(β) and using this relation for δ(β) and substituting it in (25), we get on rearrangement: Dψ(β)ψ(t) + (1 − Dψ(1))[βψ(t) + tψ(β) − ψ(1)tβ] = ψ(tβ). If we set σ(t) = equation :

(26)

Dψ(t) + (1 − Dψ(1)), then (26) reduces to the functional t σ(tβ) = σ(t)σ(β),

t, β > 0

which has the trivial solution σ(t) = 0 or the nontrivial one σ(t) = tz = ez log t , for some fixed z ∈ C and t > 0. Since D 6= 0 now, the trivial case leads to ψ(t) = (A + iC)t, considered already in (13). The general case thus gives the solution ψ(t) = (A + iB)tz+1 + (P + iC)t, A + iB 6= 0,

t > 0.

(27)

Replacing t by tβ here, one gets ψ(tβ) = (A + iB)β z+1 tz+1 + (P + iC)tβ, t > 0, β > 0.

(28)

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3 Random Measures Admitting Controls

Using (20) we have for real d(β) and δ(β) ψ(tβ) = d(β)(A + iB)tz+1 + d(β)(P + iC)t − iδ(β)t.

(29)

Equating the right sides of (28) and (29), we get z ∈ R so that (27) reduces to: ψ(t) = (A + iB)tp + (P + iC)t, A + iB 6= 0, for some real p, and d(β) = β p from (28) and (29). Since ψ(t) ↓ 0 as t ↓ 0, we must have 0 < p < ∞. If p = 1, then ψ(·) will be of the form (13) and if p 6= 1, then (12) obtains. From (28) and (29) with d(β) = β and varying t one can have both equations satisfied only if P = 0. Thus (12) holds as given. This completes the proof of the direct part of (a) which is the main result. The converse is a straightforward verification of weak stability and will be left to the reader. However, part (b) is discussed as a problem with most details, since that result is not used for the work to follow. 2 An important application of this result is for random measures Z : B0 (R) → Lp (P ), 0 < p < ∞ which are weakly stable. Suppose that Z(·) is also symmetric, whence Z(A) and −Z(A) have the same distribution for every A ∈ B0 (R). Then by the fact that Z(·) is independently valued on disjoint sets, one has for Ai ∈ B0 (R), i = 1, 2, with A1 ∩ A2 = ∅, (cf. Definition 3.2.3) ϕZ(A1 ∪A2 ) (t) = ϕZ(A1 ) (t)ϕZ(A2 ) (t).

(30)

Hence by Theorem 3.2.4(a) and the uniqueness of representation (11)–(13), ϕZ(A1 ∪A2 ) (t) = exp{−ψA1 ∪A2 (t)}

= exp{−(αA1 ∪A2 + iβA1 ∪A2 )tp + iCA1 ∪A2 t}, 0 < p < 1, 1 < p < ∞, and

= exp{−(αA1 ∪A2 + iβA1 ∪A2 log t)t + iCA1 ∪A2 t},

α + iβ 6= 0, α ≥ 0, β ∈ R. (31)

Moreover (also αA , βA , CA are later written as α(A) , β(A) , C(A) to be used) ϕZ(A1 ) (t)ϕZ(A2 ) (t) = exp{−[(αA1 + αA2 ) + i(βA1 + βA2 )]tp + i(CA1 + CA2 )tp ]} = exp{−[(αA1

p > 0, p 6= 1, and + iβA1 log t)t + (αA2 + iβA2 log t)t

+i(CA1 + CA2 )t]}, αAi + βAi 6= 0, t > 0.

(32)

From (30)–(32) we conclude that α(·) , β(·) and C(·) are additive on B0 (R). Moreover , if Z(An ) → 0 in probability, then ϕZ(An ) (t) → 1 so that ψAn (t) → 0. This implies that 0 ≤ α(An ) ց 0, and β(An ) → 0, C(An ) → 0, as An ↓ ∅. Thus if µ(A) = α(A) + |β|(A) + |C|(A) , A ∈ B0 (R), then µ is a finite measure on B0 (R) and Z(An ) → 0 in probability implies µ(An ) → 0 and conversely if µ(An ) → 0, then ϕZ(An ) (t) → 1 for each t ∈ R, implying Z(·) is σ-additive in probability and hence is a measure. Thus Z(·) is controlled by the real measure µ. This may be recorded as the following important observation:

3.3 Integral Representations of Stable Classes by Random Measures

83

Proposition 3.2.5 If Z : B0 (R) → Lp (P ), 0 < p < ∞, is a weakly stable, symmetric random measure, then there is a measure µ : B0 (R) → R+ , which controls Z(·). Reviewing the concept of weak stability in Definition 3.2.3, the symmetry of Z(·) is not really used, and may be dropped in the above proposition. We shall leave this as an exercise to the reader to verify it. The point of the above propoR sition is that one can define the integral R f (t) RdZ(t) for all f ∈ LRp (µ) ∩ C0 (R). Then an analog of Theorem 3.1.4 holds. Hence R f1 (t) dZ(t) and R f2 (t) dZ(t) are independent if f1 · f2 = 0, and Z(·) has independent values under the weakly stable condition. It must be observed that a weakly stable Z(·) need not be p-stable, so that the two conditions of (weak and p-) stability cover different sets of functions (and integrals) with possibly different controlling measures which however enable one to study the corresponding (perhaps different) sets of stochastic integrals. They are seen to agree on the common classes of functions. We discuss the converse problem of processes that have integral representations relative to such random measures in the next section.

3.3 Integral Representations of Stable Classes by Random Measures At the very beginning, it is understood that a random measure having some (positive) control measure defines a stochastic integral which is not pathwise. It is given in terms of their finite dimensional distributions. This means, that the uniqueness is understood for such integrals in somewhat weaker terms, since two random variables defined on different probability spaces can have identical distributions and hence will only have the same characteristic functions. The pathwise concept is included but the thus admitted class is far wider. This is especially true of all stable classes. From the point of view of applications, this weakening is generally adequate and the representation theory is accordingly equally useful. It may be recalled that defining processes on different probability spaces is also not a hindrance since by the classical adjunction procedure, the space can be enlarged and the process can be defined on the same sufficiently large probability triple. The really new weakening is to settle for the distributional equivalence. Thus we start with the p-stable class. Definition 3.3.1 Let X = {Xt , t ∈ T } be a (scalar) p-stable stochastic process on (Ω, Σ, P ). Then X is said to satisfy condition S (or a type of separability condition) if there is a countable set T ′ ⊂ T such that, for each t ∈ T, Xt can be approximated in probability by elements of L = sp{Xt , t ∈ T ′ } ⊂ Lp (P ) where L is the linear span (i.e, all finite real linear combinations over T ′ ) of P the given process. Thus for each t ∈ T and ε > 0, there is an element nε Xε = i=1 ai Xti , ti ∈ T ′ in L, such that P [|Xt − Xε | > ε] < ε, or simply Xt = p limn→∞ Xεn , as εn ↓ 0 (continuity in probability).

84

3 Random Measures Admitting Controls

Note that if T = N = {1, 2, . . .}, then T ′ = T (any ε ≥ 0) and Xt = Xn . In fact the following result contains the finite dimensional case (T = finite) first established by Schilder (1970), and was extended by Kuelbs (1973) for 0 < p ≤ 2, T = T ′ = N, and independently by Kanter (1972) for 1 < p ≤ 2. The work below exemplifies the role played by the random measures in integral representations of many processes or fields which are (semi-)stable or stationary. Here the condition S of Definition 1 is not well-motivated. However, it can and will be dropped later for p-stable classes, concentrating on the stability hypothesis for the analysis that follows. First we establish the representation of symmetric stable classes. It contains the key elements of this new and nontrivial theory in Fr´echet spaces, Lp (P ), p > 0. Here is the desired result: Theorem 3.3.2 Let {Xt , t ∈ T } be a symmetric p-stable process satisfying condition S. Then there exists a symmetric p-stable random measure Z : B0 ([0, 1]) → Lp (P ) with a controlling measure µ and a class f (t, ·) ∈ Lp (µ) such that if Z 1 Yt = f (t, λ) dZ(λ), t ∈ T, (1) 0

then one has P [Xt = Yt ] = 1 for all t ∈ T. The family {f (t, ·), t ∈ T } is not necessarily unique, but any other {g(t, ·), t ∈ T } determining a Yt′ by (1) under condition S will have the same class of finite dimensional distributions as that of the Yt (hence Xt ) process.

Proof. We present the long and detailed argument in steps for clarity. Step I Recall that, from Section 1 above, a p-stable symmetric random vector X = (X1 , . . . , Xn ) has the characteristic function which is a multivariate extension, due to Rva˘cena (1962), of the one dimensional case obtained by P. L´evy (1938), with 0 < p ≤ 2, as : Pn

ϕX1 ,...,Xn (u1 , . . . , un ) = E(ei i=1 uj Xj ) R = exp{− Sn |u1 λ1 + · · · + un λn |p dGn (λ1 , . . . λn )} (2)

where Gn (·)Pis a finite measure supported by the unit sphere S n = {x ∈ n Rn : kxk2 = j=1 |xi |2 = 1}, which when n = 1 reduces to the form: and where cp = 1 1∧ p

R

ϕXi (u1 ) = exp{−|u|p cp },

|λ|=1

cp ≥ 0,

(3)

|λ|p dG1 (λ) ≥ 0. Consider the functional k · kp : X 7→

. It was already seen in the proof of 3.1.2 above that k ·kp is a translation |cp | invariant (or Fr´echet) metric, and remember that it would be a norm only if p ≥ 1 (hence homogeneous). It was also noted there that if X1 , X2 are independent, then kX1 + X2 kp = kX1 kp + kX2 kp . To simplify (2) and proceed with analysis, we map (S n , dGn ) onto ([0, 1], µ), the Lebesgue unit interval by means of the relation T : λ 7→ (f1 , . . . , fn )(λ), 0 ≤

3.3 Integral Representations of Stable Classes by Random Measures

85

λ ≤ 1 in a one-to-one and measurable as well as measure-preserving manner from Lp (P ) → Lp (µ). The existence of such a T has been (nontrivially) established by various authors. The general theory with detail has been given in Royden (1988, Chapter 15). [The result is due to Kuratowski, since S n is separable but an uncountable set.] The particular case here, using the positive definiteness of ϕ, is in the work of Bretagnolle, Dacunha-Castelle, and Krivine (1966) for 1 ≤ p ≤ 2 and extended by Schriber (1972) for 0 < p < 1 both for real spaces. An additional argument for the complex (even quaternion) case was given by Hardin (1981). These works attest to the nontrivial nature of this mapping T. Thus for symmetric p-stable processes the relation (2), with T introduced above, becomes: Z |u1 λ1 + · · · + un λn |p dGn (λ1 , . . . , λn )} ϕX1 ,...,Xn (u1 , . . . , un ) = exp{− Sn 1

= exp{−

Z

0

= exp{−

Z

1

0

|u1 f1 (λ) + · · · + un fn (λ)|p d(Gn ◦ T −1 )(λ)} n p X uj fj (λ) dF (λ)},

(4)

j=1

where F = Gn ◦ T −1 is a measure on the Borel σ-algebra B([0, 1]). This key representation was obtained by Schilder (1970). Step II The motivation for the general case is as follows. If F itself is Lebesgue measure, then the derived extension is the classical Wiener construction for the Brownian motion (which is simply 2-stable), which we denote as {Yt , 0 ≤ t ≤ 1} on some probability space where E(Yt1 − Yt2 ) = 0, E(|Yt1 − Yt2 |2 ) = a|t1 − t2 | = aµ([t1 , t2 ]), a > 0, with µ as the Lebesgue measure and the Yt -process has independent increments. If µ is replaced by F as here, one can construct, using the classical Kolmogorov existence theorem, a p-stable process {Yt , 0 ≤ t ≤ 1} with independent increments (hence is a Markov process) i.e., for t0 < t1 < t2 < 1, then Yt2 − Yt1 and Yt1 − Yt0 are independent p-stable symmetric random variables since the finite dimensional characteristic functions, by (2), are seen to be a consistent 1∧ 1

family. From this one gets ||Yt2 ||p p = F (t2 ), where a ∧ b = min(a, b). Using the additivity property of this metric for independent stable random variables, we have for 0 < t1 < t2 < 1 (Yt1 and Yt2 − Yt1 being independent) 0 < p < 2 1 1∧ p

F (t2 ) = ||Yt2 ||p

1 1∧ p

= ||Yt2 − Yt1 ||p

1 1∧ p

+ ||Yt1 ||p

= F (t2 − t1 ) + F (t1 ),

(5)

which implies 1∧ p1

||Yt2 −Yt1 ||p

1 1∧ p

= F (t2 −t1 ) = F (t2 )−F (t1 ) = ||Yt2 ||p

1 1∧ p

−||Yt1 ||p

≤ F (1) < ∞, (6)

86

3 Random Measures Admitting Controls

so that if Z([t1 , t2 ]) = Yt2 − Yt1 , then Z(·) is additive on the semi-ring of halfopen sub-intervals of [0, 1] and (by classical methods) has a unique extension to the algebra generated by such a collection. Since F (·) is monotone increasing, it can be taken left-continuous, and then it is easily verified that Z(An ) → 0 in probability (which is a metric) as An ↓ ∅. It follows that Z(·) extends to a random measure on the Borel σ-algebra B([0, 1]) of [0, 1]. By construction Z(·) R is controlled by F. Hence W (A) = A f (t) dZ(t) is defined for all A ∈ B([0, 1]). Also if Y˜t = W ([0, t]), then Y˜t and Xt will have the same finite-dimensional distributions where f (·) is obtained by the transformation T. In fact let Wj = R1 Wj ([0, 1]) = 0 fj (t) dZ(t), j = 1, . . . , n, in which the fj are as in (4). Then Pn

ϕW1 ,...,Wn (t1 , . . . , tn ) = E(ei j=1 tj Wj ) n

1∧ p1 o n X

tj Wj = exp − = exp

= exp = exp = exp

n

n

n

n

j=1 n Z 1

X

−

−

− −

Z

j=1 1

0

Z

0

Z

0

0

p

1∧ p1 o

tj fj (v) dZ(v) p

n p X o tj fj (v) dF (v) j=1

o |t1 f1 (v) + · · · + tn fn (v)|p d(Gn ◦ T −1 )(v)

1

o |t1 v1 + · · · + tn vn |p d(Gn (v1 , . . . , vn ) ,

by (4), = ϕX1 ,...,Xn (t1 , . . . , tn ).

(7)

Thus {X1 , . . . , Xn } and {W1 , . . . , Wn } are identically distributed. This is the basic step in representing the X-sequence, shown next. Step III In {Xt , t ∈ T }, let T be countable. This extension is due to Kuelbs (1973). Note that the preceding finite dimensional case, with the consistency condition, may be extended using the projective limit theory of Kolmogorov and Bochner, for any T , and in this procedure there are several intermediate technical calculations to be verified. But the following alternative argument is direct. Thus using the work of the above two steps, with T : λ 7→ (f1 , . . . , fn )(λ) which maps [0, 1] into Rn , consider its natural extension replacing Rn by ℓ2 , the sequence Hilbert space. We show that the desired Z : B([0, 1]) → Lp (P ) with a controlling measure µ : B([0, 1]) → R+ , and a R1 suitable sequence fn ∈ Lp (µ) such that Wn = 0 fn (t) dZ(t) gives the desired representation where the Xn ’s and Wn ’s have the same finite-dimensional distributions (or equivalently characteristic functions) which give (1) in this case as well.

3.3 Integral Representations of Stable Classes by Random Measures

87

Since Xn ∈ Lp (P ), considerPthe mapping T : ω 7→ (a1 X1 , a2 X2 , . . .)(ω) for some ai > 0, and such that E( ni=1 |ai Xi |q ) < ∞, n ≥ 1, where 0 < q < p ≤ 2. This holds since E(|Xn |q ) < ∞ and also because ℓq ⊂ ℓp ⊂ ℓ2 the mapping T : Ω → ℓ2 is well-defined, measurable for (Σ, B) where B is the Borel (cylinder) σ-algebra of ℓ2 . Hence ν = P ◦ T −1 : B → R+ is a probability. Consider the (finite) projection πn : ℓ2 → Rn . Then ν ◦ πn−1 = P ◦ (πn ◦ T )−1 , is a measure which is a symmetric p-stable probability that governs the set (X1 , · · · , Xn ), and this holds for all finite projections of the same dimension. Consequently ν is a symmetric p-stable probability. Its characteristic function (or Fourier transform) denoted νˆ has the L´evy form supported by the unit sphere of ℓ2 . In fact (ˆ ν (x))k = lim[ν\ ◦ πn (x)]k , ν being the projective limit of (ν ◦ πn−1 ), n

1

= lim[ˆ ν (xn )]k = lim[ˆ νn (k p xn )], n

n

since νn is symmetric p-stable, xn = πn (x), 1

1 p

= lim[ν\ ◦ πn (k xn ) = νˆ(k p x)). n

(8)

Hence there is a unique (finite) Borel measure Γ supported by the unit sphere S = {x ∈ ℓ2 : kxk = 1} such that Z p |(x, t)| dΓ (t) , (9) νˆ(x) = exp − S

by the L´evy representation. Note also that S(⊂ ℓ2 ) is itself a complete separable (and uncountable) metric space, in the ℓ2 -metric, which supports Γ. This is a renowned theorem due to Kuratowski, as noted earlier, and so (S, Γ ) can be put in a one-to-one and onto correspondence, denoted by ξ, with a similar conveniently chosen metric space, which we take it as the closed unit interval [0, 1] with its Euclidean metric so that ξ : [0, 1] → S is a Borel equivalence, meaning that ξ and its inverse ξ −1 are Borel measurable between ([0, 1], B([0, 1]), µ) and (S, B(S), Γ ), the Borel spaces. [See e.g., Royden (1988), p.406 and p.409 for a proof.] Now each point of S(⊂ ℓ2 ) is an infinite vector ξ : λ 7→ (a1 , a2 , . . .)(λ) and the measure on B([0, 1]) is ν ◦ ξ −1 = P ◦ T −1 ◦ ξ −1 = P ◦ (ξ ◦ T )−1 . However, ξ˜ = ξ ◦ T being Borel, each ai (·) is a Borel function on [0, 1]. These facts can be used along with the symmetric p-stable property of ν, a p-stable random measure on [0, 1] into Lp (P ), to define an extension of the procedure given in Step II. Let the Yt process be as in that step, and set Z([t1 , t2 ]) = Yt2 − Yt1 for 0 < t1 < t2 < 1, so that Z(·) has controlling measure F˜ (·), whence this definition produces a random ˜ 1 , t2 ])1∧ p1 ), 0 < p ≤ 2, (cf. measure Z(·). Here we can take F˜ ([t1 , t2 ]) = Γ ((ξ[t (9)). Then n p o n Z X P i n tj Yj j=1 tj sj dΓ (s) , s = (s1 , s2 , . . .) E(e ) = exp − S

j=1

88

3 Random Measures Admitting Controls

= exp = exp

n

n

−

Z

1 0

Z

−

n p X o tj aj (λ) dF˜ (λ), F˜ = Γ ◦ ξ −1 j=1 n 1X

0 j=1

o

tj aj (λ) dZ(λ) p

(10)

R1 which implies that Wt = 0 at (λ) dZ(λ) and Xt will have the same finite dimensional distributions, given by (1). Since there are many spaces and the respective mappings ξ, giving Borel equivalences to (S, B(S)), the representation is unique only in the distributional sense. [The result of this extension was also proved, as a generalization of Step I, by Kanter (1972) with a different argument.] Step IV. Suppose T is general but the process satisfies condition S. Let tj ∈ T be an arbitrary sequence for condition S. Then by Step III, Xtj has a representation such that for a suitable ftj ∈ Lp ([0, 1], ν)(= Lp (ν)). Z 1 (11) ftj (s) dZ(s), Xtj = 0

where the equality stands for equivalence in distribution. Let u ∈ T −{t1 , t2 , . . .}. By condition S, there is some sequence {tn,u , n ≥ 1} such that for some {Yn,u , n ≥ 1} ⊂ L = sp{X ¯ t , t ∈ T }, one has ||Xt − Yn,t ||Lp (P ) → 0 or Yn,t → Xt in probability, which defines a metric. But Xt − Yn,t is p-stable by hypothesis, so that cn,t > 0, (12) ϕXt −Yn,t (u) = exp{−cn,t |u|p }, 1∧ 1

and ||Xt − Yn,t ||Lp (P ) = cn,tp in the earlier notation. Since this tends to zero by the definition of Yn,t , it follows that {Yn,t , n ≥ 1} is Cauchy in probability. R1 But Yn,t = 0 fn,t (s) dZ(s) for some fn,t and the random measure Z(·) constructed there. So fn,t ∈ Lp (ν), and ||fm,t − fn,t ||Lp (ν) → 0. Now by the comR1 pleteness of Lp (ν), fn,t → f˜t ∈ Lp (ν). It follows that Yt = 0 f˜t (s) dZ(s), t ∈ T. Finally we need to verify that the Xt and Yt -processes have the same (finite-dimensional) distributions. Thus if τ1 , . . . , τm ∈ T , consider on recalling Xt = p limn Yn,t (probability limit) by the above, we can calculate the characteristic function as: Pm i j=1 uj Xτj ) ϕXτ1 ,...,Xτm (u1 , . . . , un ) = E(e i

= lim E(e n

m P

j=1

uj Yn,τj

)

Pkn bn,tj Xtr }) = lim E(exp{i j=1 uj r=1 n o n R1 Pkn Pm bn,tj 0 ftr (s) dZ(s)k = lim exp − k j=1 uj r=1 n Pkn P = lim exp{−k m r=1 bn,tj ftr kLp (ν) } j=1 uj n

Pm

3.3 Integral Representations of Stable Classes by Random Measures

89

Pm

= exp{−k j=1 uj fτj kLp (ν) } o n P R1 m = E(exp i j=1 uj 0 fτj (s) dZ(s) ) Pm = E(exp{i j=1 uj Yτj }).

Hence the representation holds on T if condition S is present. The preceding result also implies the following useful consequence:

2

Corollary 3.3.3 Let {Xt , t ∈ T } be a symmetric p-stable process where T is a topological space with a dense denumerable set and Xt be P -continuous, or only limt→t0 P [|Xt − Xt0 | ≥ ε] = 0 for each t0 ∈ T and ε > 0. Then the process admits a representation of the form (1), i.e., Yt =

Z

0

1

f (t, s) dZ(s),

t ∈ T,

(13)

where {Yt , t ∈ T }, and {Xt , t ∈ T } are distributionally equivalent. Here ft ∈ Lp (µ), t ∈ T and µ can be taken as Lebesgue measure on [0, 1]. Proof. Let {tn , n ≥ 1} ⊂ T be a dense countable set. By the P -continuity of the Xt -process, for each t ∈ T , there is a subsequence {tj , j ≥ 1} ⊂ {tn , n ≥ 1} such that Xt = p limj Xtj . Hence by the theorem we can find a set {ft ∈ Lp ([0, 1], ν)} ⊂ Lp ([0, 1]) since ν is a finite measure absolutely continuous relR1 ative to Lebesgue measure. Then Y˜t = 0 ft (s)dZ(s) gives {Y˜t , t ∈ T } and {Xt , t ∈ T } to be distributionally equivalent versions of each other as asserted. 2 Since the representation of a process as an integral is not unique but any two can have isomorphic “kernels” (the ft ’s of (1)), it raises the question of selecting an f (·, ·) using some “efficient” procedure so that it may be regarded as unique in a “narrower sense”. If no such essential uniqueness is insisted on, does there exist a representation, such as (1), on some “suitable” measure space? Positive answers to both questions can be given. They will be sketched based on Hardin’s (1982) work. Another consequence of the above theorem is to specialize the symmetric p-stable process, whose representing random measure may not have independent values on disjoint sets, but may have an integral representation relative to one that has the independent value property, just as the case of Theorem 2.3.2 for general shift invariant L2 (P )-valued random measures. The last question can be answered quickly as follows. It moreover implies a “change of variables” property. [See Section 5.3 for further analysis with stable random measures.] Corollary 3.3.4 Let Z : B0 (S) → Lp (P ) be a jointly p-stable symmetric, [not necessarily independently valued on disjoint sets] σ-additive regular function, so for A ∈ B0 (S), there is a compact Cε ⊂ A with P [|Z(A − Cε )| > ε] < ε), where S is a separable topological space and B0 (S) is the δ-ring of bounded Borel sets. Then there exists an independently valued p-stable symmetric random

90

3 Random Measures Admitting Controls

measure Z˜ : B([0, 1]) → Lp (P ) with control µ : B([0, 1]) → R+ such that for some ‘measure’ kernel f : B(S) × [0, 1] → C, with f (A, ·) ∈ Lp (µ), one has Z 1 ˜ Z(A) = f (A, s)dZ(s), A ∈ B0 (S) (14) 0

where f (·, s) : B0 (S) → C is σ-additive in the metric of the range space C, and Z˜ is controlled by µ, 0 < p ≤ 2, so that the integral (14) is well-defined. Moreover, the following change of variables formula holds: Z

S

g(s)dZ(s) =

Z

0

1

Z

S

˜ g(t)f (dt, s)dZ(s),

g ∈ B(S).

(15)

Proof. Consider {Z(A), A ∈ B0 (S)} as a random process (or field) indexed by T = B0 (S) which is separable, by hypothesis on S. So representation (13) of CorollaryR3.3.3 is applicable and hence there is a kernel f (A, ·), A ∈ T , for which ˜ for a random measure Z˜ which takes independent Z(A) = S f (A, s)dZ(s), values on disjoint sets of B0 (S). We need to verify the σ-additivity of f (·, s) for s ∈ S. It may be recalled that {Z(A), Z(B)} is jointly p-stable since Z(A ∪ B) = Z(A) + Z(B) − Z(A ∩ B) by additivity of Z(·), and each of the right side terms is p-stable which follows from the additivity property. Generally the statement is false even if p = 2, because the sum of a pair of Gaussian random variables need not be Gaussian! Thus the additivity of the p-stable Z(·) is essential here. Now using Theorem 3.3.2 (and the definition of a metric following (3)), one has the basic isomorphism kZ(A)kLp (P ) = kf (A, ·)kLp(µ) where µ is the con˜ trolling measure of Z(·) and also of Z(·), and since Z(·) is σ-additive in Lp (P ), it follows that f (A, s) is also σ-additive in A for each s. Next observe Pn that (15) holds for g = χA , A ∈ B0 (S) and by (14) this extends if g = i=1 ai χAi is a simple function by the linearity of the integral. Using the isometry again one deduces (15) in general for f ∈ Lp (µ), and the remaining standard details can now be left to the reader. 2 In order to extend the isomorphism to get a type of uniqueness noted above, we introduce a condition on the kernel f (·, ·) appearing in (1) and (13) as follows (cf. Corollary 3.3.3). Definition 3.3.5 The mapping t 7→ f (t, ·) of Corollary 3.3.3 is termed minimal if the following two properties hold: (i) f (t, ·) = 0 for all t ∈ T only on a µ-null set, ([i.e., f (t, ·) 6= 0 a.e.[µ]]), (ii) if Σ1 = σ(f (t, ·)/f (t′ , ·) : t, t′ ∈ T ), the σ-algebra generated by the ratios f (t,·) f (t′ ,·) then f (t, ·) is a constant a.e., on the atoms of Σ1 ( relative to µ), and each atom belonging to an equivalence class of Borel sets. Using this unmotivated technical concept, the following type of uniqueness has been established, by Hardin (1982) which we state for completeness.

3.3 Integral Representations of Stable Classes by Random Measures

91

Theorem 3.3.6 Every symmetric p-stable process {Xt , t ∈ T } satisfying condition S, and in particular if the index set T is a separable topological space and the Xt -process is continuous in probability, has a representing kernel {f (t, ·), t ∈ T } which is unique up to an isometry on Lp ([0, 1], µ) → Lp ([0, 1], µ) with µ as Lebesgue measure. Thus if g(t, ·) is another such kernel then g(t, ·) = Af (t, ·) for a linear isometric operator A so that kg(t, ·)kp = kf (t, ·)kp , t ∈ T. Discussion The proof is based on constructing isometries on the measure spaces, arising from certain measure preserving transformations and is technical. The necessary details are spelled out in Hardin (1982), and as they do not give a simple constructive procedure, we do not include them here. The point is to note the existence of such a kernel for our work, and to contrast it for all the cases considered in Chapter 2. It should be noted that the preceding result is also valid for nonseparable symmetric p-stable processes except that the Lebesgue unit interval is replaced by a general mea˜ µ ˜ S, sure space (S, ˜) and the kernels f : T × S˜ → R have the same properties as before where f (t, ·) 6= 0 a.e.[˜ µ], and there is a minimal representation relative to the new σ-algebra Σ˜1 = σ(f (t1 , ·)/f (t′ , ·) : t1 , t′ ∈ T ) which is equivalent to σ(Xt , t ∈ T ). Moreover, there is a linear isometry as before for this new collection of functions f (t, ·), and it is unique in the same sense. The main difference is thus the Lebesgue interval is replaced by an abstract measure space, completely analogous to the Kuratowski theorem. The result extends to complex p-stable process, since by definition, the real and imaginary parts are jointly p-stable and symmetric. We refer the reader to Hardin’s paper for details. In non-Banach space cases, special techniques and analysis are needed for integral representations of such processes and measures relative to certain random measures with independent values on disjoint sets. The following result is a representative of certain Banach space extensions in which one has the special kernel f (t, λ) = eitλ , namely the exponential: Theorem 3.3.7 Let {Xn , n ∈ Z} be a symmetric p-stable process. Then for a (not necessarily independently valued) random measure Z : B(−π, π) → Lp (P ) such that the following integral representation Z π Xn = einλ dZ(λ), n ∈ Z, (16) −π

holds if and only if for ak ∈ R and all ℓ > 1,

ℓ ℓ

X

n X o

a X ≤ C sup ak e−2πitnk : −π < t ≤ π

k nk k=1

p

(17)

k=1

P and (16) holds for p = 1 also if the set { ℓk=1 ak Xnk : ak ∈ R, ℓ ≥ 1} ⊂ L ⊂ L1 (P ) is relatively weakly compact in the L1 (P )-context. Further the finite di-

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3 Random Measures Admitting Controls

mensional characteristic functions of the Xn -sequence will be uniquely given by ϕXn1 ,...,Xnk (u1 , . . . , uk ) = E(ei

Pk

j=1

= exp −

Z

ui Xnj π

−π

)

k p X ˜ uj ˆhnj (λ) dG(λ)

(18)

j=1

˜ ≥ 0 and with Xnj ↔ hnj as an isometfor a bounded nondecreasing function G ric correspondence. Remark 3.4 The key condition given above as (17), called V -boundedness (V for variation) was discovered by Bochner in 1934 for the number sequences and was later generalized by Phillips in 1950, further by Kluv´anek in 1967 for Banach space valued functions, in particular for the Lp (P )-valued ones [or random processes], for p ≥ 1. This is valid if Z is replaced by R and more generally by an LCA group G. In all these cases, Z(·) is a (regular) vector ˆ = (−π, π] in ˆ (if G = Z, we have Z measure defined on the dual group G the above). When the sequence {Xn , n ∈ Z} is a p-stable symmetric process, we need to show Z(·) is actually a symmetric p-stable measure which needs an additional effort. We present a general outline of proof leaving out some routine (but nontrivial) details. [As usual L1 (Z) will denote the Z-integrable scalar functions on G.] Sketch of Proof. First some facts from functional analysis have to be recalled for use in (16). This is given in the following form. Let f be a simple function from L1 (R), the standard Lebesgue space of integrable functions on the line R with the length measure, so f ∈ L1 (R) ∩ Lp (R), and its Fourier transform fˆ exists. In this case the condition (17) is equivalent to saying that (Xt being continuous in probability) nZ o f (t)Xt dt : kfˆk∞ , f ∈ L1 (R) ∩ Lp (R) ⊂ Lp (P ), (19) R

is relatively weakly compact. Here the (vector-valued) integral is understood in the classical Bochner sense. It is a known fact that for 1 < p < ∞, the space Lp (P ) is reflexive and in that case the condition (19) is equivalent to saying that the set on the left is contained in a ball of Lp (P ). But for p = 1, the compactness condition is stronger than the set being in a ball. Thus under this restriction, R the mapping T (fˆ) = R f (t)Xt dt is defined for all f ∈ L1 (R) and is bounded by condition (19), since fˆ ∈ C0 (R) and such functions separate points of R. A familiar argument in such analysis shows that T has an extension to C0 (R) with the same bound. The relative weak compactness further implies that the extended operator, using the same symbol T for it, admits an integral representation by a generalized Riesz representation (Dunford-Schwartz (1958) VI.7.3

3.3 Integral Representations of Stable Classes by Random Measures

93

and VI.7.6) relative to a vector measure, denoted Z : B0 (R) → L1 (P ) ∩ Lp (P ), so that Z Z T (fˆ) = fˆ(t)dZ(t) = f (t)X(t) dt, f ∈ L1 (R) ∩ Lp (R), (20) ˆ R

R

and kT k = kZk(R), the semi-variation of Z which is finite, and the integral is well-defined. If x∗ is a continuous linear functional on the Banach space Lp (P ), then (19) can be simplified as, again with a standard argument, R R R x∗ Rˆ fˆ dZ(t) = R fˆd(x∗ ◦ Z)(t) = R f (t)(x∗ ◦ Xt ) dt, x∗ ∈ (Lp (P ))∗ , (21) which gives since there are enough such x∗ to separate points by the Hahnˆ = R isometrically) Banach theorem, on using the definition of fˆ, (and R Z Z ∗ f (t)x Xt − eitλ dZ(λ) dt = 0, f ∈ L1 (R) ∩ Lp (R). R

R

This implies from the density of such f , that Xt satisfies (16) where we now ˆ by Z ˆ = (−π, π]. The converse implication uses the Fubini replace R by Z and R theorem for the signed measure (x∗ ◦ Z)(·), and the weak compactness condition (17) follows from the corresponding converse parts of the result in abstract analysis used above (using the same reference). The regularity of the measure (x∗ ◦ Z)(·) is part of the abstract result, and it translates to (and implies) the regularity of Z(·) in probability. So far we have only used the boundedness (being in a ball) and Bochner integrability of Xt ’s. But it is also p-stable, 1 < p ≤ 2 and with it the corresponding stability of Z : B0 (R) → Lp (P ) has to be established. Pk Since by definition i=1 ai Xni is p-stable for ai ∈ R, k ≥ 1, the same is Rπ R π Pk true of −π ( j=1 aj einj λ ) dZ(λ), and hence also of −π hk (λ) dZ(λ), so it is p-stable for all trigonometric polynomials hk (·). Since the dominated convergence theorem is also valid for our vector measures and integrals in a Banach Rπ space (Dunford-Schwartz (1958) IV.10.9) it follows that −π χA dZ = Z(A) is p-stable for any A ∈ B(−π, π). However, Z(·) may not have independent increments on disjoint sets. Now one may use Corollary 4 above and the ‘change of variables’ to obtain a random measure Z˜ to have the independent values property relative to a kernel function f. This means (16) takes the form Z π Z π Z π ˜ ˜ fˆn (s) dZ(s), (22) einλ fn (dλ, s) dZ(s) = Xn = −π

−π

−π

where fˆn (·) is the Fourier transform of the signed measures fn (·, s). Now using the Bretagnolle, Dacunha-Castelle, Krivine, Schreiber-Kanter isomorphism, one has

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3 Random Measures Admitting Controls

ϕXn1 ,...,Xnk (u1 , . . . , uk ) = E(ei = exp

Pk

n

j=1

−

uj Xnj

)

Z X k p o ˆ n (λ) dG(λ) ˜ uj h j S

(23)

j=1

˜ Since our set is ˜ where hnj ∈ Lp (S, G). on an abstract measured space (S, G), ˜ countable, here the abstract space (S, G) can be replaced by the interval with Lebesgue measure ([−π, π], dµ). This completes the sketch. 2 It seems that under some analogous conditions Hosoya (1982) could replace R π Pk the integral in (23) by −π | j=1 uj einj λ |p dG(λ). The new restrictions are on iω ˜ ˜ ˜ the measure Z(·), called “isotropy” so that Z(λ) and Z(λ)e are identically ˜ distributed for all −π < λ, ω ≤ π and that the distribution of Z(λ) is rotation invariant. Further the p-stable process X is called strictly harmonizable if t R Xt = Zˆ eitλ dZ(λ), t ∈ Z, for an independent valued, on disjoint sets, p-stable symmetric random measure Z(·). When the process is strictly harmonizable and strictly stationary, Hosoya shows that Z is also isotropic. We shall not include the analysis of such processes. This is discussed here to exhibit the versatility of the p-stable class, 0 < p < 2. Since in Chapter 2 we treated in some detail the second order stationary and harmonizable classes and their representations, it is appropriate here to indicate some general nonstationary second order processes and their integral representations. This is done in the next section.

3.4 Integral Representations of Some Second Order Processes In Chapter 2 we saw several properties of second order random measures and their (Dunford-Schwartz type) integrals. Considering a general problem, as in the preceding section, is to continue the ideas of that work without assuming p-stability. The latter would reduce the study to Gaussian processes when p = 2. It is desired to discuss several general classes of second order processes (or fields) that admit integral representations relative to random measures into L2 (P ) but need not have orthogonal (or independent) values. The second order case has numerous possibilities for expanding the study of random fields which can fill up separate volumes. In this section we initiate analysis that leads primarily to some operator representations of such processes (and fields) using the additional structures available. An understanding of this work leads to a sharper analysis of integral representations in which both bimeasures and generalized random fields together with their local behavior (such as higher order increments processes of some known classes including stationarity, harmonizability and the like) which will enrich and highlight the role that random measures play in the resulting analysis. A specialized study of this will be postponed to the next chapter.

3.4 Integral Representations of Some Second Order Processes

95

To motivate the present considerations let us recall that a mean continuous (complex) centered process {Xt , t ∈ T } ⊂ L20 (P ) is weakly station¯ t ) = (Xs , Xt ), ary, with T = R or Z, if its covariance r : (s, t) 7→ E(Xs X [with inner product notation that connects the expectation E] is invariant under translation. Thus r(s, t) = r(s + h, t + h) for any h ∈ T so that r(s, t) = r(s − t, 0) = r˜(s − t) where r˜ is a continuous positive definite function, the last condition coming from a property of the inner product of L2 (P ). Hence by the well-known Bochner theorem on such functions r˜, it is representable uniquely as: Z ei(s−t)λ dF (λ), s, t ∈ T, (1) r˜(s − t) = Tˆ

where Tˆ is the dual of the locally compact abelian group T , and F ≥ 0 is a bounded nondecreasing (can be taken left-continuous) function, called in the present context, the spectral distribution of the process. This concept was introduced by Khintchine in the early 1930’s independently of a slightly earlier work due to Bochner. As seen in the preceding chapter, there exist many processes other than the second order ones, the latter having many special properties of independent interest, and they will be briefly discussed. If F of (1) is absolutely continuous relative to the Lebesgue measure µ, on √ ˆ ∼ ˆ ∼ Tˆ(= R) = R or = Z = (−π, π]), F ′ (t) = f (t) ≥ 0, so that g = f ∈ L2 (Tˆ, µ), then (1) can be expressed as the following integral, on taking T = R and so Tˆ = R also (with µ ˆ being the ‘dual Haar measure’ on Tˆ of µ on T , so µ ˆ = cµ): Z ei(s−t)λ f (λ) dµ(λ) r˜(s − t) = Tˆ Z = (eist g(λ))(eitλ g(λ))− dµ(λ) ˆ ZT gˆ(s + λ)g¯ˆ(t + λ) dˆ µ(λ), by Plancherel’s theorem, (2) = T

2

where gˆ ∈ L (T, µ ˆ). Then (2) motivates generalizations replacing (T, µ ˆ) with an abstract measure space (S, S, ν) as follows. Consider a centered second order process induced by a general set T (no group structure) and assume that the covariance r(·, ·) of {Xt , t ∈ T } ⊂ L20 (P ) admits a “product representation” as (2) and generalizing it: ¯ t ) = (Xs , Xt ), as before in inner product notation, r(s, t) = E(Xs X Z = g(s, λ)¯ g (t, λ) dν(λ), s, t ∈ T,

(3)

S

∼ R and ˆ= for a collection {g(s, ·), s ∈ T } ⊂ L2 (S, ν). Thus if T = R, then S = R (2) obtains with g(s, λ) = gˆ(s + λ). In both (2) and (3) as well as other cases, it is desired to have integral representations of processes {Xt , t ∈ T } relative to some family of random measures

96

3 Random Measures Admitting Controls

Z : S → L2 (P ). To answer this, we start with certain other forms of (2) that are of equal interest in both the theory and applications. In fact (1) is a special case of Z Z ′ eist−itλ β(dλ, dλ′ ), (4) r(s, t) = R

R

where β(·, ·) is a positive definite bimeasure, so that (4) reduces to (1) if β concentrates on the diagonal of the plane R × R, and is bounded. In general, the integral of (4) is defined in the Lebesgue-Stieltjes sense on R × R only if β(·, ·) has (locally) finite Vitali variation as seen in Section 2 of Chapter 2 and more often β can only have (locally) finite Fr´echet variation as defined and discussed in detail in that section. Let us again recall these variations here. A bimeasure β : B0 (R) × B(R) → C has (locally) finite Vitali variation if on each set A ∈ B0 (R), |β|(A × A) < ∞ or A = R, where |β|(A × A) = sup

n X n nX i=1 j=1

|β(Ai , Bj )| : Ai , Bj ∈ B(A), disjoint

o

(5)

with B(A) ⊂ B0 (R) as the trace algebra {A ∩ C : C ∈ B(R)}, A ∈ B0 (R), and it has (locally) finite Fr´echet variation, kβk(A × A) < ∞, on each (bounded) set A ∈ B0 (R), where for positive definite β, ai ∈ C, by definition: kβk(A × A) = sup

n X n nX i=1 j=1

o ai a ¯j β(Ai , Aj ) : Ai ∈ B0 (A), disjoint, |ai | ≤ 1 . (6)

In general kβk(A × A) ≤ |β|(A × A) ≤ ∞ with usually a strict inequality between the first two terms. Since a bimeasure β determines a vector mea˜ = β(·, )(∈ ca(R, B(R))), and a vector measure is bounded, has finite sure β(·) semi-variation (cf. Proposition 1.2.4) and since B(A) is a σ-algebra for every A ∈ B0 (R), it follows that β(·, ·) always has locally finite Fr´echet variation on B0 (R) × B0 (R), whereas this is not true for the Vitali case. So to use the Lebesgue-Stieltjes integration one has to assume that β has the latter property or the more natural form, using the (locally) finite Fr´echet variation, the integration must necessarily be the weaker Morse-Transue concept (cf. Definition 2.2.3). We have obtained an integral representation in Theorem 2.2.8 of a “separable” second order process {Xt , t ∈ T } ⊂ L2 (P ) where the closed linear span of the process is assumed to be a separable subspace when T is a general index set. The conclusion is that such a process is of Karhunen class. To gain more information on its “kernel” and the associated random measure we now present different types and forms of second order processes admitting similar representations with more precise information. To separate classes of second order, let us introduce some classifications in the following manner, extending the concept of weak or (K-) stationarity:

3.4 Integral Representations of Some Second Order Processes

97

Definition 1. (a) Let (S, B0 ) be a measurable space where S is a locally compact space and B0 is the δ-ring of its bounded Borel sets. If {Xt , t ∈ T } ⊂ L20 (P ) ¯t) = is a (second order centered) process with covariance r : (s, t) 7→ E(Xs X (Xs , Xt ), then it is (locally) weakly of class (C), when Z Z g(s, λ)¯ g (t, λ′ )β(dλ, dλ′ ), s, t ∈ T, (7) r(s, t) = S

S

as a strict M T -integral relative to a (locally finite) positive definite bimeasure β : B0 × B0 → C (necessarily of finite Fr´echet variation) and a family {g(t, ·), t ∈ T } of measurable functions, g(t, ·) : S → C. If β has finite Vitali variation, then it is of class (C), and if β concentrates on the diagonal of S × S, then the resulting integral reduces to Lebesgue-Stieltjes type and the process is of Karhunen class. [Here class (C) stands for the Cram´er class introduces by Cram´er in 1951, extending the Karhunen concept.] (b) If S = Rn , g(s, λ) = ei(s·λ) in (7), then the weak Cram´er class becomes weakly harmonizable and the Cram´er class (C) becomes strongly harmonizable [and the Karhunen field reduces to the weakly stationary one]. Using this nomenclature we can present a general result as: Theorem 3.4.1 Let (S, B0 ) be a measurable space with S locally compact and β : B0 → C a positive definite bimeasure for (S, B0 ). If {Xt , t ∈ T } is a centered (second order) weakly class (C) process relative to a family {g(t, ·), t ∈ T } of locally β-integrable Borel functions, then there exists a random measure Z : B0 → L20 (P ) such that Z Xt = g(t, λ) dZ(λ), t ∈ T, (8) S

where (Z(A), Z(B)) = β(A, B), A, B ∈ B0 and the above symbol stands for the vector integral (in Dunford-Schwartz sense), well-defined since Z is a vector measure into Hilbert space (hence has a controlling measure). Conversely, if {Xt , t ∈ T } is given by the representation (8), then it is weakly of class (C) relative to (S, B0 ) and β. Proof. Since β : B0 × B0 → C satisfies the conditions of Theorem 2.3.5 first applied to the trace σ-algebra B0 (A), A ∈ B0 (S), and employing the standard ˜ Σ, ˜ P˜ ) and a stochasgluing procedure, we can find a probability space (Ω, 2 ˜ 1 ˜ ∼ 2 ˜ tic measure Z : B0 → L0 (P ) such that L (Z) = L (β), and an isomorphism j : Z˜ → β so that for each B0 -step function f one has Z Z ˜ f (u) dZ(u) = j f (u) dβ(u, ·) ∈ L20 (P˜ ). (9) S

S

But the definition of weak class (C) membership relative to g(t, ·) and β implies g(t, ·) ∈ L2 (β). Using an approximation of g(t, ·) by step functions f , (9) im˜ i.e., (9) holds if f is replaced by g(t, ·) there since plies that j(g(t, ·)) ∈ L1 (Z),

98

3 Random Measures Admitting Controls

a dominated convergence theorem holds for the strict M T -integrals. Therefore we have, using the inner product notation for the RKHS space Hβ of Theorem 2.3.5, the following simplification: Z Z ˜ ˜ g(t, u) dZ(a) g(s, u) dZdu, S

= =

Z

S

S Z Z S

g(s, u) dβ(u, ·),

Z

S

L20 (P˜ )

g(t, v) dβ(v, ·)

g(s, u)¯ g(t, v) dβ(u, v) = r(s, t).

Hβ

(10)

S

R ˜ ˜t = g(t, u)dZ(u), then (10) Using the reproducing property of Hβ , if X S gives ˜s , X ˜t) 2 ˜ , (Xs , Xt )L20 (P ) = r(s, t) = (X s, t ∈ T. (11) L P) 0

˜ t , t ∈ T } ⊂ L2 (P˜ ). Then by Let K1 = sp{Xt , t ∈ T } ⊂ L20 (P ) and K2 = sp{X 0 ˜ s → Xs is an isomorphism of K1 into K2 . If Z = τ ◦ Z˜ (11) the mapping τ : X then Z : B0 → L20 (P ) is a random measure, and τ preserves inner products so that it is continuous implying the σ-additivity of Z as well as an isometry: ˜ ˜ (Z(A), Z(B))L2 (P ) = (Z(A), Z(B)) 2 ˜ = β(A, B), 0

L 0 (P )

and this easily gives, using the approximation valid for the Dunford-Schwartz integration, the following relation: Z ˜ ˜ Xt = τ (Xt ) = τ g(t, u) dZ(u) Z S ˜ g(t, u) d(τ ◦ Z)(u), by E. Hille’s classical theorem, = S Z = g(t, u) dZ(u), t ∈ T. S

˜ Σ, ˜ P˜ ) need not be the same This establishes (8). Since (Ω, Σ, P ) and (Ω, space, the argument through τ is necessary. For the converse direction, if Xt is defined by (8), then using the properties of the Dunford-Schwartz integral, it is deduced immediately that r(s, t) = R R ¯t) = g(s, u)¯ g (t, v) dβ(u, v) and by Definition 1, {Xt , t ∈ T } is of E(Xs X S S weak class (C), as asserted. 2 The preceding result can be specialized to obtain the corresponding representations of both the harmonizable, K-stationary as well as the Karhunen process and fields. The proof is simply a specialization of the bimeasure β in each case in (7). A separate characterization of weak harmonizability as V -bounded process will be given later. For now we have Corollary 3.4.2 Let {Xt , t ∈ T } ⊂ L20 (P ) be a Karhunen process (or field) relative to a family {g(t, ·), t ∈ T } ⊂ L2 (S, S, µ) on some measure space (S, S, µ), [as in (3)]. Then it admits an (D − S) integral representation as:

3.4 Integral Representations of Some Second Order Processes

Xt =

Z

g(t, u) dZ(u),

S

t ∈ T,

99

(12)

where Z : S → L20 (P ) is orthogonally valued and satisfies (Z(A), Z(B)) = µ(A ∩ B), A, B ∈ S, and conversely (12) always defines {Xt , t ∈ T } ⊂ L20 (P ) to be a Karhunen process (or field). If T = Rn , and (S, S) is given as S = Tˆ = Rn (the adjoint or dual group) and the covariance is given by (4), then (12) becomes Z Pn ei(t·u) dZ(u), t ∈ Rn , (t · u = j=1 tj uj ), (13) Xt = Rn

with (Z(A), Z(B)) = β(A, B), for A, B ∈ B0 (Rn ). Conversely every process or field given by (13) as a Dunford-Schwartz integral, always defines a weakly (and strongly if β has finite Vitali variation) harmonizable field. Moreover, in (13), Z(·) has orthogonal values if and only if Xt is weakly stationary. It is perhaps of interest to restate with a separate demonstration of weakly harmonizable process, emphasizing the group structure of the index set and its close affinity with Fourier transform, called V -boundedness here. We have already used the method in Theorem 3.3.7, but it will be given here in a more general context when T is an LCA group: Theorem 3.4.3 Let {Xt , t ∈ G} ⊂ L20 (P ) be a process or a field where the index set G is a locally compact abelian (LCA) group. Then it is weakly harmonizable if and only if it is V -bounded and its covariance function r(·, ·) is continuous. Proof. Recall that a process {Xt , t ∈ G} ⊂ L20 (P ) is V -bounded if it is in a ball of the space (i.e., it is uniformly norm-bounded) and the following set with µ as a Haar measure of G R (14) C = G f (t)Xt dµ(t) : kfˆk∞ ≤ 1, f ∈ L1 (G, µ) ⊂ L20 (P )

is bounded where the integral is in the strong (or Bochner) sense. This is where we need the “weak continuity” of t 7→ Xt and µ to be the Haar measure of G. [For the general case that L20 (P ) is replaced by Lp (P ), p ≥ 1, one needs to stipulate the (relative) weak compactness of the set, and for reflexive spaces (1 < p < ∞), this is equivalent to boundedness or be in a ball.] Here R ˆ (dual group) defined fˆ(s) = G hs, tif (t) dµ(t), is the Fourier transform, s ∈ G in the usual manner with Lebesgue integration. For the direct part, suppose that the given Xt is (weakly) harmonizable so ˆ Thus that (13) holds with Rn replaced by G. Z t ∈ G, (15) Xt = h t, λi dZ(λ), ˆ G

ˆ is a “character” (like eit·λ ) and Z : B0 (G) ˆ → L20 (P ) is a where ht, ·i ∈ G random measure. By Proposition 1.2.4, Z(·) has finite semi-variation, and hence

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3 Random Measures Admitting Controls

Z

ˆ ≤ kZk(G) ˆ < ∞, kXt k2 = ht, λi dZ(λ) ≤ kht, ·ik∞ kZk(G) 2

ˆ G

and the process lies in a ball of L20 (P ). To see that (14) holds, let x∗ ∈ L20 (P ) and consider the Bochner integral for (f X) as Z Z ∗ x f (t)x∗ ◦ Xt dµ(t), x∗ ∈ (L20 (P ))∗ , f (t)Xt dµ(t) = ˆ G G Z Z f (t) ht, λi d(x∗ ◦ Z)(λ) dµ(t), since x∗ ◦ X is the = ˆ G

G

=

Z

ˆ

ZG

Fourier transform of the measure (x∗ ◦ Z) Z f (t)ht, λi dµ(t) d(x∗ ◦ Z)(λ), (Fubini’s thm.) G

fˆ(λ) d(x∗ ◦ Z)(λ) = ˆ G Z ∗ fˆ(λ) dZ(λ) . =x ˆ G

Since x∗ ∈ (L20 (P ))∗ is arbitrary, this easily implies Z Z fˆ(λ) dZ(λ) ∈ L20 (P ). f (t)Xt dµ(t) = ˆ G

G

It follows that

Z

f (t)Xt dµ(t) ≤ kfˆk∞ kZk, by Proposition 1.2.4 ,

G

(16)

2

≤ C0 kfˆk∞ ,

f ∈ L1 (µ).

(17)

This proves (14), since L20 (P ) is reflexive, and the direct part is established. For the converse, let the process (or field) be V -bounded so that (14) holds and

Z

f (t)Xt dµ(t) ≤ Ckfˆk∞ , f ∈ L1 (G) = L1 (G, µ). (18)

G

2

R ˆ by an For f ∈ L1 (G), λ 7→ fˆ(λ) = G ht, λif (t) dµ(t) shows that fˆ ∈ C0 (G), analog of the Riemann-Lebesgue lemma, and such functions (transforms) form ˆ forming an algebra and also separating points a uniformly dense set in C0 (G) so that the Stone-Weierstrass theorem applies. Thus τ : f 7→ fˆ is a one-toThis fact and (18) imply that the operator T : fˆ 7→ R Rone contractive mapping. 2 ˜ f (t)Xt dµ(t) ∈ L0 (P ) is well-defined, and T = T ◦ τ : f 7→ G f (t)Xt dµ(t) G ˆ µ) → L20 (P ) which is therefore (by reis a bounded linear mapping on L1 (G, 2 flexivity of L0 (P )) relatively weakly compact. Such an operator T˜ has a vector integral representation—an extended Riesz representation—given in Dunford and Schwartz (1958, VI, 7.3) if G were a compact set. It can be extended to the ˜ˆ ˆ identified ˆ say G, so that T˜ maps C(G) one point compactification at ‘∞’ of G,

3.4 Integral Representations of Some Second Order Processes

101

˜ ˆ : f (∞) = 0} which is an “abstract M-space”. By a classical with {f : C(G) (Kelley-Nachbin-Goodner) theorem, there is a projection Q of norm one on ˜ ˆ onto C0 (G), ˆ and T¯ = T˜ ◦ Q is a continuous norm preserving extension of C(G) T˜ to which the preceding generalized Riesz theorem applies and therefore one has Z ˜ˆ ˜ f (t) dZ(t), f ∈ C(G), T˜f = (19) ˆ G

¯ ˆ for a unique vector measure Z, ˜ and the integral is well˜ G) with kTˆk = kZk( ¯ ˜ ˆ ˜ ¯ defined. Hence T f = T f, f ∈ C0 (G) with kT k ≤ kT k ≤ kT˜k. So Z ˆ f (t) dZ(t), f ∈ C0 (G), (20) T˜f = ˆ G

˜ G ˜ ∩ A), A ∈ B0 (G). ˆ It follows from this fact that where Z(A) = Z( Z ˆ kgk∞ ≤ 1} = kZk(G). ˆ g(t) dZ(t)k : g ∈ C0 (G), kT˜k = sup{k ˆ G

Then T˜ and Z(·) correspond to each other uniquely. Since the restriction 1 (G, µ) = T , we get T˜ | L\ Z Z ˆ ˆ Tf = f (t)Xt dµ(t), f ∈ L1 (G, µ). f (t) dZ(t) = ˆ G

G

Hence one can repeat the same argument as in the direct part to conclude that (15) holds for all t ∈ G by (the weak) continuity of the process or field as asserted in the statement of the theorem. 2 Remark 3.5 While both Theorem 3.4.1, Corollary 3.4.2 and the above theorem give the integral representations of the process (cf.(8), (12) and (15)), in the earlier results the form of g(t, λ) is not explicit while in (15) it is the character ht, λi, that is completely specified. For instance if G = Rn , then ht, λi = eit·λ . On the other hand, the former representations are applicable for larger classes of processes. These distinctions are worth noting. In all cases the random measure Z(·) has finite semi-variation, and usually determines just a bimeasure. For the strongly harmonizable case and its extension, the Cram´er class (C), the bimeasure determines a measure on G × G, but this is not true in general. Looking at (1) and (2), the continuous covariance r(s, t) of a stationary (centered) process {Xt , t ∈ R}(⊂ L20 (P )) has the form r(s, t) = r˜(s − t), and the process is termed weakly stationary. From (1) and (4) we deduce that the bimeasure β satisfies β(A, B) = (Z(A), Z(B)) = F (A ∩ B) so that Z(·) is orthogonally valued. In this case the integral representation can be obtained in a different way, as noted by Kolmogorov (1941) which leads to a deep relation between the random measure Z(·) and a certain

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3 Random Measures Admitting Controls

operator class, and this connection, specifically for second order processes, is important for our study. It will be discussed here as it shows the special roles played by these random measures and their analyses in Hilbert space. Observing that the concept of (weak) stationarity is in terms of translation invariance of its covariance function we can associate a mapping Ut : Xs → Xs+t , s, t ∈ T where T is R or Z (more generally can be an LCA group). We then have the mapping Ut well-defined and extended to L20 (P ), so that kUt Xs k22 = (Ut Xs , Ut Xs ) = E(|Ut Xs |2 ) = E(|Xs+t |2 ) = r(s + t, s + t) = r(s, s), by stationarity, = E(|Xs |2 ) = kXs k22 .

(21)

Hence kUt Xs k2 = kXs k2 , and on the closed linear span L = sp{Xt , t ∈ G} ⊂ L20 (P ), it is seen to be a linear operator of unit norm. [We concentrate on G = Rn or Zn here for simplicity.] Moreover (Ut Xs , Xv ) = (Xs+t , Xv ) = r(s + t, v) = r˜(t + s − v) = r˜(s − (v − t)) = (Xs , Xv−t) = (Xs , U−t Xv ),

(22)

so that on L which is a Hilbert subspace of L20 (P ), one has that (Xs , Ut∗ Xv ) = (Xs , U−t Xv ), s, v ∈ G. Hence Ut∗ = U−t and thus {Ut , t ∈ G} is a unitary group of operators on L which extends to L20 (P ) with preservation of this property (simply set U0 = I on L⊥ = L20 (P ) ⊖ L). Hence {Xt = Ut X0 , t ∈ G} and the Ut family is a weakly continuous (since r˜(·) is continuous) unitary group of operators on the Hilbert space L20 (P ), for which the classical Stone integral (or spectral) representation can be invoked to obtain a vector integral representation as (cf. Edwards (1965), Sec.10.3): R Xt = Ut X0 = ( Gˆ ht, λi dEλ )X0 , t ∈ G, R R (23) = Gˆ ht, λi d(Eλ X0 ) = Gˆ ht, λi dZX0 (λ),

ˆ is the resolution of the identity and symbol here is again where {Eλ , λ ∈ G} a spectral integral so that, by the often recalled theorem due to E.Hille, X0 ˆ → L2 (P ) is then identified can be brought inside the integral and ZX0 : B0 (G) 0 as orthogonally valued random measure on the delta ring. This is a new form, in the stationary case, of (15). The argument does not cover the (weak or strong) harmonizable case. The above result can be summarized for reference as follows: Proposition 3.4.4 Let {Xt , t ∈ T } be a weakly stationary process (or field) with a continuous covariance r and T as an LCA group. Then Xt is representable as : Z (24) ht, λi dZX0 (λ), t ∈ G Xt = U t X0 = ˆ G

3.4 Integral Representations of Some Second Order Processes

103

where {Ut , t ∈ G} is a weakly continuous unitary group on L20 (P ), and ˆ → L2 (P ) is an orthogonally valued random measure and conversely. ZX0 : B(G) 0 (Here again the integral is in the D-S sense.) A similar operator representation for the harmonizable case is somewhat deeper. We have seen in Example 2 of Section 2.2 that for an orthogonally valued Z : B → L20 (P ) and also a bounded linear operator V : L20 (P ) → L20 (P ), it was seen that Z˜ = V ◦ Z : B → L20 (P ) to be a vector measure that does not have orthogonal values. It has exactly the same (or similar) type behavior as the representing measure in (15) above for a harmonizable process. Our purpose now is to obtain an extension of (24) by finding a suitable family of operators that replaces the unitary class {Ut , t ∈ G} and still useful here. To this end we establish the following “dilation” result of independent interest: Theorem 3.4.5 Let {Xt , t ∈ T } ⊂ L20 (P ) be a weakly harmonizable random field where T = G an LCA group. Then there exists a super Hilbert space, ˜ Σ, ˜ P˜ ) ⊃ L2 (Ω, Σ, P ) = L2 (P ), a weakly stationary realized as L20 (P˜ ) = L20 (Ω 0 0 2 ˜ field {Yt , t ∈ T } ⊂ L0 (P ) such that Xt = QYt , t ∈ T where Q : L20 (P˜ ) → L20 (P ) is the orthogonal projection from L20 (P˜ ) onto L20 (P ). In fact if {Xt , t ∈ ˜ Σ, ˜ P˜ ) can be chosen such that {Yt , t ∈ T } spans T } spans L20 (P ), then (Ω, 2 ˜ L0 (P ). Proof. One direction is simple. In fact if {Yt , t ∈ T } ⊂ L20 (P ) is weakly stationary, then Z ht, λiZ(dλ), t∈G Yt = ˆ G

and by Hille’s theorem invoked many times before, if Q is an orthogonal projection from L20 (P ) into itself, then Z < t, λ > (Q ◦ Z)(dλ), Xt = QYt = ˆ G

ˆ → L2 (P ) is a random (and a vector) measure which has and Z˜ = Q ◦ Z : B(G) 0 finite semi-variation, since this property is valid for all Banach space valued measures (cf., e.g., Dunford-Schwartz (1958), IV.10.4). So the Xt -field defined above is weakly harmonizable by Theorem 3.4.1 (cf.(15)). The converse direction lies deeper as it depends on an explicit construction of a larger Hilbert space L20 (P˜ ). We now sketch this (tedious but useful) construction. Thus let X : G → L20 (P ) be weakly harmonizable. So by the last theorem, Z ht, λi dZ(λ), t ∈ T = G. (25) Xt = ˆ G

That dominating measure µ (which typically is not unique) can be chosen ˆ B0 (G)), ˆ ˆ → C, the following on (G, to satisfy, for each bounded Borel f : G inequality:

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3 Random Measures Admitting Controls

k

R

ˆ G

f (x) dZ(x)k22 ≤

R

ˆ f ∈ C0 (G)

|f (t)|2 dµ(t),

ˆ G

(26)

This is a nontrivial result, and the method uses some deep results from the works of Grothendieck and Pietsch as extended by Lindenstrauss and Pelezy´ nski, the details of which as needed are, in the form (26), in the author’s Measure Theory book (Rao (2004), Theorem 7.4.7). They will not be ˆ × G) ˆ → R+ as ν(A, B) = reproduced here. Now define a measure ν : B(G ˆ and for any bounded continuous f : G ˆ×G ˆ → C, we µ(A ∩ B), A, B ∈ B(G), let Z Z Z f (s, s)dµ(s) (27) f (s, t)dν(s, t) = ˆ G

ˆ G

ˆ G

ˆ × B(G) ˆ → C is a bimeasure of and if F (A, B) = (Z(A), Z(B)), then F : B(G) finite Fr´echet variation (or of finite semi-variation). Using the D-S and MTintegrations one obtains

Z

2 Z Z

(28) 0≤ f (s) dZ(s) = f (s)f (t)F (ds, dt) 2

ˆ G

ˆ G

ˆ G

and if α = ν − F , then the above computation in (26)–(28) implies Z

2 Z Z

Z

2 f (s) dZ(s) = f (s)f (t) dα(ds, dt). |f (s)| dµ(s) − 0≤ ˆ G

ˆ G

2

ˆ G

ˆ G

(29)

It follows that α(·, ·) is positive (semi-)definite and = 0 only if ν = F. Excluding the trivial case that α = F , (29) is strictly positive and it then results that ˆ Cb (G) ˆ → C defined as (Cb (G) ˆ is the space of bounded continuous [·, ·]′ : Cb (G)× ˆ : scalar functions on G) Z Z ′ ˆ f (s)¯ g(t)α(ds, dt), f, g ∈ Cb (G), (30) [f, g] = ˆ G

ˆ G

ˆ and H1 = is a semi-inner product. If N0 = {f : [f, f ]′ = 0, f ∈ Cb (G)} ˆ Cb (G)/N0 , the factor space, let [·, ·] = H1 × H1 → C be defined as [(f ), (g)] = [f, g]′ ,

f ∈ (f ) ∈ H1 , g ∈ (g) ∈ H1 .

ˆ → H0 be the canoniIf H0 is the completion in [·, ·] of H1 , let π0 : Cb (G) cal projection, then (H0 , [·, ·]) is a nontrivial Hilbert space which can even be nonseparable. It may be realized isometrically as a subspace of L20 (P ) on a probability space (Ω ′ , Σ ′ , P ′ ), as follows. Let {hi , i ∈ I} ⊂ H0 be a complete orthonormal set, and (Ωi , Σi , Pi ) be a Gaussian probability based on the complex plane C with the Gaussian distribution given by : Z √ |t|2 1 Pi (A) = A ⊂ C, (t = t1 + −1t2 ), e− 2 dt1 dt2 , 2π A and let (Ω ′ , Σ ′ , P ′ ) = ⊗i∈I (Ωi , Σi , Pi ), determined by the Fubini-Jessen theorem. If now we take the coordinate variable Xi (ω) = ω(i) ∈ Ω(= C I ),

3.4 Integral Representations of Some Second Order Processes

105

then E(Xi ) = 0, E(|Xi |2 ) = 1 and L = sp{Xi , i ∈ I} ⊂ L20 (P ′ ). The correspondence τ : hi → Xi is one-to-one. Extending it linearly takes H0 onto L and kτ (hi )k22 = E(|Xi |2 ) = 1 = [hi , hi ], so that by the usual po¯ j ) and L ⊂ L20 (P ) as desired. Thus larization one gets [hi , hj ] = E(Xi X τ : H0 → L is a one-to-one and onto mapping preserving norms. Let ˆ → L20 (P ′ ) be a mapping and if et : s 7→ ht, si, where π = τ ◦ π0 : Cb (G) ˆ ˆ define X ˜ t = π(et ) and ht, ·i ∈ G is a character of G at t, so et ∈ Cb (G), ′ 2 ′ ˜ ˜ H = sp{Xt , t ∈ G} ⊂ L0 (P ). Then X0 = π(e0 ) = π(1) = τ (1) since π0 (1) = 1, the constant function, (1 ∈ / N0 ), E(|τ (1)|2 ) = 1. Now define ′ ˜ t , t ∈ G so that identiK = H0 ⊕ H , the direct sum and Yt = Xt + X ′ fying H0 as H0 ⊕ {0} in K and similarly H as {0} ⊕ H′ , we see that if Q : K → H0 = {H0′ ⊕ {0}} as the orthogonal projection so that Xt = QYt , ˜ 0 = 0 only if no enlargement is needed. Hence {Yt , t ∈ G} ⊂ K, and X ˜ Σ, ˜ P˜ ) = (Ω, Σ, P ) ⊗ which may be considered as a subspace of L20 (P˜ ), (Ω, ′ ′ ′ (Ω , Σ , P , ) the (weak) stationarity of Yt -process or field can be verified as follows : ˜s , X ˜ t ), since X ⊥ X ˜ r(s, t) = (Ys , Yt ) = (Xs , Xt ) + (X Z Z Z Z hs, λi ht, λ′ iα(dλ, dλ′ ), hs, λi ht, λ′ iF (dλ, dλ′ ) + = ˆ G

= = =

Z Z Z

ˆ G

ˆ G

ˆ G

ˆ G

ˆ G

both these being MT-integrals,

hs, λi ht, λ′ iν(dλ, dλ′ ), since α = ν − F,

ˆ

hs, λi ht, λiµ(dλ), by (27),

ˆ G

hs − t, λiµ(dλ), by the multiplication rule of characters. (31)

ZG

˜ t are weakly continuous, so is Yt and (31) implies that Yt -is Since Xt and X stationary while Xt = QYt , t ∈ G, as asserted. 2 Remark 3.6 The converse part of the above result depends on constructing the Yt -process on a larger Hilbert space that contains L = sp{Xt , t ∈ G} ⊂ L20 (P ) and this construction depends on the inequality (26) which is the crucial part of the argument. This in turn uses the positive definiteness of functions (covariance here) that may be dilated. This is a fundamental result originally due to M. A. Na˘imark and extended by B. Sz.-Nagy. Our alternate procedure is also basically on the same idea. Some extensions and consequences will now be obtained. ˆ → L20 (P ), the process or field Xt = Given a random measure Z : B(G) ˆ ht, λi dZ(λ), defines a weakly-harmonizable class and hence by the above G 2 ˜ 2 theorem we R can find a weakly stationary field Y : G 2→ L0 (P ) 2⊃ L0 (P )2 such ˜ that Yt = Gˆ ht, λidZ(λ), where Q ◦ Z˜ = Z and Q : L0 (P˜ ) → L0 (P )(⊂ L0 (P˜ )) is the orthogonal projection. Thus we have the following useful: R

106

3 Random Measures Admitting Controls

Corollary 3.4.6 Every random measure Z˜ : B(G) → H has an orthogonally valued dilation Z˜ : B(G) → K where the set K ⊃ H is a (super) Hilbert space. An interesting consequence of the theorem is that one can obtain an operator representation of a weakly harmonizable field. Indeed if {Xt , t ∈ G} ⊂ L20 (P ) is weakly harmonizable, then there is a stationary field {Yt , t ∈ G} on a super space L20 (P˜ ) ⊃ L20 (P ) such that Xt = QYt , t ∈ G where Q : L20 (P˜ ) → L20 (P ) is the orthogonal projection. But we already have seen that {Yt = Ut Y0 , t ∈ G} for a unitary group {Ut , t ∈ G} on L20 (P˜ ) so that Xt = QUt Y0 = Vt Y0 (say). The {Vt , t ∈ G} need not be unitary, but kVt k ≤ 1 and is positive definite in the sense that for si ∈ G and xi ∈ L20 (P ) n X n X (32) (Vs−1 si xi , xj ) ≥ 0, j

i=1 j=1

which can be directly verified. The precise version is as follows:

Proposition 3.4.7 Let {Xt , t ∈ G} ⊂ L20 (P ) be weakly harmonizable where G is an LCA group. Then there is a Hilbert space L20 (P˜ ) ⊃ L20 (P ), random variable Y0 ∈ L20 (P˜ ), weakly continuous contractive family {Vt , t ∈ G} of linear operators from L20 (P˜ ) into L20 (P ) with V0 =identity on L20 (P ) (the identity element of G is denoted ‘0’ also) such that when restricted to L20 (P ), the Vt -family satisfies (32), and then Xt = Vt Y0 , t ∈ G. Conversely, every weakly continuous contractive family Vt : L20 (P˜ ) → L20 (P˜ ) satisfying (32) also and defines a process XtY0 = Vt Y0 for each Y0 ∈ L20 (P˜ ), a weakly harmonizable random field where V0 = identity. We omit the proof which follows easily from the preceding work. Note that in the converse direction, L20 (P˜ ) is possible whereas in the other direction an enlargement of the space to L20 (P˜ ) is necessary and the asymmetry is inherent in this representation. The preceding representation has the following interesting consequences. Since Z : B0 (R) → L20 (P ) is a random measure, it follows that for each bounded linear operator V : L20 (P ) → L20 (P ), V ◦ Z : B0 (R) → L20 (P ) is also σ-additive, hence a random measure, so that E. Hille’s theorem (used many times) and the preceding proposition together imply that Yt = V Xt , t ∈ R (or LCA G) is harmonizable if {Xt , t ∈ R} is either a (weakly) stationary or harmonizable process or a field. Thus the class of weakly harmonizable processes (or fields) is closed under bounded linear transformations. But this statement does not hold for the stationary class. Consider the Karhunen process (or field), {Xt , t ∈ T } relative to a family {g(t, ·), t ∈ T } in L2 (S, S0 , µ), so that (12) obtains with a random measure Z : S → L20 (P ), of orthogonal values. It is a rather large class containing all the stationary as well as the harmonizable fields and many others, as we now analyze this large family in a somewhat detailed manner. Integral representations relative to (orthogonally valued and other) random measures naturally play a vital role.

3.4 Integral Representations of Some Second Order Processes

107

Proposition 3.4.8 Let {Xt , t ∈ T } ⊂ L20 (P ) be a weakly harmonizable field with T = G (or Rn or Zn ) an LCA group. Then it is of Karhunen class in ˆ → L2 (P˜ ) where that there is an orthogonally valued random measure Z : B0 (G) 0 2 ˜ 2 L0 (P ) ⊃ L0 (P ) as a subspace and a family {f (t, ·), t ∈ G} in L2 (G, ν) where ˆ → R+ given by (Z(A), Z(B)) = ν(A ∩ B) for all A, B ∈ B0 (G), ˆ ν : B0 (G) relative to which the field is representable as: Z f (t, λ) dZ(λ), t ∈ T (= G). (33) Xt = ˆ G

Proof. By the preceding proposition, there exists a space L20 (P˜ ) ⊃ L20 (P ), and an orthogonal projection V : L20 (P˜ ) → L20 (P ) such that Xt = V Yt where {Yt , t ∈ G} ⊂ L20 (P˜ ) is a stationary field which therefore has an integral representation as: Z (34) Yt = ht, λi dZ(λ) ˆ G

ˆ → L20 (P˜ ) has the description of the statement. Now by an where Z : B0 (G) important observation due to Kolmogorov (1941), that LY = sp{Yt , t ∈ G} ˆ µ), the space determined by the measure µ(A ∩ B) = hZ(A), Z(B)i, and L2 (G, ˆ → R+ is a regular measure [the inner product on L20 (P˜ )] where µ : B0 (G) ˆ µ), representing this mapare isometrically isomorphic. Thus τ : LY → L2 (G, ˜ ∈ LY a unique g ∈ L2 (G, ˆ µ) such that kgkL2(µ) = ping, gives for each X 2 ˜ ˜ ˜ kτ Xk2 = kXkL20 (P ) and the mapping V : L0 (P ) → L20 (P ) has its corresponˆ µ) → L2 (G, ˆ µ), a continuous (and norm nonincreasing) dent V˜ = τ ◦ V : L2 (G, mapping. This important property implies Z Z f (t, λ) dZ(λ), t ∈ G (35) ht, λi dZ(λ) = Xt = V Yt = V ˆ G

ˆ G

ˆ µ) where Z : B0 (G) ˆ → L20 (P˜ ) is orthogonally valued. for a unique f (t, ·) ∈ L2 (G, 2 But this means that {Xt , t ∈ G} ⊂ L0 (P ) ⊂ L20 (P˜ ) is a Karhunen field relative ˆ µ), which is (33). to {ft , t ∈ G} ⊂ L2 (G, 2 Remark 3.7 It should be noted that this important connection depends on the dilation property of the harmonizable process which in turn uses some relatively deep properties of the Grothendieck inequality as elaborated by Lindenstrauss and Pelezy´ nski, and used crucially in Theorem 6. The distinctions between the representing random measures of Xt as in (15) and that in (35), which have different properties with values in different spaces, should be noted. The results of Theorem 3.4.5 and Proposition 3.4.8 above, in comparison with the representation of Proposition 3.4.4, prompts us to seek an analogous operator representation also for Karhunen processes. The following are some possibilities dealing with the general and specialized forms in all of which the representing random measure necessarily takes orthogonal values where the

108

3 Random Measures Admitting Controls

available geometric properties of the underlying Hilbert space will be exploited fully. Our key representation is the following: Theorem 3.4.9 Let {Xt , t ∈ T } ⊂ L20 (P ) be a Karhunen process relative to a family {g(s, ·), s ∈ T } ⊂ L2 (S, S0 , α) on some measure space (S, S, α) and thus its covariance r(·, ·) is given by: Z g(s, u)¯ g(t, u) dα(u), s, t ∈ T. (36) r(s, t) = (Xs , Xt ) = S

Then there exists an abelian set of bounded operators {Bs , Bt∗ , s, t ∈ T } on L20 (P ) such that Xt = Bt Xt0 where with {EA , A ∈ S} as a resolution of identity, and Z g(t, λ) Edλ (37) Bt = g(t, τ ) = S

for an operator τ on LX → LX determined uniquely by the g(t, ·)-family with the correspondence LX ↔ L2 (α), so that for the orthogonally valued ZXt0 (A) = EA Xt0 , A ∈ S, kXt0 k2 = 1, one has the representation: Xt = Bt Xt0 =

Z

S

g(t, λ)ZXt0 (dλ),

t0 , t ∈ T.

(38)

Moreover, LX = sp{Xt , t ∈ T } is isometrically isomorphic to L2 (S, S, α) whenever {g(t, ·), t ∈ T } ⊂ L2 (α) is a dense subset. Proof. In establishing this result and its consequences below, we have to use some aspects of general spectral representations from Dunford-Schwartz ((1963), Chapter X). First assume that {g(t, ·), t ∈ T } forms a basis in L2 (α). Then LX = sp{Xt , t ∈ T } ⊂ L20 (P ) and L2 (α) are isometrically isomorphic determined by (36), i.e., (Xs , Xt )L20 (P ) = (g(s, t), g(t, ·))L2 (α) , so that Xt ↔ g(t, ·) are in correspondence as stated. Since only the isomorphic images are used here, we may replace (S, S0 , α) by a regular (lo˜ S˜0 , α cally compact) space (S, ˜ ) and hence L2 (α) with L2 (˜ α). This can always be done (for a proof, see Rao (1987), 10.2.5 or the second edition (2004), 11.2.5). Dropping the ‘tilde’ notation, we use L2 (α), for simplicity, and let A ∈ S0 (A is compact) and taking χA as a multiplication operator on L2 (α), let the corresponding operator on LX be denoted by EA so that χA f ↔ EA x where f ↔ x are in (the above) correspondence. Since χA χB = χA∩B = χB χA , and χ2A = χA one will have EA EB = EB EA = 2 EA∩B , EA = EA and {EA , A ∈ S0 } is increasing and forms a resolution of the identity in B(LX ), the set of all bounded linear operators on the Hilbert space LX . Consider the collection of operators from B(LX ), the(Banach) space of bounded linear operators on LX into itself, defined as C = {τ ∈ B(LX ) : τ ⌣ EA , τ ∗ ⌣ EB , A, B ∈ S0 }

(39)

3.4 Integral Representations of Some Second Order Processes

109

where τ1 ⌣ τ2 denotes that τ1 , τ2 commute, τ ∗ being the adjoint of τ. Let A be the B ∗ -algebra generated by the commutative class C. This means A is a Banach subalgebra of B(LX ) which is closed under involution ∗ : A → A, (here ‘*’ is the adjoint operation), i.e., (A1 A2 )∗ = A∗2 A∗1 , (aA∗1 ) = a ¯A∗1 (a ∈ C, ∗ ∗ ∗ complex numbers), (A1 + A2 ) = A1 + A2 , Ai ∈ A. Then by Dunford’s general spectral theorem (established in 1950, which appears as Theorem X.2.1 in the book Dunford-Schwartz (1963)) each V ∈ A is representable as Z f (λ) Edλ , f ∈ C(S), (40) Vf = S

where S can be identified with the ‘spectrum’ of A, usually called the ‘maximal ideal space’ of A and since we are assuming I ∈ C, the space S is also compact. The integral in (40) is defined just as the Dunford-Schwartz integral (cf. Section X.1 for review in their book (1963)). One can now use a standard extension argument to show that (40) holds for all f ∈ L2 (α). Thus we have Z g(t, λ) Edλ , t ∈ T, (41) Bt = g(t, τ ) = S

˜ ˜ so that Z(·) where τ ∈ A. If X0 is a fixed element, set EA X0 = ZX0 (A) = Z(A), is an orthogonally valued random measure in L20 (P ), and using the isomorphism again we get Z Z ˜ g(t, λ) dZ(λ), t ∈ T, (42) g(t, λ) Edλ Xt0 = Xt = Bt Xt0 = S

S

and the covariance function r(·, ·) of the Xt -field is given by Z r(s, t) = (Xs , Xt ) = g(s, λ)¯ g (t, λ) dα(λ)

(43)

S

˜ ˜ where α(A ∩ B) = (Z(A), Z(B)). It therefore follows that, by (42), we have an operator representation of the field {Xt , t ∈ T }. From the construction {g(t, ·), t ∈ T } is dense in L2 (α), and if g(t, ·)’s are not linearly independent (basis) we can replace it by a basis by eliminating the redundant ones. Thus (37) and (38) hold, giving a complete analog of (24) for stationary fields. If the g(t, ·)’s are not dense, then one can adjoin a suitable collection of h(s, ·)(⊥ g(t, ·)) to proceed as above, and then discard the correspondents of the adjoined h-family. 2 Remark 3.8 If S = Rn , the procedure can be simplified, by not using the general spectral theorem due to N. Dunford which was needed for the above extended case without using B ∗ -algebras. But then S will be locally compact and the operators Bt will be closed densely defined – not necessarily bounded. We briefly discuss this special case for illustration. [The above analysis holds if A is adjoined by addition of the unit element as in the ‘generalized’ harmonic analysis, and then can obtain this result.] If {Xt , t ∈ Rn }

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3 Random Measures Admitting Controls

R g (t, λ)dα(λ), as before, let LX is a Karhunen field with r(s, t) = Rn g(s, λ)¯ and L2 (Rn , B0 (Rn ), α) or L2 (α) be the spaces that are isometrically isomorphic so that (Xs , Xt )L20 (P ) = (g(s, ·), g(t, ·))L2 (α) and the correspondence Xs ↔ g(t, ·) extended as before. For each A ∈ B0 (Rn ), consider EA and χA so that EA X ↔ χA f if X ↔ f in the isomorphism. Since A is a bounded Borel set and EA is idempotent, define τ : LX → LX as τ X = λf if λf ∈ L2 (α) when X ↔ f. This relation defines τ as a closed densely defined linear operator (usually unbounded) and one gets in the same manner as above: Z Z (44) g(t, λ) dZXt0 (λ). g(t, λ) Edλ Xt0 = Bt Xt0 = g(t, A)Xt0 = Rn

Rn

This special case was sketched by Getoor (1956), extending the original argument due to Karhunen (1947). But the details are a specialization of Dunford’s argument, albeit independently outlined in the above paper. Since the Karhunen class contains both the weakly stationary and harmonizable fields, one admitting a shift and the other not, and both have operator representations, it will be of interest to obtain general analogs in the broader class. This will be discussed to understand the problem better, and to elucidate the key role of Hilbertian geometry in our study. If τs : LX → LX is a linear operator, s ∈ R, which is also a shift on the space in that τs Xt = Xt+s , then for the K-stationary process, {τs , s ∈ R} is a family of unitary group of operators which fact was the basis of integral representations of such processes. If the Xt is a more general L2 (P )-valued process, then it admits a family of shifts provided the following pair of conditions hold P for every finite collection of “times” t1 , . . . P , tn and constants P a1 , . . . , an so that ni=1 ai Xti ∈ LX and τs acts on it, or τs ( ni=1 ai Xti ) = ni=1 ai Xti +s ∈ LX : Pn Pn (i) k i=1 ai Xti k2 = 0 ⇒ k i=1 ai Xti +s k2 = 0 and τs is moreover bounded on L a constant C0 > 0, so that PXn provided there isP n (ii) kτs i=1 ai Xti k2 ≤ C0 k i=1 ai Xti k2 , n ≥ 1, s ∈ R.

Clearly (ii) implies (i) and makes τs a bounded operator as well. Moreover, (ii) makes {τs , s > 0} a family of bounded semi-group of operators. If τ1 is written as τ and τ k = τ · · · τ , so X1 , . . . , Xp ∈ LX , p ≥ 1, then for m, n ≥ 0, one has: p X

(τ n Xm , τ m Xn ) =

p X

(Xm+n , Xm+n ) =

m,n=0

m,n=0

m,n=0

p X

kXm+n k22 ≥ 0.

(45)

When (45) holds and kτ k ≤ C0 , then it was shown by Bram (1955) that we also have: p X

(τ n+1 Xm , τ n+1 Xn ) ≤ C0

m,n=0

p X

(τ n Xm , τ m Xn ),

m,n=0

(46)

3.4 Integral Representations of Some Second Order Processes

111

for the same constant C0 > 0 which may be taken as kτ k2 . A linear mapping τ satisfying (46) is called in the literature a subnormal operator and it has a bound preserving normal extension τ1 to a larger Hilbert space containing LX (here) which we can assume to be L20 (P ) by a standard enlargement by adjunction procedure, familiar in probability theory and we also used it earlier. Denote this extended operator τ1 or (τ1 )s by Ns so that {Ns , s > 0} is a strongly or weakly (the same in the present context) continuous normal semi-group on L20 (P ). Thus kNs x − Nt xk2 → 0 as s → t, for all x ∈ L20 (P ) and Nt∗ Nt = Nt Nt∗ since Nt is normal (this is a defining condition). Note that the unitary class is included here since Ut∗ = (Ut )−1 = U−t , t > 0, so that U0 being identity, it automatically satisfies the normal semi-group definition and hence the (weak) stationary case is included. More importantly, the spectral structure of normal operators is also well-known and the work in the classical monograph of Hille and Phillips (1957, Chapter XX II) becomes applicable and allows us to draw useful conclusions that extend the stationary case in many ways. It is known that the measurability of t 7→ Nt x ∈ L20 (P ) for x ∈ L20 (P ) implies the continuity of the mapping and the operator norm kNt k = eω0 t for a real ω0 < ∞. Also the limit limt↓0 Nt x = Jx exists where J 2 = J. Then by a fundamental theorem of E. Hille and Sz-Nagy (cf., Hille-Phillips (1957), p.595) ZZ ZZ t > 0, (47) eλt dZX0 (λ), eλt E(dλ)X0 = Xt = Nt X0 = ∆

∆

where ∆ = {λ : Re (λ) ≤ ω0 } ⊂ C(∼ = R2 ), and {E(A), A ∈ B(∆)} is a unique resolution of the identity, ZX0 (·) = E(·)X0 : B(∆) → L20 (P ) orthogonally valued random measure, B(∆) being the Borel σ-algebra of the set ∆, of the complex plane. The integral is defined in the Dunford-Schwartz (or spectral) sense that is common in this application. The preceding discussion implies the following precise representation of the random field under consideration: Theorem 3.4.10 Let {Xt , t > 0} be a mean continuous process in L20 (P ) admitting a (bounded right) shift transformation. The underlying (Ω, Σ, P ) can be assumed rich enough, so that one can associate a strongly continuous semigroup of normal operators on L20 (P ) such that Xt = Nt Xt0 , t > t0 ≥ 0, and Xt admits an integral representation as (47), with its covariance r(·, ·) given by ZZ ¯ r(s, t) = esx+tλ dα(λ), s, t > 0, (48) ∆

¯ where α : B(∆) → R+ is a measure for which one has α(A) = kZ(A)k22 , λ being the complex conjugate of λ. Thus the process {Xt , t > 0} is in Karhunen class. As shown in Hille-Phillips (1957, p.598), the case of (weak) stationarity, hence the extension of Nt = Ut the unitary family, is automatically included in

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3 Random Measures Admitting Controls

the above case and one gets Xt = U t X0 =

Z

eiλt dZ(λ),

R

t ∈ R.

(49)

In fact the Rinfinitesimal generator of the {Ut , t ∈ R}, the unitary group is given by Ax = i R λ dE(λ)x, x ∈ domain(A), such that Z o n 2 |λ|2 dαx (λ) < ∞ . dom(A) = x ∈ L0 (P ) : R

The reader is referred to Hille-Phillips (1957, Theorem 22.4.4, p.599)for details. This result shows that the Karhunen class includes large sets of random fields that are also shift invariant. But it also includes random fields that do not admit such classes, the harmonizable fields being immediate examples (cf. the exercises section as well). An interesting specialization of the above result is when the normal operator is self adjoint, i.e., Nt = Nt∗ , t > 0 so that ∆ ⊂ R. In this case, the resulting (Karhunen) random process is called “symmetric”, by Getoor (1956). In such cases the representations (47) and (49) reduce to: Corollary 3.4.11 Let {Xt , t ≥ 0} ⊂ L20 (P ) be a mean continuous process so that Xt = At Xt0 where {At , t ≥ t0 ≥ 0} is a semi group of bounded self-adjoint operators (so kAt k = eω0 t , limt↓0 At x = Jx, x ∈ L20 (P ), J 2 = J = J ∗ ). Then there exists an orthogonally valued ZXt0 : B(R) → L20 (P ), a random measure such that Z ω0 t > 0, (50) eλt dZXt0 (λ), Xt = At Xt0 = −∞

and its covariance r is given by Z ω0 eλ(s+t) dα(λ) = r(s + t, 0), r(s, t) = (Xs , Xt ) =

s, t > 0,

(51)

−∞

where α(A) = kZXt0 (A)k22 , A ∈ B0 (R). The process is (still) nonstationary but is of Karhunen class admitting a (right) shift operator. The result is deduced from the preceding theorem by using a classical consequence as in Hille and Phillips ((1957), Theorem 22.3.1). The details will not be included here. However, there is an interesting and direct connection in its operator representation of Karhunen processes (or fields) given in the above Theorem 3.4.9 and a similar result for (weakly) harmonizable fields as characterized in Theorem 3.4.1. We present it as the following (final) item of this section for comparison: Proposition 3.4.12 Suppose that a mean continuous Karhunen field {Xt , t ∈ T } ⊂ L20 (P ) with its operator representation Xt = Bt Xt0 , t ∈ T is given for a class of bounded commuting operators B = {Bt , Bs∗ , s, t ∈ T } as in

3.5 Complements and Exercises

113

Theorem 3.4.9. If T is a locally compact abelian group, and B also forms a group, then Z t ∈ T, (52) Xt = ht, λi dZ(λ), Tˆ

relative to a not necessarily orthogonally valued random measure Z on Tˆ into L20 (P ) so that the field {Xt , t ∈ T } is weakly harmonizable, Tˆ being the dual group of T.

Proof. By Theorem 3.4.9, B is an abelian collection of bounded operators on LX , the space spanned by the Karhunen field. With the additional hypothesis that B is a group, it follows that with each Bt , Bt−1 is also in B. But then by a special case of a result of Wermer as given in Dunford-Schwartz (1971, XV.6.1) there is a bounded self adjoint operator C with a bounded inverse such that Ut = CBt C −1 is unitary. Hence using Stone’s spectral representation, (and Hille’s theorem used often) one has Z t ∈ T, (53) Bt = C −1 Ut C = ht, λi dC −1 Eλ C, Tˆ

Then by Theorem 10 above, writing G for T , Z Xt = Bt Xt0 = ht, λi dZXt0 (λ), ˆ G

t ∈ G,

ˆ → L20 (P ) is a not necessarily orwhere ZXt0 (A) = (C −1 E(A)C)Xt0 : B0 (G) ˆ Consequently by thogonally valued random measure on the δ-ring B0 (G). Theorem 3.4.1 (and also representation (15)), the process (or field) is weakly harmonizable. 2 The analysis in this section shows that, with the use of finer aspects of Hilbertian operator structure and geometry, many new and deeper properties of second order processes and fields wider than the stationary ones can be investigated. Some additional results will be indicated as complements. Other aspects of random measures will be considered in the following chapters, and then the present work can and will be extended further.

3.5 Complements and Exercises 1. Let Z : B0 (R) → Lp (P ), 0 < p ≤ 2 be an infinitely divisible, especially p-stable, random measure so that the L´evy-Khintchine formula of Z(A) has the parameters γA , βA and CA where γA ∈ R, |βA | ≤ 1, CA ≥ 0. Show that γ(·) , C(·) are σ-additive, but ∞ X

n=1

CAn βAn = C∪n An β∪n An (= (Cβ)(∪n An ) )

for disjoint An ∈ B0 (R) with ∪n An ∈ B0 (R). Argue that β(·) itself is not necessarily σ-additive.

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3 Random Measures Admitting Controls

2. This problem describes a case where a vector X = (X1 , X2 ), X1 , X2 are (infinitely divisible) p-stable, and each linear combination is also pstable but X itself is not infinitely divisible, hence not p-stable. Consider the mapping ψp : (t1 , t2 ) √ 7→ ψp (t1 , t2 ) = exp{−rp + iρr cos 3θ}, where t1 = r cos θ, t2 = r sin θ, i = −1, 0 0 is small enough, then ψp is a characteristic function (Fourier transform) of (X1 , X2 ). [This needs a nontrivial proof.] For any a1 , a2 real, Y = (a1 X1 + a2 X2 ) is pstable.pFirst verify that, E(eitY ) = exp{−r1p |t|p + it(ρr1 cos 3θ′ )} where r1 = a21 + a22 > 0, and θ′ = tan−1 ( aa12 ). This shows that Y is p-stable. But X is not p-stable for 0 < α < 1, and not even infinitely divisible. If X were p-stable, then for X ′ , X ′′ identically distributed as X, there is by definition of stability a vector d = (d1 , d2 ) ∈ R2 , such that 1 X ′ + X ′′ = 2 p X + d. But using a (nontrivial) computation show that this X gives a contradiction for all 0 ≤ θ < π. [The reader should find the details in D.J Marcus (1983), or in the book by Samorodnitsky and Taqqu (1994, p.63). Such an example is eliminated by a result of Gini and Hahn (1983), according to which if X is an infinitely divisible vector and all linear combinations of its components are p-stable, then X will be p-stable also.] 3. (a) In our analysis, we have considered for the most part the concrete spaces Lp (P ), 0 < p < ∞. These are contained in L0 (P ), the space of random variables metrized by convergence in probability, the inclusion being a continuous embedding where the invariant metric d : (f, g) → dp (f, g) is, as seen in Chapter 1, given by Z |f − g|   dP, for p = 0, f, g ∈ L0 (p)  1 + |f − g| Z dp (f, g) = h Ω (∗) i1∧ p1  p  , 0 < p < ∞. |f − g| dP  Ω

This fact is also a consequence of the following more general result due ˜ to Okazaki (1979). Let (Y, d(·)) be a linear metric space with an invariant ˜ ˜ = d(·, 0). The metric metric d(·), the above (∗) being an example with d(·) ˜ is called accessible if there exists a function ϕ : Y → R, continuous and d(·) positive definite, such that for each ε > 0 there is cε > 0 satisfying the condition: ˜ > ε ⇒ ϕ(0) − ϕ(x) > cε d(x)

˜ ≤ ε). (or, ϕ(0) − ϕ(x) ≤ cε ⇒ d(x)

(†)

A consequence of this concept is that a linear (invariant) metric space ˜ is embeddable isomorphically into L0 (P ), on some probability space (Y, d) ˜ is accessible, i.e., satisfies (†). In particular (Ω, Σ, P ), if and only if d(·) Y = Lp (P ), p > 0 of (∗) is included, since the functional ˜

p

ϕp (f ) = e−dp (f ) ,

f ∈ Lp (P ) = Y, 0 < p ≤ 2

3.5 Complements and Exercises

115

satisfies (†), ϕp (·) being the desired positive definite functional. Also when {(Yn , d˜n (·)), n ≥ 1} is a sequence of L0 (P )-embeddable (such) invariant linear metric spaces, then their cartesian product as well as their projective limits are likewise embeddable. This fact depends on the results of characteristic functionals on (Ya , Σ, P ) where Ya is the algebraic dual of Y and Σ is the smallest σ-algebra relative to which the functionals hx, ·i are measurable for x ∈ Ya , hx, ·i being the duality pairing of (Ya , Y). The work is nontrivial and depends on a result due to Bochner (cf. Bochner (1955), Theorems 5.4.3 and 5.4.4). This is needed for the analysis here (cf. also Kuelbs (1973) who raised the question which is now solved in Okazaki (1979) as noted above). (b) The preceding part shows that, given a p-stable characteristic function on Lp (S, S0 , µ), one can construct a corresponding random measure Z : S0 → L0 (P ) on some probability space (Ω, Σ, P ). This is a kind of converse to our work in Section 1 above which complements the analysis based on it. Thus let 0 < p ≤ 2, c ≥ 0, |β| ≤ 1, α ∈ R and A ∈ S0 . Consider the p-stable characteristic function using the parameters as: ϕA (t) = exp{iαµ(A)t − cµ(A)|t|p [1 − iβ sgn t · h(t, p)]} where h(t, p), is as in (4) of Section 1, given by ( if p 6= 1, 0 < p ≤ 2 tan pπ 2 , h(t, p) = 2 − π log |t|, if p = 1. Show that there is a probability space (Ω, Σ, P ) and a (general) p-stable random measure Z : S0 → L0 (Ω, Σ, P ), having the parameters (α, c, β, µ) as given above, with µ(·) bounded in case p = 1, using the following hints: [For each A ∈ S0 , ϕA (·) is positive definite being a characteristic function. Use Bochner’s theorem as follows: Let Ω = Yap (= Lp (S, S0 , µ)a ), the algebraic dual. As in (a), there is a probability triple (Ω, Σ, P ) such that for each f ∈ Yp , Z ϕf (ξ) = eihξ,f i dP (f ), h·, ·i : hξ, f i → Yap × Yp , Ω

where h·, f i is measurable for (Σ) (this is a functional, f ∈ Yp ) by Bochner’s theorem noted for characteristic functionals. Let Z : S0 → Lp (P ) be defined by Z(A) = hχA , ·i. It is to Rbe verified that Z(·)R is the desired random measure, since ϕZ(A) (t) = Ω eihtχA ,ωi dP (ω) = Ω eitZ(A) dP = E(eitZ(A) ), the random variable Z(A) has the given characteristic function. Now we need to verify that Z(·) is σ-additive in P -measure, and has independent values on disjoint sets of S0 . This is obtained with the following standard computation. The S σ-additivity is established on noting that for An ∈ S0 , disjoint with n An ∈ S0 , consider for a given ε>0

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3 Random Measures Admitting Controls

1 Pn P ω ∈ Ω : | j=m Z(Ai )(ω)| ≥ ε = ε

Z

≤ 7ε

|x|≥ 1ε

Z

0

1 ε

dFA1 ,··· ,An (x1 , . . . , xn )

Re (1 − ϕA1 ,...,Am (t)) dt,

by the truncation inequality (cf. Lo`eve (1955), p.196) and this can be shown to tend to zero as (m < n) m, n → ∞, for our ϕAi (·) given. This implies that the {Z(An ), n ≥ 1} is convergent in probability, hence in L0 (P ) and Z(·) is p-stable, as desired. The result also follows from Kolmogorov’s existence theorem, but this alternative is of interest. The argument is detailed in Y.Okazaki (1979), pp.7–10, and discussed further in Chapter 5 later.] 4. In the main text we defined integrals relative to symmetric p-stable random measures, using a controlling measure obtained from the L´evy-Khintchine formula. Here we indicate the general case without symmetry restrictions for the random measures given by the preceding problem which is thus demanding more computation. Let (S, S0 , µ) be a δ-ring and Z : S0 → L0 (P ) be the associated p-stable random measure, as discussed in the preceding problem. P it n i=1 aj Z(Aj ) ) = If Yp is the linear metricR space defined there, so that E(e R ϕA1 ,...,An (t) = exp{−iα S tf dµ + c S |f |p dµ|t|p [1 − iβ sgn th(t, p)]} where n R R Pn f = j=1 aj χAj , and hence ϕf (t) = exp − (αt S f dµ + c S |f |p dµ|t|p × o R [1 − iβ sgn t h(t, p)] extends to all f ∈ Yp and the mapping T f 7→ S f dZ

is uniquely defined where T : Yp → L0 (P ) is an isomorphism into Lp (P ) ⊂ L0 (P ). [The result is immediate for simple functions f , with T f ∈ Lp (P ), and the general case of f ∈ Yp depends on an estimate with the truncation inequality (and this is what makes it different from the D-S approach), discussed in Problem 3(b). Details can again be found in Okazaki (1979), p.15–18.] 5. Complete the proof Theorem 3.2.4(b) on minimally positive definite functions using the following detail. Note that even the minimal positive definiteness implies a number of properties for ϕ. Using the 3 × 3 positive determinental condition already given, show that |ϕ(α + β) − ϕ(α)ϕ(β)|2 ≤ (1 − |ϕ(α)|2 )(1 − |ϕ(β)|2 ),

α, β ∈ R.

Taking α = β and noting |ϕ(α)| ≤ 1, the above inequality gives, |ϕ(2α) − ϕ(α)2 |2 ≤ (1 − ϕ(α)2 )2 ≤ (1 − |ϕ(α)|2 ).

(†′ )

Using the fact that |ϕ(2α) − ϕ(α)2 | ≥ |ϕ(α)|2 − |ϕ(2α)|, (the reverse triangle inequality) these two parts of (†′ ) imply (take a root in the first) |ϕ(2α)| ≥ 2|ϕ(α)|2 − 1. Squaring and using the preceding inequalities, we obtain

3.5 Complements and Exercises

1 − |ϕ(2α)|2 ≤ 4|ϕ(α)|2 1 − |ϕ(α)|

2

≤ 4 1 − |ϕ(α)|2 .

117

(∗)

Now let ϕ(α) = exp{−(A + iB)αp − iCα}, p > 0, A > 0, and using it in (∗), we get 1 − exp(−2A2p αp ) ≤ 4 1 − exp(−2Aαp ) .

Dividing by αp and letting α ↓ 0, this gives 2A2p = 4 · 2A or 2p ≤ 4, so p ≤ 2. Using ϕ(α) above in (†′ ) we find p

p

e−2Aα | exp{−(A + iB)(2p − 2)αp − 1| ≤ 1 − e−2Aα , and if we divide by αp and let α ↓ 0, we get |A + iB||2p − 2| ≤ 2A, or B 2 (2p − 2)2 ≤ A2 (22p − 2p+2 ). If p 6= 1, this gives 0 < p ≤ 2 and yields the inequality for |B| and A. When p = 1, ϕ(·) becomes ϕ(α) = exp{−(A + iB log α)α − iCα} and hence (†′ ) above gives e−2Aα |e−B2 log α − 1| ≤ 1 − e−2Aα . Dividing by α and letting α ↓ 0, this implies |B|2 log 2 ≤ 2A, as desired. The proof of Theorem 3.2.4 is hereby completed. 6. Consider the function ϕM,N,Q : (t1 , t2 ) → C defined by ϕM,N,Q (t1 , t2 ) = (1 − it1 )−M (1 − it2 )−N 1 +

ρ212 t1 t2 (1 − it1 )(1 − it2 )

−Q

m m where M, N, Q > 0 are given. If M = mn 2 , N = 2 and Q = − 2 , then ϕM,N,Q is the characteristic function of a bivariate gamma distribution of (X, Y ), when m, n are positive integers, |ρ12 | < 1, ρ12 being the covariance of (X, Y ) and then ϕM,N,Q never vanishes. It is even infinitely divisible. However, for M, N, Q > 0, real and Qρ212 > min(M, N ) then it is not positive definite. [Assuming it is positive definite and inverting it, W.E.Kibble (1941) found that the inverse Fourier transform is negative on some sets (intervals) of positive measure. The nontrivial computation may be found in the paper noted above.] 7. We now sketch a key fact on the structure of p-stable symmetric random measures that explains why the complications arise in defining integrals and why we need some controlling (scalar) measure but cannot ask for finite Vitali variation in this subject.

Recall that a function f on an interval J ⊂ R, f : J → R, is called completely monotone if it is infinitely differentiable on J and its derivatives

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3 Random Measures Admitting Controls

f (n) satisfy (−1)n f (n) (x) ≥ 0, x ∈ J and ≥ 0. Such functions were completely determined by S. Bernstein as Laplace transforms of nondecreasing bounded positive functions. (See Rao (2004), p.288 for a proof.) Here we need a result on the composition of a pair of such functions. However, the mappings x 7→ x1 and x 7→ x12 , 0 < x < ∞, are both completely monotone, but their composition is x → x1 ( x12 )−1 = x2 is not. Now the following restricted version is true and it suffices for our illustration here. Namely: (∗) If f : R+ → R is completely monotone, g : R+ → R, satisfies g(0) = 0 and its derivative g ′ is completely monotone in 0 < x < ∞, then f ◦ g : R+ → R is completely monotone which needs a nontrivial proof, to be given below. 2p (a) We now assert that the function t 7→ e−|t| , 0 < p ≤ 1 is positive definite and hence is a characteristic function on R. If p = 1, then it is a known result, and so let 0 < pR< 1. The function given by g(y) = y p , sat∞ −yt −p p e t dt, and is completely monotone isfies g ′ (y) = py p−1 = Γ (1−p) 0 −y 0 < y < ∞, and taking f (y) = e , (∗) implies that f (g(y)) is completely monotone, and Z ∞ p e−|y| = e−yt dF (t), 0 < y < ∞, 0

by Bernstein’s theorem for a unique 0 ≤ F ↑≤ k < ∞. Replacing y by y 2 here we get the key formula Z ∞ 2p 2 e−|y| = e−y t dF (t), 0 < y < ∞, (†”) 0

which is well-defined, the integral can be approximated by finite Pn and2 −y 2 2 aj , aj > 0. But e−y aj is positive defisums of the form j=1 bj e nite, as the characteristic function of a normal distribution (with range R) and the finite linear combination with positive coefficients is positive definite. The same holds for its pointwise limit so that (†”) implies that 2p f : t 7→ e−|t| , 0 0) Y with E(eitY ) = e−σ |t| , 0 < p ≤ 1, t ∈ R, and Y is symmetric 2p-stable on R (cf. Theorem 3.1.2). To establish (∗), using Bernstein’s characterization, itPsuffices to show that h(ρ) = f (g(ρ)) is the limit of (finite) sums of the form n e−g(ρ)tn (α(tn+1 )− α(tn )), giving the (desired) integral representation. But the sum, product and (pointwise) limits of completely monotone functions are completely monotone again. So it is enough to replace g(ρ)tn by g(ρ) and show that e−g(ρ) is completely monotone, since h(·) is a limit of finite sums noted above. But if g ′ (·) has P this monotonicity property, it may be approximated by the finite sums n e−ρtn (γ(tn+1 ) − γ(tn )), for an increasing bounded γ(·). So we may assume that g ′ (ρ) = e−ρt whence

3.5 Complements and Exercises

119

e−ρt . t

1 e−ρt − 1 = e− t exp t R∞ P −nρt −ρt Since exp( e t ) = n≥0 en!tn = 0 e−ρµ dβ(u), it is completely monotone 1 by Bernstein’s converse. In particular taking g(·) as g(ρ) = (ρ+ν 2 ) 2 −ν, ν > 0, which is admissible since now Z ∞ 2 1 1 1 2g ′ (ρ) = (ρ + ν 2 )− 2 = √ e−ρt e−ν t · t− 2 dt π 0

g(ρ) = exp

and is completely monotone, for f (completely monotone), we can take 1

h(ρ) = f ((ρ + ν 2 ) 2 − ν), it is again completely monotone for ρ ≥ 0, and establishes (∗). (b) Using (a) show that a symmetric p-stable random variable with parameter c > 0, has the same distribution as Yp Z where Yp ≥ 0 is a random variable independent of a centered normal random variable with variance determined by c > 0 above. [Sketch: The characteristic function of X is ϕX (·) where 2p

ϕX (t) = E(eitX ) = e−|σt| ,

σ 2 = c > 0, 0 0, and Z be an independent random variable, from Y˜p , which is N (0, σ 2 ), i.e., normally distributed with mean 0, and variance σ 2 = c > 0. Then e−|σt|

2p

˜

= E(eitYp ) = E(eitYp Z ) = E[E(eitYp Z |Yp )] 2

= E(e−t

= E(e

2

Yp2 σ2

), since Z is N (0, σ 2 ) distributed,

2 −t2 2Y˜p · σ2

) = E(e−(σ

t )Y˜p

2 2

).

2p Thus Y˜p has the characteristic function as e−|σt| , 0 < p ≤ 1 and Y˜p and 2 Z are independent, Z being N (0, σ ).] (c) A complex random variable W = U + iV (U, V real) is symmetric pp p stable if for each complex number z, E(ei Re (zW ) ) = e−σ |z| , (U, V need not be independent if 0 0) of (b) again. (d) Using (b) and (c) show that if Z : B0 (R) → L20 (P ) is a p-stable random measure 0 0 is a random variable independent of WA which is a centered Gaussian random

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3 Random Measures Admitting Controls

measure. This shows that, since W(·) can be Brownian random measure, Z(·) cannot be assumed to have finite (Vitali) variation even locally. Thus a separate analysis of Z(·), with its controlling measure must be considered. In fact, the Dunford-Schwartz method of integration of scalar functions relative to such a random (or vector-valued) measure, having a dominating measure, seems indispensable, and it can be extended for more general functions later on, as we shall see in the following chapters. 8. Show that a weakly (or strongly) harmonizable random field does not generally admit a shift operator, on its range, using the following example. Let {Xn , n = 0, ±1, ±2, · · · } be a complete orthonormal set in a space L20 (P ) and T : L20 (P ) → L20 (P ) be a (linear) bounded operator. Then Yn = T Xn , n = 0, ±1, ±2, · · · is a second-order process, since {Xn , n = 0, ±1, ±2, · · · } is weakly stationary, the Y -process is (weakly) harmonizable. If τk Yn = Yn+k , n = 0, ±1, ±2, · · · , it does not define a shift operator for the harmonizable sequence {Yn , n = 0, ±1 ± 2, · · · }. Take T to be (for example) given by Yn = T Xn = X2n , = 0 otherwise, so that Yn is the even-numbered sequence. For this kτ Yn k2 6= kYn k if τk is a shift to odd indices. But observe that the Y -sequence is in the Karhunen class by Proposition 3.4.8. 9. This problem exemplifies Theorem 3.4.10, and shows how large is the class of Karhunen fields which are associated with families of bounded commuting (linear) operators on a Hilbert space. Thus let {Bt , t ∈ T } be a collection of commuting bounded operators from B(L20 (P )) which is not necessarily a semi-group but each Bt is normal (i.e., Bt ⌣ Bt∗ ). If {Xt = Bt Xt0 , t, t0 ∈ T } ⊂ L20 (P ), then {Xt , t ∈ T } is a Karhunen field in the sense that there is a compact set M , and an orthogonally valued Z : B(M ) → L20 (P ) such that we have the following representation: Z Xt = f (t, λ) dZ(λ), t ∈ T, [Z(·) = ZXt0 (·)] M

for a class of complex-valued measurable functions {f (t, ·), t ∈ T } on M which satisfy Z r(s, t) = (Xs , Xt ) = f (s, λ)f¯(t, λ) dα(λ) M

where α(A) = kZ(A)k22 , A ∈ B(M ) the Borel σ-algebra of M. [Sketch: The basic idea is similar to the proof of Theorem 3.10 but is simpler. Let A be the commutative B ∗ -algebra generated by {I, Bs , Bt∗ , s, t ∈ T }. We can then apply the (by now classical) theorem of Gel’fand and Na˘imark, and conclude that A is isometric and ⋆-isomorphic to C(M ), the continuous (complex) function space on the set M which is the space of ‘maximal ideals’ of A endowed with a ‘hull-kernel’ topology relative to which M is a compact Hausdorff space (cf. Loomis (1953)). If τ : A → C(M ) is this

3.5 Complements and Exercises

121

mapping then for each f ∈ C(M ) there is a unique τ f ∈ A, such that kf ku = kτ −1 f k, the former is the uniform norm and the latter is the operator norm. Thus for each Bt ∈ A, there is f (t, ·) ∈ C(M ) such that the correspondence Bt ↔ f (t, ·) is the ⋆-isomorphism. By the Dunford generalized spectral theorem (cf. Dunford-Schwartz (1963), Theorem X.2.1) there is a unique spectral measure E(·) such that Z f (t, λ) dE(λ), t ∈ T, Bt = M

and then Xt = Bt Xt0 gives the desired representation as in our Theorem 3.10. Taking Z(·) = E(·)Xt0 , one gets the integral for r(s, t) as given since Z(·) is orthogonally valued, following from the corresponding property of E(·) : B(M ) → A. The argument is abstract but conceptually simpler than that of Theorem 3.4.9. 10. Following the analogy of weak stationarity and harmonizability, one shows that the Karhunen processes can be similarly generalized. This was shown by Cram´er (1951), and we indicate a generalization of it here since the corresponding random measure should not be orthogonally valued. If (S, S) is a measurable space and Z : S → L20 (P ) is a random measure (not assumed to have orthogonal values), let β(A, B) = (Z(A), Z(B))L20 (P ) so that β(·, ·) : S × S → C is a bimeasure. If g(t, ·) : S → C is measurable and strictly β-integrable in the sense of Morse-Transue (or M T ), t ∈ S, then R R g(s, λ)¯ g (t, λ′ )dβ(λ, λ′ ) is defined and by our earlier analysis (see TheA B R random field orem 3.4.1), {Xt = S g(t, λ) dZ(λ), t ∈R S} R is a well-defined g (t, λ′ ) dβ(λ, λ′ ), as the and its covariance r is given by r(s, t) = S S g(s, λ)¯ MT-integral. The field {Xt t ∈ S} so obtained is the (weak) Cram´er process (or field) denoted class (C). Here the term “weak” will be omitted (and called “strong” for comparison if β(·, ·) has finite Vitali variation as originally studied by Cram´er (1951)). With Corollary 3.4.6, when T = S = G, an LCA group now, show that there is a larger space L20 (P˜ ) ⊃ L20 (P ) and an orthogonally valued random measure Z˜ : S → L20 (P˜ ), such that Z = Q◦ Z˜ where Q is the orthogonal projection of L20 (P˜ ) onto L20 (P ), which ˜ ˜ has a finite dominating measure µ, given by µ(A ∩R B) = (Z(A), Z(B)). If ˜ is well{g(t, ·), t ∈ S} is a bounded family, then Yt = S g(t, λ) dZ(λ) defined and is a Karhunen field. Show that, since g(t, ·) is Z-integrable, ˜ and also Z-integrable, we have Xt = QYt , t ∈ S, using the analog of Hille’s theorem (cf. Dunford-Schwartz (1958), IV.10.8(f)). If β has finite Vitali variation, can we take µ to be the variation measure, and conclude that a β-integrable g(t, ·) is also µ-integrable to assert that every class (C) field has a dilation to a Karhunen class? In the harmonizable case S = T = G, an LCA group, and g(t, λ) = ht, λi being a character of G, is always bounded, but g(t, ·) of class (C) need not be so bounded presenting some difficulty! (The general statement without boundedness however need not hold.)

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3 Random Measures Admitting Controls

Bibliographical Notes The analysis presented in this chapter highlights the distinction to be recognized between the treatments of vector-valued measures taking values in Fr´echet (or invariant linear metric) spaces and those with ranges in Banach spaces, to be termed vector measures to maintain the differences in their classical use and introduction of integrals relative to them. The basic problem, to define a secure integral of (even) a scalar function for these measures, is to find a controlling measure so that the general procedure in Dunford-Schwartz (1958, IV.10), which is based ultimately on the Vitali-Hahn-Saks theorem, may be followed. It was shown there (nontrivially, see Theorem I.3.4 for a different argument) that such a control always exists for vector (or Banach space valued) measures. That method fails, however, for Fr´echet space valued ones. If the vector (or random) measures have their values in Lp (P ), 0 < p < 1 and more generally 0 < p < ∞, one can find different classes of measures having controls, and hence useful integrals are definable. We have devoted sections 3.2 and 3.3 for this problem and obtained several general results if the random measure is either symmetric p-stable (0 < p ≤ 2) or more generally p-weakly stable classes. In the former case the classical L´evy-Khintchine representation (infinite divisibility) and in the latter the weak stability (applicable for 0 < p < ∞) are used. Here (in weak stability) positive definiteness is essentially dropped, and the corresponding Bochner extension of weak stability (called ‘formal approach’ to stability by him) plays a special role as it extends the concept to all Lp (P ) spaces, 0 < p < ∞. In all the cases of integration of scalar functions relative to a vector-valued set function (or random measure here), the Dunford integral takes the prominent place when a controlling measure is found, and the work of Sections 2 and 3 is devoted to solving this problem. In the cases where Z(·) takes values in Lp (P ), p > 0, one uses some special results from Probability Theory and the analysis on the former, which seems unavoidable. The results depending on stable and semi-stable classes are taken primarily from Bochner (1975), Schilder (1970) and Hardin (1981), especially Theorem 3.2 follows their work and of Kuelbs (1973), (cf. also Kanter (1972)). The material on generalized stable (or semi-stable) class follows Ramachandran and C.R. Rao (1968). We are also influenced by Marcus and Pisier (1984), particularly regarding Problem 7(b). Theorem 3.1.2 is taken from the author’s recent work (cf. Rao (2007)), and the second order processes of Section 4, have been considered by many authors. A special case of Corollary 3.3.4 and some applications have also been discussed in Soltani and Parvardeh (2005). The (weakly or K-) stationary analysis has its roots in Kolmogorov’s (1941) classical representation theory, extended by Masani (1978) several of whose earlier results are also cited there. A general account with new types of (operator) representations for both (weakly) harmonizable and more inclusive Karhunen classes, originally started in Karhunen (1947), extended much further in the

Bibliographical Notes

123

present author’s recent work (cf. Rao (2006)), is followed here. Also bimeasure theory has been treated by Dinculeanu and Muthiah (2000) to Banach space valued cases for further analysis and applications. In this connection, it may be noted that much of the (representation) theory based on positive definite bimeasures allows an extension via the Grothendieck inequality (even omitting positive definiteness) and using the theory of “p-summing operators” in Banach spaces. These will admit analogs of the Karhunen theory of second order fields expounded in the last section above. However, such ideas will only be touched on here as they have little probabilistic content. A good introduction and several applications of Grothendieck’s inequality may be found in Pisier (1986). We detail some aspects of the latter in Chapters 4, 5 as well as 6 and continue integration without this inequality where Z(·) may be valued in L0 (P ). Since most of the references of the subject are included in the text, we merely point out that several different applications of p-stable processes are discussed in the book by Samorodnitsky and Taqqu (1994). Our concentration here is on random measures and their role in several integral representations of processes and fields, keeping close to the vector measure theory of Banach/Fr´echet spaces omitting any special considerations that may have particular applicational relevance to areas such as point processes. There are books devoted completely for such applications but these ideas are not emphasized here. Our considerations already lead to important and different questions on vector measures themselves treated in the following chapters.

4 Random Measures in Hilbert Space: Specialized Analysis

Motivated by the last part of the preceding chapter, we study several special properties of random measures coming from the available geometry of Hilbert space and the ensuing interplay between measures and bimeasures arising from the randomness present in the problems. Utilizing some work on the structure of the well-known vector harmonic analysis we obtain special properties of random fields in such a space using their integral representations of the related covariances. The work is aided by some structural decompositions of processes and classifications into different types. The availability of a well-developed geometric and algebraic analysis is exploited in the structural aspects of processes and fields connecting it with the probabilistic interpretation. Extensions to the multivariate random fields of second order, including some infinite dimensional cases, are discussed to explain the far reaching points of this type emphasizing the special role of positive definiteness property in the context. A bimeasure is a signed measure in each component but it need not determine a (signed) measure on the product space, and naturally this makes a significant difference. In some specific applied studies, ‘random measures’ are used only to denote ‘point processes’. Such a restricted interpretation is not taken in our analysis. The general vector measure and functional analysis view prevails in the following study. Some applications are included however to show several interesting representations of nonlinear but simply additive (operator) functions as integrals of random measures which are valid. Moreover different kinds of (nonlinear) kernels that include certain ‘Poincar´e-type’ additive functionals are discussed. Also novel applications of Karhunen dilation of Cram´er class of processes are included in the last section.

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4 Random Measures in Hilbert Space: Specialized Analysis

4.1 Bilinear Functionals Associated with Random Measures Let X be a locally compact space, and C0 (X) be the space of scalar continuous functions on X vanishing at infinity. Let L20 (P ) be the standard Lebesgue space on a probability triple (Ω, Σ, P ) of centered random variables, so that their expectations vanish. Let F : C0 (X) → L20 (P ) be a linear mapping, also termed a random linear functional, and consider the bilinear functional B defined by B(f, g) = (F (f ), F (g)) = E(F (f )F (g)), f, g ∈ C0 (X),

(1)

where E(·) is the expectation symbol as usual. It is clear that B(·, g) and B(f, ·) are linear and under the usual continuity conditions, the Riesz representation theorem implies that, without the assumption that E(F (f )) = R 0, E(F (f )) = X f dν for a scalar measure ν, for all f ∈ C0 (X) where ν is a scalar (Baire) measure on the (Baire) σ-algebra B(X) of X. A more useful and interesting result will be to find a random measure Z : B(X) → L20 (P ) such that Z f (λ) dZ(λ), f ∈ C0 (X), (2) F (f ) = X

where the integral is defined as in the preceding chapters, and one would expect that ν(A) = E(Z(A)), A ∈ B0 (X). Here we find conditions for (2) to hold and with an extension of the bimeasure β to obtain [extending B(·, ·) to bounded Baire functions] ˜ 1 , A2 ) = E(Z(A1 )Z(A ¯ 2 )), A1 , A2 ∈ B0 (X), B(χA1 , χA2 ) = β(A

(3)

˜ called a bimeasure associated with Z(·), so that ν(·) may be termed with β, the mean (or expected value) measure of Z(·). For centered Z, we denote β˜ as β and observe that β, as so defined extends and that β : B0 (X) × B0 (X) → C is a bimeasure on the (product) δ-ring. Let us take X = R for simplicity, motivation and some special analysis in this case to get a good insight of the problem which also aids the work of the preceding chapter. From this point of view, (2) implies that, for all bounded Borel functions B(R) and random measures Z : B0 (R) → L20 (P ) where B0 (R) is the δ-ring of (bounded) Borel sets of R, the following relation holds: Z f (t) dZ(t), f ∈ B(R), (4) F (f ) = R

and if fn → 0 boundedly and pointwise, then F (fn ) → 0 in probability for the Dunford-Schwartz integral, of (4), for which both the bounded and dominated convergence theorems hold. Moreover for the bilinear functional B : (f, g) 7→ (F (f ), F (g)), one has B(f, ·) and B(·, f ), f, g ∈ B(R),

4.1 Bilinear Functionals Associated with Random Measures

127

to be anti-linear and linear functionals which are continuous on B(R), the latter having uniform norm. We can replace R by Rn , n ≥ 1 here and below. Instead of (4), one may consider a mapping F : B(Rn ) → L20 (P ) and hope to characterize the representation (4) under suitable assumptions, so that one seeks conditions in order that such a result is true. These functionals F are termed generalized random mappings. As we saw briefly in Chapter 2, such functionals exist and we consider a general class here. Note that, as a motivation, if {Xt = F (ft ), t ∈ T } for a class of functions {ft , t ∈ T } ⊂ B(Rn ), we get a second order random field. But subspaces of B(Rn ) that carry a desirable structure including σ-finite, or the Radon character of measures on Rn , we start with the well-structured and popular Schwartz spaces which are vector spaces and carry a locally convex topology. They are known to be very useful in (partial) differential equations. These are spaces K(⊂ B(Rn )) of infinitely differentiable functions with compact supports whose topology is determined by the conditions : fn ∈ K and fn → 0 as n → ∞ to mean that all the fn of the sequence vanish outside of a fixed compact set (determined by the sequence) m fn on which the partial derivatives ∂X m∂1 ...∂X mk (x), m = m1 + · · · + mk , and 1

k

fn ∈ K(⊂ B(Rk )), converge uniformly in x as m → ∞. This defines a locally convex topology J under which K will be complete. In fact a more inclusive class with weights were proposed by Gel’fand and his associates (Shilov, Vilenkin and others) and the resulting space, denoted Φ(⊃ K), is called the space of test functions. This space is found to be of great interest in this analysis. These will be briefly considered later. The discussion is intended to be a preview. Let us introduce the desired random functional F extending the representation (4). [A generalization of this work to random currents will be treated in Chapter 8.]

Definition 4.1.1 Let K be the Schwartz space of infinitely differentiable scalar functions f : Rn → C vanishing off compact sets endowed with the topology J (noted above and it is locally convex) relative to which K is (known to be) a complete space. Then a linear mapping F : K → L20 (P ) is called a generalized random field (process if n = 1) abbreviated g.r.f (or g.r.p) (i) in the sense of I.Gel’fand if fm → f in the topology J, as m → ∞, then F (fm ) → F (f ) in probability, or (ii) in the sense of K. Itˆ o if for the sequence fm → f (as above in J), then F (fm ) → F (f ) in L20 (P )-mean, as m → ∞. That both notions are equivalent is perhaps not evident, although (ii) ⇒ (i) is clear. However, if each F (fm ) ∈ L20 (P ), then we now see that (i) ⇒ (ii) also. Indeed since fm → 0 in J, the {fm , m ≥ 1} vanish off a compact set K0 and by Fatou’s lemma, they or an infinite subset of {F (fm ), m ≥ 1} will be bounded. [This can be shown in an indirect way or use the argument as in the classical Vitali-Hahn-Saks theorem on noting that K(K0 ), the restrictions of K to K0 , is a complete metric space, and the desired boundedness follows

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4 Random Measures in Hilbert Space: Specialized Analysis

from the Baire category theorem.] But then the set {F (fm ), m ≥ 1} is relatively weakly compact in L20 (P ) because bounded sets in reflexive spaces have such a property. Since F (fm ) → 0 in probability, and boundedly, it follows by the classical Vitali convergence theorem that F (fm ) → 0 in norm so that it is true in Itˆ o’s sense also, and thus both are equivalent concepts. Thus we can invoke either according to convenience in the following analysis under appropriate hypotheses. Using the topological structure of Rn on which K is defined the finer properties of the g.r.f. F : K → L20 (P ) will be discussed. Of particular interest here are the concepts in place of σ-finiteness, stationarity, harmonizability and others. These are set out in a way that they are applicable in both Gel’fand and Itˆ o definitions. They will be presented in the form of the following: Definition 4.1.2 (a) A measure ρ : B0 (Rn ) × B0 (Rn ) → C is said to be tempered, of order q ≥ 0, if Z Z d2 |ρ|(x, y) (5) (i) q < ∞, 2 2 Rn Rn [(1 + |x| )(1 + |y| )] 2 and there is a Borel function (or ‘kernel’) k : Rn × Rn → C such that Z Z ¯ x) d2 ρ(s, t) = b(x), x ∈ Rn , (ii) k(s, x)k(t, Rn

(6)

Rn

exists with b(·) being bounded on bounded sets of Rn . (b) A bilinear functional B : K × K → C is said to be of class (C), or of Cram´er class, relative to a tempered measure ρ and a kernel k, as in (a), if Z Z B(u, v) = u ˜(s)v¯˜(t) d2 ρ(s, t), u, v ∈ K, (7) Rn

Rn

exists where u˜ (and similarly v˜) is the k-transform of u, i.e., Z u˜(t) = k(t, x)u(x) dx.

(8)

Rn

[If k(t, x) = eit·x then u ˜ = uˆ, is of course the Fourier transform.] (c) A random field F : K → L20 (P ) is of class (C) if its bilinear form B : (f, g) 7→ (F (f ), F (g)), f, g ∈ K, which is its covariance, is of class (C) relative to some tempered measure ρ and a kernel k. (d) The random field F and its bilinear functional B(·, ·) are harmonizable (weakly) if k(t, x) = eit·x in (7), and the integrals in (6) and (7) are in strict Morse-Transue (or sMT-)sense. If moreover, (i.e., in addition to k(t, x) = eit·x ), B(u, v) = B(τh u, τh v) where (τh u)(x) = u(x − h), τh being a translation operation, so that B(·, ·) is translation invariant, then F (·) and B(·, ·) are called weakly or K(for Khintchine)-stationary, and then in (6) and (7) the integrals are automatically in the Lebesgue (-Stieltjes) sense.

4.1 Bilinear Functionals Associated with Random Measures

129

It will be seen that in the weakly stationary case, the tempered measure ρ is real, concentrating on the diagonal x = y of Rn × Rn , so that (7) becomes Z u ˆ(t)v¯ˆ(t) dσ(t), u, v ∈ K, (9) B(u, v) = Rn

where σ(·) then is a tempered measure on Rn , (ρ is usually replaced by σ). A basic representation of random fields F (·) of class (C)(and harmonizable ones) will now be presented and then it is specialized to (weak) stationarity and harmonizability, showing the basic role played by the random measures Z(·) of (2) and (4). Theorem 4.1.3 Let Φ be a test space on Rn , such as K (and more precisely, Φ is a complete countably normed space or a countable union of such spaces whose topology has the property that fm ∈ Φ, fm → 0 in its topology implies fm → 0 pointwise in Rn ; but the reader may restrict to K here). If F : Φ → L20 (P ) is a class (C) random field relative to the kernel k and a tempered measure ρ : B0 (Rn ) × B0 (Rn ) → C, as in (8), then there exists a random measure Z : B0 (Rn ) → L20 (P ) whose covariance bilinear functional B(·, ·) is of class (C) relative to B and ρ, in terms of which one has the representation: Z F (f ) = f˜(t) dZ(t), f ∈ Φ, (10) Rn

where f˜ is the k-transform as in (8) and the integral in (10) is defined in the Dunford-Schwartz sense. Conversely, if k and ρ are as in Definition 2, with a random measure Z : B0 (Rn ) → L20 (P ) whose covariance bimeasure is given by (7) relative to ρ and K(⊂ Φ densely), then the random field F defined by (10) is of class (C) on K and has a unique continuous extension to all of Φ.

Proof. For convenience the essential argument is presented in four steps. I. Let F : Φ → L20 (P ) be of class (C) and consider B(f1 , f2 ) = (F (f1 ), F (f2 )) so that B(·, ·) is positive definite. It can be assumed for simplicity to be non-degenerate, i.e., B(f, f ) = 0 ⇒ f = 0. [Otherwise we can consider the quotient space Φ˜ = Φ/N where N = {f ∈ Φ : B(f, f ) = 0} and set ˜ g) = B(f + N, g + N ) where f + N ∈ Φ˜ ; and consider B ˜ in place of B.] B(f, Using (7) define an inner product as f1 , f2 ∈ Φ ⇒ (using sMT-integrals, with ‘star’) Z Z ∗ (f1 , f2 )ρ = B(f1 , f2 ) = f˜1 (s)f¯˜2 (t) d2 ρ(s, t), (11) Rn

Rn

= (F (f1 ), F (f2 ))L2 (P ) ,

so that the mapping f 7→ F (f ) is an isometry of L2 (ρ) into L20 (P ), where L2 (ρ) is the completion of Φ under (·, ·)ρ of (11). If A ∈ B0 (Rn ) then χA ∈ L2 (ρ) and

130

4 Random Measures in Hilbert Space: Specialized Analysis

let Z(A) be its correspondent in L20 (P ), whence Z : B0 (Rn ) → L20 (P ) defined as (Z(A1 ), Z(A2 ))L20 (P ) = (χA1 , χA2 )L2 (ρ) implying that Z(·) is additive. It will now be verified that Z(·) is σ-additive in the norm of L20 (P ) so that it is a random measure. S∞ II. Let A, Am ∈ B0 (Rn ), A = m=1 Am , disjoint union. Then

m m

X X

S Z(Ai ) = χA − χ Ak = χ

Z(A) − Ak

2

k≥m+1 L2 (ρ) 2 i=1 k=1 L0 (P ) L (ρ) [ = ρ˜( Ak × Ak ) ≤ ρ˜(A × A) < ∞, (12) k≥m+1

and since ρ˜ is a tempered measure by hypothesis it follows that the right side tends to zero as m → ∞. Hence Z(·) is a random measure. Moreover for A1 , A2 ∈ B0 (Rn ), ¯ 2 )) = (Z(A1 ), Z(A2 ))L2 (P ) = (χA1 , χA2 )L2 (ρ) E(Z(A1 )Z(A 0 Z Z ∗ = d2 ρ˜(x, y), A!

A2

as desired. [Note that we extend the measures from Φ to B(Rn ), the space of bounded Borel functions under the uniform norm and the subspace restricted to functions with bounded supports is dense in L2 (ρ).] III. To derive the representation (10), consider Φ˜ = {f˜ : f ∈ Φ} ⊂ L2 (ρ) and by hypothesis Φ˜ is dense in L2 (ρ) since the mapping f → f˜ is an isometry and Φ is dense in L2 (ρ) by the fact that ρ is (regular and) tempered. Hence each f is approximable (in norm)Pby step functions fk → f as k → ∞ and so k ˜ ˜ f˜k → f˜ in the same way. If f˜k = m i=1 aki χAi then fk → f in (quadratic mean 2 2 of) L (ρ) and {F (fk ), k ≥ 1} is Cauchy in L0 (P ) which will have a limit F (f ). Since Z mk X f˜k (t)dZ(t) aki Z(Ai ) = F (fk ) = Rn

i=1

is well-defined, by the definition and properties of D–S integral, this clearly implies (10). IV. Conversely if the random field F (·) is given by (10), then it is of class (C). The random measure Z(·) and the tempered measure ρ have their associated bilinear functionals B (or bimeasures) connected by Z Z ∗ B(u, v) = u ˜(s)v¯˜(t)d2 ρ(s, t), u, v ∈ Φ, (13) Rn

Rn

and should be shown to be continuous. By hypothesis K ⊂ Φ densely. Let fm ∈ K and fn → 0 in the topology of the latter. Then the fm ’s are all supported on a fixed compact set and then fn → 0 pointwise as well as boundedly. The fact that ρ is tempered implies that one can invoke the Vitali convergence

4.1 Bilinear Functionals Associated with Random Measures

131

theorem to conclude that B(fn , fm ) → 0 as m → ∞ so that F (fm ) → 0 in L2 (P ). Hence F (·) is continuous in K and by its density in Φ, the same holds on Φ which has a unique extension to all of the latter. It then follows that the g.r.f. F (·) is of class (C) as desired. 2 In the case that F : Φ → L20 (P ) is (weakly or strongly) harmonizable, then k(t, x) = eit·x and ρ is tempered, and the rest R of the hypothesis of ˆ(t), the above theorem holds. In this formulation u˜(t) = Rn eit·x u(x) dx = u since for u ∈ K, u ˆ always exists. Consequently the above theorem implies: Corollary 4.1.4 Let K ⊂ Φ, densely where Φ is a test space. If F : Φ → L20 (P ) is (weakly or strongly) harmonizable relative to a tempered positive definite measure ρ on B0 (Rn ) × B0 (Rn ) into C, then there exists a random measure Z : B0 (Rn ) → L20 (P ) with a bimeasure which is tempered relative to ρ, so that the representation of F is given by Z fˆ(t)dZ(t), f ∈ Φ. (14) F (f ) = Rn

Conversely every functional given by (14) is a generalized harmonizable random field F whose covariance is given by (13), with f˜ = fˆ, relative to the tempered measure ρ. It is of interest to find conditions in order that in (9) and (10) the k-transform becomes k(t, x) = eit·x so that u˜(t) = u ˆ(t) obtains. That this is the case for a K-stationary g.r.f.’s is seen from the following: Proposition 4.1.5 Let K ⊂ Φ be the test space as before and F : Φ → L20 (P ) be a g.r.f. on Rn which is translation invariant (or K-stationary) in that (F (τh f ), F (τh g))L20 (P ) = (F (f ), F (g))L20 (P ) for all h ∈ Rn where (τh f )(x) = f (x + h). Then there is an orthogonally valued random measure Z : B0 (Rn ) → L20 (P ) tempered relative to a measure σ (i.e., ρ(·, ·) concentrated on the diagonal of R2n ) such that one has: Z F (f ) = fˆ(t) dZ(t), f ∈ Φ, (15) Rn

where fˆ is the Fourier transform of f ∈ Φ. Conversely every g.r.f. F (·) given by (15) is weakly (or K-)stationary on Φ. Proof. The new property to be established here is that, under the translation invariance condition, we need to deduce that the kernel k(t, x) is the exponential eit·x . First observe that, if F (·) is given by (15) where Z(·) has orthogonal increments, (with values in L20 (P )) then the covariance bilinear form is given by Z fˆ(t)g¯ˆ(t) dµ(t) B(f, g) = (F (f ), F (g))L2 (P ) = 0

Rn

132

4 Random Measures in Hilbert Space: Specialized Analysis

where µ(A) = (Z(A), Z(A)), A ∈ B0 (Rn ), and µ is a tempered measure. In this case that B(τh f, τh g) = (F (τh f ), F (τh g)), is a consequence (nontrivial result!) of the fundamental kernel theorem of L.Schwartz’s, and the details are spelled out in Gel’fand and Vilenkin (1964, Chapter II, Sec. 3.5) when f, g ∈ K, and the extension to Φ follows from the work of this same reference. We therefore omit the reproduction of details, referring the reader to this volume and prove the other direction here under our somewhat more general conditions. We now need to show that the kernel k(·, ·) is the exponential, by reducing it to the Bochner characterization of positive definite functions. Thus let f, g ∈ K (we can restrict to K, which is dense in Φ) and suppose that B(τh f, τh g) = B(f, g) = (F (τh f ), F (τh , g)) as above without assuming a priori that R F (·) is K-stationary, but F (·) satisfies the integral representation F (f ) = R f˜(t)dZ(t), where f˜ is the ‘k-transform’ of f. Now using the fact that Z(·) takes orthogonal values, one has after a rearrangement Z Z Z k(t, x)k(t, y) dσ(t)] dx dy u(x)v(y)[ Rn Rn Rn Z Z Z ¯ k(t, x − h)k(t, y − h) dσ(t) dx dy (16) = u(x)v(y) Rn

Rn

Rn

for all u, v ∈ K. Since all are Lebesgue integrals here, we can identify the expressions in [ ] for almost all (x, y). If σA (·) = σ(A ∩ ·), the restriction n of R σ to B0 (A) for A ∈ B0 (R ), which is a finite measure, let WA (x, y) = ¯ k(t, x) k(t, y) dσ (t), and observe that WA (·, ·) is positive definite. In fact, A Rn for ai ∈ C, n X

i,j=1

ai a ¯j WA (xi , xj ) =

Z

n X

¯ yj ) dσA (t) ai , a ¯j k(t, xi )k(t,

Rn i,j=1

2 n X ai k(t, xi ) dσA (t) ≥ 0. = Rn Z

i=1

Since WA (·, ·) is (Lebesgue) measurable, it follows from (16) that on taking h = y there, WA (x, y) = WA (x − y, 0), x, y ∈ Rn . Hence WA (·, 0) being a positive definite measurable function, by the classical Bochner-Riesz theorem we must have Z eit·(x−y) dσA (t), (17) WA (x − y, 0) = Rn

uniquely for a·a·(x, y) ∈ R2n . Since σ(·) is tempered it follows that (16) reduces to

4.2 Local Classes of Random Fields and Related Measures

B(u, v) =

Z

Rn

= =

Z

ZR

n

Z

u(x)¯ v (u)

Rn

Z

Rn

Z

133

eit·(x−y) σA (t) dx dy

Rn

Z u(x)eit·x dx

u ˆ(x)ˆ v (x) dσ(t),

Rn

v(u)eit·y dy dσA (t)

Rn

where σ|B0 (A) = σA , A ∈ B0 (Rn ). This implies (15) and B(·, ·) is translation invariant. 2

4.2 Local Classes of Random Fields and Related Measures The preceding section motivates several questions to be answered if the random measures Z are defined on (Borel) δ-rings of Rn regarding their local structure, or the structure of higher order increments of the corresponding (stochastic) integrals related to the generalized random fields (g.r.f.’s). Particular interest here is to analyze the local structure of g.r.f. of respective local classes of K-stationary, (weakly) harmonizable and even of class (C) fields. This implies that the representing random measures must satisfy some supplementary conditions in addition to those of Theorem 4.1.3 and its consequences. These will be analyzed after formulating the concept “mth order increment fields”, following the basic analysis of such for K-stationary classes due to A.M. Yaglom (1957). Note that if {Xt , t ∈ Rn } is a random field and τh′ : Xt 7→ Xt−h (i.e., τh′ f )(x) = f (x − h)) is a (backward) shift of h units, then the new process or field {Yt = Xt − Xt−h = (I − τh′ )Xt , t ∈ Rn } is the increment or “a local” field for each such h ∈ Rn . Motivated by this observation, we introduce the desired general concept as follows [before shift is denoted τ−h = τh′ , but the change is just convenient to explain]: Definition 4.2.1 Let the Schwartz space K ⊂ Φ be a dense subspace as beR fore and Φ0 = {f ∈ Φ : Rn f (x) dx = 0}. Then a g.r.f. F : Φ → L20 (P ) is locally of class (C) with Th = I − τh′ (τh′ being a ‘shift’ as above) and Φh = {Th f : f ∈ Φ}(⊂ Φ0 ), h ∈ Rn , the functional Fh = F |Φh , the restriction of F to Φh , is locally of class (C) [weakly or strongly] relative to some tempered covariance bimeasure ρ in the sense of Definition 4.1.2 above. [Thus Fh is simply of class (C) on Φh for each h.] Similarly F is locally harmonizable [respectively weakly or strongly] if k(t, x) = eit·x in the above, and it is locally K-stationary (or homogeneous) if k is again the exponential, and the tempered measure ρ concentrates on the diagonal of Rn × Rn . We shall now characterize these classes in terms of their bilinear covariance functionals as was done for the general classes in the preceding section and

134

4 Random Measures in Hilbert Space: Specialized Analysis

obtain integral representations for these local classes. [If ∆h Xt = τh′ Xt , we can define inductively ∆nh Xt = ∆h [∆n−1 Xt ], for the nth order local or increment process or field.] One of the reasons to study local properties, as set forth in the above definition is seen from the following classical example. Let {Xt , t ∈ R} be a Kstationary process and X0 , an arbitrary random variable. If {Yt = X0 + Xt , t ∈ R} then the Yt -set is not K-stationary, if |E(X0 )| = +∞ for instance, but letting Zt = Yt − Yt0 , t ∈ R, and Yt0 is fixed so that the local process {Zt , t ∈ R} is K-stationary, hence the local properties of {Yt , t ∈ R} can be studied. A more natural example is when {Xt , t ∈ R} is the Brownian motion which is not K-stationary . But now Xt − Xs is normally distributed with mean zero and variance c|t − s|, c > 0, so that it has stationary increments which are centered normally distributed with variance equaling the length |t − s| if c = 1 above. Thus one can analyze and derive valuable information by studying the increments (or local) process. Note however that if the process is a sequence, then Yn = Xn+1 − Xn is stationary implies that the original {Xn , n ≥ 0} will be stationary, and the local analysis is unnecessary. Thus the continuous parameter (or non-discrete) processes are of interest for our analysis, and as noted already, Yaglom’s (1957) work is the basic study and new points emerge when it is considered from the g.r.f. (generalized random fields) point of view. Thus let F : K → L2 (P ) be a g.r.f. on the Schwartz space K on Rn and then (ϕ) ∂ϕ ∂m ϕ ∂mF m define ∂F ∂xj = −F ( ∂xj ) and similarly ∂xm (ϕ) = (−1) F ( ∂xm ) for ϕ ∈ K, each j

j

ϕ with a compact support and is infinitely differentiable, (the minus signs comR ing from integration by parts applied if F (ϕ) = Rn Yt ϕ(t)dt and the general case is abstracted from this property). If F (τh ϕ) = F (ϕ) so that F is translation invariant where (τh ϕ)(x) = ϕ(x + h), h, x ∈ R, then M (ϕ) = E(F (ϕ)), ϕ ∈ K is a continuous linear functional and M (τh ϕ) = M (ϕ) is also translation invariant. Then by Riesz’s classical representation one has for a unique Radon measure µ on Rn , Z Z ϕ(x + h) dµ(x) ϕ(x) dµ(x) = M (τh ϕ) = M (ϕ) = n Rn ZR ϕ(x)d(µ ◦ τh−1 )(x), ϕ ∈ K. (1) = R

Hence µ ◦ τh−1 = µ, a translation invariant measure on Rn and then (by uniqueness of its ‘Haar property’) µ = αλ, α ∈ R, which is a constant multiple of the Lebesgue measure λ on Rn . Hence Z ϕ(x)dλ(x), ϕ ∈ K. (2) M (ϕ) = α Rn

Considering the m

th

derivative of M (·) one has

E(F (m) (ϕ)) = (−1)m E(F (ϕ(m) )) = (−1)m M (ϕ(m) ) = M (m) (ϕ),

(3)

4.2 Local Classes of Random Fields and Related Measures

135

where M (m) is the ‘distributional derivative’ of M. Thus if F (·) is locally stationary so that (2) implies Z ′ M (ϕ) = α ϕ(t)dλ(t) (4) Rn

or M ′ = α and its formal solution is M (ϕ) = (αt, ϕ) = α n = 1, and in the general case it is a vector given at ϕ as: M (ϕ) =

n X

k=0

αk

Z

tk ϕ(t)dλ(t) =

R

n X

R

R

ak (t(k) , ϕ)

tdλ(t) in case

(5)

k=0

t(k) being the k th partial derivative. A similar argument applies for higher order moments of a stationary g.r.f., but it is not as simple. For instance the bilinear form is given as : ′ B(ϕ(m) , ψ (n) ) = E(F (ϕ(m) )F¯ (ψ (m ) )), ϕ, ψ ∈ K,

(6)

and using the fact, established in Section 1, that in the stationary case, where M (ϕ) = 0 is stipulated but one can consider F (ϕ) − M (ϕ) in the argument which will be assumed in this illustration here, Z ¯ˆ dσ(λ), (7) ϕ(x) ˆ ψ(x) B(ϕ, ψ) = E(F (ϕ)F (ψ)) = Rn −{0}

for some (unique) tempered measure σ on the δ-ring of bounded Borel sets of Rn − {0}. Replacing ϕ, ψ by ϕ(n) , ψ (m) the mth order partial derivatives, (7) becomes using (6) under local stationarity to get the covariance functional of mth order as: B(ϕ(n) , ψ (m) )    Z m−1 m−1 j j X X λ λ ϕ(λ) ϕˆ(j) (0)  ψˆ − β(λ) = ˆ − α(λ) ψˆ(j) (0)  dσ(λ) j! j! n R −{0} j=0 j=0 +a2n αn β¯n +

m−1 X j=0

αj ℓ¯j (ψ) +

m−1 X j=0

β¯j ℓj (ϕ) +

m−1 X m−1 X

cjk αj β¯k

(8)

j=0 j=0

where α(·), β(·) are entire analytic functions on Rn , α(λ) − 1 and β(λ) − 1 have a zero of order m − 1 at λ = 0, αj = (tj , ϕ), βj = (tj , ψ), a2n ≥ 0, cjk ∈ R, 0 ≤ j, k ≤ n − 1, ℓj : K → C are linear functionals, and √ αj = i−j ϕˆ(j) (0), βj = i−j ψˆ(j) (0). (Here i = −1.) As is common in the multivariate problems, j! = j1 ! . . . jn !, and |j| = j1 + · · · + jn ≤ k − 1 in the k-differential situation. These expressions are computed using Fourier transforms of functions from K. The complicated notation is unavoidable, although the ideas are generalizations of the familiar case n = 1 and order k = 1. The

136

4 Random Measures in Hilbert Space: Specialized Analysis

necessary detailed computations can be found in Gel’fand and Vilenkin(1964, Section 3.3). We omit them here and discuss increments of order one, calling them local g.r.f.’s for convenience. The preceding discussion leads us to present a locally stationary field, following Yaglom (1957), concentrating on the random aspect. We can then consider the class (C) fields to complete the discussion at hand. Theorem 4.2.2 If F : K → L20 (P ) is a locally stationary g.r.f. with covariance functional B(ϕ1 , ϕ2 ) = E(F (ϕ1 )F¯ (ϕ2 )) so that B(τh ϕ1 , τh ϕ2 ) = B(ϕ1 , ϕ2 ), ϕi ∈ K, h ∈ Rn , then B is representable as: Z (9) B(ϕ1 , ϕ2 ) = ϕˆ1 (x)ϕ¯ˆ2 (x)dG(x) + A(∇ϕˆ1 )(0) · (∇ϕˆ2 )(0), Rn −{0}

¯ + is a tempered measure of order p > 0, (i.e., where G : B0 (Rn ) → R R |x|2 dG(x) 1+p < ∞), and A = (aij ) is a constant positive definite n × n herRn (1+|x|2 )

2

∂ mitian matrix, ∇ = ( ∂x , . . . , ∂x∂ n ) being the gradient operator on Rn with the 1 ‘dot product’ notation. Conversely a bilinear functional given by (9) is the covariance functional of some locally stationary random field F : K → L20 (P˜ ), on some probability space ˜ Σ, ˜ P˜ ), representable as: (Ω, Z ϕ(λ)dZ(λ) ˆ + iA · (∇ϕ)(0), ˆ ϕ ∈ K, (10) F (ϕ) = Rn −{0}

where A is an n-vector of random variables from L20 (P˜ ) such that A and Z(A) are uncorrelated for each A ∈ B0 (Rn − {0}), Z(·) is a random measure of orthogonal values and ∇ is the gradient operator as defined above. Proof. We include the essential ideas of proof so that the result can be extended to the local g.r.f.’s of class(C) whose format can be better appreciated. The key point is to reduce the local analysis to the general (or global) case on a suitable subspace for an infinite collection (or an infinite vector) of g.r.f.’s and apply the known theory. Here are the general ideas to be followed. For each h ∈ Rn , let Fh (f ) = F (Th f ) = F (f − τh f ), and we analyze Fh : K → L20 (P ). If r1′ , r1′′ ∈ Rn , then τr′ τr′′ = τr′ +r′′ (= τr′′ τr′ ). Using this commutativity and the relation between Fr (·) and F (·) we have: Fr1′ +r1′′ (f ) = F (f ) − F (τr1′ (τr1′′ f ))

= Fr1′ (f ) + Fr1′′ (τr1′ f ) = F (f ) − F (τr1′ f ) + F (τr1′′ (τr1′ f ))

= F (f ) − F (τr1′ f ) + F (τr1′ f ) − Fr1′ (τr1′′ f )) = Fr1′′ (f ) + Fr1′ (τr1′′ f ).

(11)

4.2 Local Classes of Random Fields and Related Measures

137

A similar computation shows Fr1′ +r1′′ (f ) = Fr1′ (f ) + Fr1′′ (τr1′ f ).

(12)

Since F (·) is a stationary g.r.f., so the same is true of Fr′ +r′′ (·), we have its covariance bilinear functional representable as in (7) for a suitable (unique) tempered measure depending on the parameters r1′ and r1′′ of the formulations (11) and (12). Since ((τˆA f )(λ) = e−irλ fˆ(λ), we get from (7), (11) and (12) the following key representation for any r1′ , r1′′ , r2 ∈ Rn where we write the tempered measures as σ(·; r1 , r2 ), r1 , r2 ∈ Rn : (λr is now λ · r for vectors below) Z fˆ1 (λ)ϕˆ2 (λ)σ(dλ; r1′ + r1′′ , r2 ) n R −{0} Z ′ fˆ1 (λ)fˆ2 (λ){σ(dλ; r1′ , r2 ) + e−iλr1 σ(dλ; r1′′ , r2 )}by (12), (13) = n R −{0} Z ′′ = fˆ1 (λ)fˆ2 (λ){σ(dλ; r1′′ , r2 ) + e−ir1 λ σ(dλ; r1′ , r2 )}by (12). (14) Rn −{0}

Since f1 , f2 ∈ K are arbitrary and fˆ1 , fˆ2 ∈ L2 (σ) the left side of (13) and (14) being equal, we can identify the right sides and their measures because the products fˆ1 · fˆ2 separate points of respective spaces. This gives an important relation for A ∈ B0 (Rn − {0})1 , r2 ∈ Rn : Z Z ′ −iλr1′′ ′ ′ (15) (1 − e )σ(dλ; r1 , r2 ) = (1 − e−iλr1 )σ(dλ; r1′′ , r2 ). A

A

We now apply the same argument to Fr2 (·) in (11) and (12) to obtain: Z Z ′′ ′′ ′ ′ (1 − e−iλr1 )(1 − eiλr2 )σ(dλ; r1′ , r2′′ ) = (1 − e−iλr1 )(1 − eiλr2 )σ(dλ; r1′′ , r2′′ ). A

A

(16) Since r1′ , r2′ , r1′′ , r2′′ can vary independently, these define a unique tempered measure σ : B0 (Rn − {0}) → [0, ∞] by the equation Z σ(dλ; r1 , r2 ) , (17) σ : A 7→ −iλr1 )(1 − eiλr2 ) A (1 − e

which is independent of r1 , r2 where we need to separate the point 0 ∈ Rn so that the integral in (17) is well defined. From this we may deduce that B(f1 , f2 ) = E(Fr1 (f1 )Fr2 (f2 )) Z ¯ = fˆ1 (λ)fˆ2 (λ)(1 − e−iλr1 )(1 − e−iλr2 )σ(dλ) Rn −{0}

¯ + fˆ1 (0)fˆ2 (0)σ(0; r1 , r2 ).

(18)

138

4 Random Measures in Hilbert Space: Specialized Analysis

But now we use the fact that (r1 , r2 ) 7→ σ(0; r1 , r2 ) is a positive definite function, and the relations (13)and (14) imply that σ(0; ·, ·) is also a bilinear functional in the two variables. Consequently by another well-known Riesz theorem it can be expressed as σ(0; r1 , r2 ) = (A0 r1 , r2 ) where A0 is a hermitian positive definite matrix in Rn . To evaluate fˆ1 (0) and fˆ2 (0) in terms of r1 , r2 , we note that Tr f ∈ K1 , the subspace of K determined by these functions as f and r vary. Thus Z fˆ1 (λ)fˆ2 (λ)σ(dλ)+ A∇\ Tr2 f2 )(0) , Tr1 f1 (0), ∇(\ E(F (Tr1 f1 )F (Tr2 f2 )) = Rn −{0}

(19) which establishes (9). For the converse, it is immediate that the bilinear functionalRB(·, ·) given by (9) is positive definite and hermitian. Taking K1 = {f ∈ K : Rn f dx = 0} as the index set, the basic Kolmogorov existence theorem shows, with K as the index set, that there exists a probability space (can be taken to be Gaussian) and a random field {F (ϕ), ϕ ∈ K1 } having B(·, ·) as its covariance functional. The bilinearity of B(·, ·) then can be used to show that F (·) is linear and continuous in the topology of K1 . Since K1 is also dense in K, it is routine to verify that F : K → L20 (P ) satisfies (10). Here the random measure Z : B0 (Rn − {0} → L20 (P ) is constructed using the isomorphism of K into L2 (σ). The necessary detail is as follows. Given the bilinear functional, the tempered measure σ and a hermitian positive definite matrix A˜ satisfying (9) let {fn , n ≥ 1} be a sequence of elements of K1 that converge pointwise to χA , A ∈ B0 (Rn − {0}), and also in mean, i.e., B(fn , fn ) → B(χA , χA ) = σ(A). We employ the inner product in L2 (σ) given by (f, g)σ = B(f, g) of (9) and the completion of simple functions will determine this L2 (σ). We then let Z(A) ∈ L20 (P ) correspond to χA so that E(Z(A1 )Z(A2 )) = (Z(A1 ), Z(A2 ))L20 (P ) = (χA1 , χA2 )L2 (σ) and as usual extend this correspondence. It will determine (10) with the random measure Z : B0 (R − {0}) → L20 (P ), on the probability space given by the Kolmogorov existence theorem invoked before. Although the details of the above sketch are not entirely trivial, the computations are completely standard. Thus the following properties on the behavior of Z(·) at the origin are used. Since by construction Z({0}) does not yet have a meaning we take it as P [Z({0}) = 0] = 1 and to compensate for the second term of (9), let A = (a1 , . . . , an ), ai ∈ L20 (P ). Set E(Z(A)aj ) = 0 for all j and let their covariance (A′ , A)L20 (P ) = A˜ which is possible. In fact we can again ˜ take P them to be centered Pn Gaussian 2with covariance matrix A. Then for each n f = k=1 αk χAk + k=1 βk fk ∈ L (σ), with some real constants αk , βk and Ak ∈ B0 (Rn − {0}) one gets the mapping (with the dot product) Z f (λ)Z(dλ) + β · A (20) f→ Rn −{0}

4.2 Local Classes of Random Fields and Related Measures

139

where β = (β1 , . . . , βk ) ∈ Cn and A is the matrix given above so that this correspondence is an isomorphism. Hence taking f ∈ K1 ⊂ K, we then find Z f (λ)Z(dλ) + i A ·(∇(Td f ))(0), (21) F (f ) = Rn −{0}

where A is the vector of random variables in L20 (P ) to satisfy (20), and the differentiated transform (∇(Td f ))(0) corresponds to the constant vector β. Thus (21) is the desired representation (10). Once this is obtained the uniqueness follows from the fact that K1 and K are dense in L2 (σ). 2 There has been some simplification in the above demonstration because of the (local) stationarity. An extension of the above result is not difficult, although it needs some care to replace the Fourier transform here with the “k-transform” in the class (C) case, which should include the (weak and strong) harmonizable g.r.f.’s. We now sketch this extension to complete the set of ideas introduced here. It may be observed that for (weakly) harmonizable g.r.f.’s the above work extends with the exception that Z(·) will not have orthogonal values, but still k(t, x) = eit·x . With straight forward procedure using the (strict) MTintegration in place of the L-S procedure one can present the following correspondent of the above result: Theorem 4.2.3 Let F : K1 (⊂ K) → L20 (P ) be a locally harmonizable g.r.f. on K1 . Then there exists a tempered covariance (scalar) measure ρ : B0 (Rn − {0}) → C, a random measure Z : B0 (Rn − {0}) → L20 (P ) and a random vector A = (a1 , . . . , an ), E(Z(A)ai ) = 0, i = 1, . . . , n, such that for some p ≥ 0, Z Z |x||y||d2 ρ(x, y)| (22) p+1 < ∞ Rn −{0} Rn −{0} (1 + |x|2 )(1 + |y|2 ) 2 and

F (f ) =

Z

Rn −{0}

fˆ(t)dZ(t) + iA · (∇fˆ)(0), f ∈ K, i =

√ −1,

(23)

where ∇ = ( ∂∂x , · · · , ∂x∂ ), x = (x1 , . . . , xn ) is the gradient operator, and the n 1 dot product notation is again used in (23). Further the covariance functional B(·, ·) of F (·) is given by Z Z ∗ ˜ fˆ1 )(0), (∇fˆ2 )(0)), (24) B(f1 , f2 ) = fˆ1 (s)fˆ2 (t)d2 ρ(s, t) + (A(∇ Rn −{0}

Rn −{0}

with A˜ as the covariance matrix of A, and fˆ1 , fˆ2 are Fourier transforms of f1 , f2 ∈ K, the integrals being in (strict) MT-sense. On the other hand if a positive definite bilinear functional B satisfies (24) relative to a covariance bimeasure ρ satisfying (22) and a hermitian positive definite n × n matrix A˜ as in (24), then on some probability space (Ω, Σ, P )

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4 Random Measures in Hilbert Space: Specialized Analysis

there is an L20 (P ) valued g.r.f. F which is weakly harmonizable with the covariance bilinear functional B given by (24) relative to the strict MT-integral representation. The difference between this result and that of class(C) or Karhunen fields, is the function ‘k’ of the k-transform in lieu of the Fourier operation in which the kernel is k(t, x) = eit·x having many useful properties. The MT-integration is the same. In order to find a tempered covariance measure ρ, it will be necessary to restrict the kernel k(t, x), analogous to the exponential to develop local analysis of the class (C). Thus a possible condition to consider is to demand that ¯ y), t, x, y ∈ Rn . k(t, x − y) = k(t, x)β(t, (25)

This is clearly satisfied if k(t, x) = eit·x with |β(t, 0)| = 1 for all t. The condition (25) is strong enough that k(t, x) is close to an exponential, (the ‘character’ of Rn ), and the class (C) with this k may be some extension of the weakly harmonizable class. It will be of interest to detail the analysis for some group cases. This enables us to find the tempered measure generalizing (17) in case Fr (f ) = F (Tr f ), f ∈ K, Tr = I − τr , so that one has the k-transform giving again a tempered measure ρ as above from the relation: B(f1 , f2 ) = E(F (Tr1 f1 )F (Tr2 f2 )) Z Z ^ 2 = (T^ r1 f1 )(t)(Tr2 f2 )(s)d ρr1 ,r2 (t, s) Rn

(26)

Rn

and proceeding as in the previous case. Thus one gets the desired positive definite covariance bimeasure (with strict MT-integration) as: Z Z ρr1 ,r2 (dt, ds) ρ˜(A × B) = . (27) ¯ A B (1 − β(s1 r1 ))(1 − β(t1 , r2 ))

From this point on one can complete the calculations and obtain an analog of Theorem 4.2.3. We leave the tedious details to the reader.

Remark 3. From the above analysis, it is clear that Yaglom’s(1957) fundamental result on locally K-stationary fields, in the context of g.r.f.’s extends to weakly harmonizable g.r.f.’s where the (strict) MT-integration replaces the more familiar LS-procedure. There is also an aspect of isotropic g.r.f.’s and their local extension can be considered. This analysis of harmonizable fields is also nontrivial but useful for some applications as well as theoretical questions. This has been briefly discussed by the author (cf. Rao (1969) with some suggestions from A. M. Yaglom) and also see the review in Swift (1997).

4.3 Bilinear Forms and Random Measures In the preceding two sections it was seen that random measures into a Hilbert space, such as L20 (P ), induce bilinear forms and bimeasures that are positive

4.3 Bilinear Forms and Random Measures

141

definite. In the representation theorems the converse implication was also true in many cases. Here we present a general method based on Aronszajn (or Reproducing Kernel Hilbert) space techniques to show that it is possible to associate a Hilbert space with each positive bilinear form and produce a random measure that represents the given form. This will explain and motivate other ideas in the ensuing work. In Section 2.2, especially in Definition 2.2.5, we considered a bimeasure β : B0 (S) × B0 (S) → C where (S, B0 (S)) is a (topological) measurable space (with B0 (S)) as a δ-ring (of bounded Borel R Rsets) of S, and an integral in the (strict) Morse-Transue sense denoted as A B (f, g)β(ds, dt) for A, B ∈ B0 (S) and f, g bounded B0 (S)-measurable complex functions on S. Then the corresponding functional Z Z (f, g)β(ds, dt) (1) B : (f, g) 7→ B(f, g) = S

S

is a bilinear mapping (or form) on B(S) × B(S) → C where B(S) is the space of bounded B0 (S)-measurable scalar functions. It is clear that B(χA , χB ) = β(A, B) forA, B ∈ B0 (S), and the symbol in the right side of (1) is the (strict) MT-integral (the ‘star’ on it is dropped). This correspondence and (1) imply that the bilinear form B(·, ·) on B(S) and the bimeasure β on B0 (S) × B0 (S) are in one-to-one relation for simple functions. Using the dominated convergence property of the (strict) MT-integral one deduces from the (separate) continuity of the bilinear form B(·, ·) on B(S) × B(S) → C with the uniform metric, the key fact that B(·, ·) and β(·, ·) determine each other uniquely. Several important consequences result from this correspondence which we now discuss and connect two different aspects of analysis, due to S. Bochner (1956) and A. Grothendieck (1956) formulated independently and differently. Definition 4.3.1 Let (S, B0 (S)) be a measurable space where B0 (S) is a δring, and consider Z : B0 (S) → L20 (P ), a random measure into a Hilbert space of centered (for convenience) square integrable functions (or random variables) on a probability space. Then Z(·) is termed L2,2 -bounded relative to a (σ-finite) measure µ : B0 (S) → R+ , if there is an absolute constant C > 0 such that for each B0 (S)-simple function f : S → C one has Z Z |f |2 dµ (2) E(| f dZ|2 ) ≤ C R

Pn

S

S

Pn

where S f dZ = i=1 ai χAi , Ai ∈ B0 (S) disjoint, i=1 ai Z(Ai ) for f = this integral (the finite sum!) being well-defined since it clearly does not depend on the representation of f as a simple function. [We take µ(S) < ∞ for simplicity.] R From (2) it follows that the mapping f 7→ S f dZ is continuous on L2 (S, B0 (S), µ), or L2 (µ), into L20 (P ). If Z1 , Z2 are a pair of random measures

142

4 Random Measures in Hilbert Space: Specialized Analysis

each is L2,2 -bounded on L2 (µ1 ) and L2 (µ2 ) into L20 (P ), then the bilinear form R R B : L2 (µ1 ) × L2 (µ2 ) → L20 (P ) defined by B(f1 , f2 ) = E( S1 f1 dZ1 S2 f2 dZ2 ), then satisfies the inequality (C1 , C2 , K below are as C in (2)): Z Z 2 ! 2 ! 2 |B(f1 , f2 )| ≤ E f1 dZ1 E f2 dZ2 , by the CBS inequality, S1 S2 Z Z ≤ C1 C2 |f1 |2 dµ1 |f2 |2 dµ2 , by(2), S

S2

= Kϕ1 (f1 , f1 )ϕ2 (f2 , f2 ), fi ∈ L2 (µi ), i = 1, 2,

(3)

where ϕ1 , ϕ2 are quadratic forms on L2 (µ1 ), L2 (µ2 ) defined by the respective integrals. Since B : L2 (µ1 ) × L2 (µ2 ) → C is a continuous bilinear form also, it can be represented as (1) for a unique bimeasure β : B0 (S1 ) × B0 (S2 ) → C, so that we have, since B(Si ) ⊂ L2 (µi ), i = 1, 2, can be taken as a dense subspace, using (·, ·) as an inner product of L20 (P ), Z

S1

2 Z Z 2 f2 dZ2 f1 dZ1 , f1 (s1 )f2 (s2 )β(ds1 , ds2 ) = |B(f1 , f2 )| = S2 S1 S2 Z Z |f2 (s2 )|2 dµ′2 (s2 ), |f1 (s1 )|2 dµ′1 (s1 ) ≤

Z

S1

S2

(4)

by (3) where µ′i = Ci µi , i = 1, 2. (Thus Ci can be absorbed into µi .) This may be presented for a convenient reference as: Proposition 4.3.2 Let Zi : B0 (Si ) → L20 (P ), i = 1, 2, be a pair of L2,2 -bounded random measures relative to σ-finite measures µi : B0 (Si ) → R+ 0 . Then the associated bimeasure β : (A, B) 7→ (Z1 (A), Z2 (B)), (A, B) ∈ B0 (S1 )×B0 (S2 ) admits the integral inequality (4) [or equivalently the associated bounded bilinear form B(·, ·) satisfies the corresponding inequality (3)]. It is interesting to observe that a converse to the above result also is true where we just assume that a bilinear form B : S(S1 ) × B(S2 ) → C is bounded (for the uniform norm) implies that (3) holds relative to a pair of measures µ1 , µ2 and this involves further analysis. The result is based on a fundamental inequality due to A. Grothendieck (1956), the L2,2 -boundedness and this inequality were obtained by these authors independently in the same year! The analysis gains insight and generality if the tensor product notion is introduced with a “tensor norm γ” on the product Banach space, B(S1 )×B(S2 ). The resulting completion is denoted B(S1 ) ⊗γ B(S2 ) so that the bilinear form B becomes an element of its adjoint space, namely B ∈ (B(S1 ) ⊗γ B(S2 ))∗ , so that a suitable form of the (extended) Riesz representation can be invoked to obtain B as an integral (of MT-type). The key point now is to dominate the bimeasure by a pair of probability measures. It involves some new ideas and several approaches with

4.3 Bilinear Forms and Random Measures

143

or without tensor products are employed in the literature. Here we present a direct construction. The desired result is given by the following important result: Theorem 4.3.3 Let Si be a compact Hausdorff space and C(Si ) be the (Banach) space of continuous scalar functions and B : C(S1 ) × C(S2 ) → C be a bounded bilinear form. Then there exist regular measures µi : B(Si ) → R+ (B(Si ) being the Borel σ-algebra), i = 1 ,2, such that Z Z |f2 |2 dµ2 , fi ∈ C(Si ), i = 1, 2, (5) |f1 |2 dµ1 |B(f1 , f2 )|2 ≤ K S2

S1

where K > 0 is an absolute constant (which can be absorbed into µi or the µi can be taken as probability measures on B(Si )) and, the resulting K = KG > 0 is termed the Grothendieck which for real spaces is found to satisfy √ constant π −1 = 1.782 · · · (KG is the same for all dimen< K ≤ π(2 log(1 + 2)) G 2 sions and the above numerical values are due to J.L. Krivine, cf. Exercise 2). Proof. It is desired to establish (5), or equivalently that the bilinear form B(·, ·), is bounded on the unit balls of L2 (µi ), i = 1, 2. By homogeneity, it suffices to show that there is an absolute constant K > 0 such that sup{|B(f, g)| : kf k2 ≤ 1, kgk2 ≤ 1} ≤ K when f, g vary on the unit balls of a Hilbert space H and B(f, g) as a matrix, i.e., the resultP is reduced to the following: to show that for (aij , 1 ≤ i ≤ n, 1 ≤ j ≤ n), sup{| 1≤i≤m aij xi xj | : |xi | ≤ 1≤j≤n

1, |xj | ≤ 1, xi ∈ R} ≤ 1 ⇒ sup{|

n X

i,j=1

aij hXi , Xj i| : kXi k = 1, kXj k = 1, Xi , Xj ∈ H} ≤ K.

(6)

Thus it suffices to establish (6) which implies (5). A short but somewhat slick argument, due to Blei (1987), follows, (in (6) the inner product and norm are of H). Let RN be the space of sequences with finitely many non zero terms so that X = {xi ∈ R, i = 1, 2, . . .} ∈ RN and for X, Y ∈ RN , consider the real valued (finite) product A(·, ·) defined as : Y A(X, Y ) = (1 + xn yn ), X, Y ∈ RN , (7) n≥1

so that we have |A(X, Y )| ≤ e

P

n

log(1+|xn yn |)

≤e

P

n

|xn yn |

≤ ekXkkY k .

(8)

We attach a convenient sequence of uniformly bounded independent centered random variables Zn (can be Rademacher functions on [0, 1]) so that

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4 Random Measures in Hilbert Space: Specialized Analysis

E(Zn ) = 0, E(Zn2 ) = 1 and |Zn | = 1, a.e, n ≥ 1. (9) Q Let F (X) = n≥1 (1 + ixn Zn ), X = (xn , n ≥ 1) ∈ RN , so that F (X) is a bounded complex random variable satisfying the inequality, since |Zn | = 1, a.e., Y 2 |F (X)|2 ≤ (1 + x2n ) ≤ ekXk , a · e, by (8) and (9). (10) n

Hence by the independence of the Zn we have for X, Y ∈ RN , Y E[(1 + ixn Zn )(1 − iyn Zn )] E(F (X)F (Y )) = n

Y = (1 + xn yn ), by (9), n

= A(X, Y ).

Now for X, Y in the unit ball of RN , we have for n ≥ 1, X X n n ajk E(F (Xj )F (Yk )) , ajk A(xj , xk ) = j,k=1 j,k=1

Xj = (0, . . . , xj , 0, . . .) and Yk = (01 , . . . , yk , 0, . . .),   n X  ≤E ajk F (Xj )F (Yk )  j,k=1

≤ e, by the hypothesis and (6), (8), and (10). (11)

Now expand the product (7), to get X A(X, Y ) = 1+(X, Y )+· · ·+

n1 >n2 >···>nJ

xn1 .xn2 . . . xnJ yn1 yn2 . . . ynJ +· · · (12)

Let {EJ′ , J ≥ 2} be a partition of N and let WJ be a J-dimensional ordered set such that WJ = {n1 , . . . , nJ ) ∈ NJ : n1 > n2 > · · · nJ } which sets up a one-to-one correspondence between EJ′ and WJ , J ≥ 2 as: n ∈ EJ′ ↔ (n1 , . . . , nJ ) ∈ WJ .

Now for X ∈ RN , kXk ≤ 1, let ϕ(X) = (ϕ(X)n , n ≥ 1) ∈ RN defined as ϕ(X)n = xn1 . . . xnJ , n ∈ EJ′ , J = 2, 3, . . . so that

kϕ(X)k2 = ≤

X X

J≥2

(xn1 . . . xnJ )2

n∈EJ′

X 1 X ( x2n )J ≤ (e − 2) = δ 2 < 1. J!

J≥2

n≥1

(13)

4.3 Bilinear Forms and Random Measures

145

Let ϕδ (X) = 1δ ϕ(X), kXk ≤ 1 and observe that ϕδ (·) maps unit length vectors into similar vectors without increasing the length. Substituting ϕδ (X) into (7) we get for any X, Y of unit lengths using the inner products of RN , (X, Y ) = A(X, Y ) − 1 − δ 2 (ϕδ (X), ϕδ (Y )). (14) Substituting (X, Y ) for A(X, Y ) of (7) with (14) recursively we get for J ≥ 2,

(X, Y ) =

J X j=0

(−δ 2 )j [A(ϕjδ (X), ϕjδ (Y )) − 1] + (−δ 2 )J+1 (ϕJ+1 (X), ϕJ+1 (Y )), δ δ

(15) where ϕjδ (X) = ϕδ (ϕδ (. . . ϕδ (X) . . .), the j th iterate. Now letting J → ∞, (14) becomes for kXk ≤ 1, kY k ≤ 1, (X, Y ) =

∞ X j=0

(−δ 2 )j [A(ϕjδ (X), ϕjδ (Y )) − 1].

(16)

With this, the expression (6) is established, for, Xi , Yk ∈ RN because n n ∞ X X X l l 2l a [A(ϕ (X ), ϕ (Y )) − 1] a (X , Y ) ≤ δ j,k j jk i k δ δ k l=0 j,k=1 j,k=1 ≤

∞ X

δ 2l (e + 1) =

l=0

e+1 , by (13). 3−e

Hence (13), consequently (5), is established. 2 Another proof of this result will be outlined in the complements section (cf. Exercise 2) because of the importance of the inequality. Many applications of the inequality, and different proofs, have been given in the literature. One of the main reasons to consider Grothendieck’s inequality is to obtain the converse direction of Proposition 2 above, namely to derive the L2,2 -boundedness of random measures in a Hilbert space, if the result of Theorem 3 above is assumed. This is established by finding an integral representation of a bounded bilinear form, given the result of the preceding theorem and the geometry of Hilbert space. The following useful integral representation is essentially due to Ylinen (1993) and will be used to get the desired converse. Theorem 4.3.4 Let (Si , B0 (Si )), i = 1, 2, be measurable spaces and β be a bounded bimeasure on them (or equivalently B : B(S1 ) × B(S2 ) → C be a bounded bilinear form so that we have Z Z B(f, g) = f (s1 )¯ g (s2 )β(ds1 , ds2 ), (17) S1

S2

146

4 Random Measures in Hilbert Space: Specialized Analysis

as MT-integral). Then there exist Hilbert spaces H1 , H2 , vectors x ∈ H1 , y ∈ H2 spectral measures Ei : B0 (Si ) → Hi , i = 1, 2 (so Ei (A) is a self-adjoint projection which is just additive on disjoint sets) and a bounded linear mapping T : H2 → H1 such that, for A ∈ B0 (S2 ), B ∈ B0 (S1 ), β(A, B) = (E1 (A)T E2 (B)y, x)H2 = (T E2 (B)y, E1 (A)x)H2 .

(18)

Here if β(·, B), β(A, ·) are σ-additive then the Ei (·) are strongly σ-additive. Proof. First observe that for each bilinear form on B(S1 ) × B(S2 ) → C, B(Si ) being the (Banach) spaces of bounded (Borel or) B0 (Si )-measurable scalar functions, we can set β(A, B) = B(χA , χB ) and get a bimeasure and conversely a bimeasure β defines a bilinear form B(·, ·) through the MT-integral (17). Hence either can be used for the result here. Now the B(Si ) are complete (complex) algebras under the uniform norm. So by the well-known isomorphism theorem (cf., Dunford-Schwartz (1958), IV, 6.18) it is equivalent to C(S˜i ), the space of continuous (complex) functions on a compact Hausdorff space S˜i for which the Grothendieck inequality (6) applies. This means there exist additive set functions µi : B(Si ) → R+ such that Z Z 2 2 |g|2 dµ2 . (19) |f | dµ1 |B(f, g)| ≤ S2

S1

Using the representation (17) as well as the fact that β(·, ·) is a bimeasure, the µi can be taken to be σ-additive in (19), seen as follows. [This additional argument is needed since a regular (countably additive) measure on S˜i typically corresponds to a finitely additive set function on Si .] Thus letting µi as these preimages, decompose µi as µci + µpi the countably additive and purely finitely additive parts by the classical Yosida-Hewitt decomposition of µi . Hence for any fixed g ∈ B(S2 ), the mapping A 7→ R R χ gβ(ds A 1 , ds2 ) defines a signed measure on B0 (S1 ) (since β(·, B) and S1 S2 β(A, ·) are such), and hence has a finite (Vitali) variation, say ν, which is a measure. Then by another result of Yosida-Hewitt, the definition of purely finitely additive notion also implies (cf. again their paper Yosida-Hewitt (1952), p.50) that for each ǫ > 0, there exist an Aǫ ∈ B0 (S1 ) such that µp1 (Aǫ ) < ǫ and ν(S1 − Aǫ ) < ǫ. Hence we have Z Z 2 Z Z Z p 2 2 c |g|2 dµ2 f | dµ |χ ≤ f )gdβ (χ |f | dµ + A A ǫ ǫ 1 1 S2 S1 S2 S1 S1 Z Z ≤ |f |2 dµc1 + ǫkf k2∞ |g|2 dµ2 . (20) S1

R

S2

R

Let m(A) = S1 S2 χA gβ(ds1 , ds2 ), A ∈ B0 (S1 ), so that (first for simple functions and then generally) we get for all f ∈ B(S1 ) Z Z Z f dm, A ∈ B0 (S1 ), f ∈ B(S1 ). (χA f )gdβ(ds1 , ds2 ) = S1

S2

A

4.3 Bilinear Forms and Random Measures

Since ǫ ≥ 0 is arbitrary this gives for (20) with µ ˜1 (= µc1 ): Z

S1

Z

S2

2 Z f gdβ(s1 , s2 ) ≤

S1

2

|f | d˜ µ1

Z

S2

2

|g| dµ2 .

147

(21)

Now apply the same argument for the second integral bound in (21) where µ2 is replaced by its σ-additive part µ ˜2 . In order to establish (18) we show that β(·, ·) can be replaced by an inner product as desired there. Thus consider the Hilbert spaces Hi = L2 (˜ µi ), i = 1, 2 obtained above, so RthatR B(Si ) ⊂ Hi is a dense subspace. Then the mapping B : (f, g) 7→ S1 S2 f gdβ(s1 , s2 ), can be extended to an inner product on H1 × H2 . By the preceding analysis B extends to a bounded bilinear functional on this product Hilbert space. But by the classical Riesz theorem (cf.,e.g.., Riesz and Sz.-Nagy (1955), p.202) there exists a bounded linear operator T : H2 → H1 such that B(g, f ) = (T g, f ). If Ei : B0 (Si ) → L(Hi ), the space of bounded linear operators on the Hilbert space Hi , where E1 (A1 )f = χA1 f, E2 (A2 )g = χA2 g, A1 ∈ B0 (S1 ), A2 ∈ B0 (S2 ), and x = 1, y = 1 on S1 , S2 , one then gets the representation: β(A1 , A2 ) = B(χA2 , χA1 ) = (T E2 (A2 )y, E1 (A1 )x) = (E1 (A1 )T E2 (A2 )x, y), for Ai ∈ B0 (Si ), i = 1, 2. This implies (18). [See also Exercise 1 on extension of a bimeasure from a product ring to its determining σ-ring.] It remains to show that Ei (·) are strongly σ-additive. But this follows from the fact that β(A1 , ·) and β(·, A2 ) are σ-additive implying that Ei (·) are weakly (hence also strongly in the present case) σ-additive. 2 In case S1 = S2 in the above, we can take H1 = H2 in (5), replacing µ1 and µ2 there by µ = 12 (µ1 + µ2 ) if necessary and absorbing the constant into µ itself. In this case T : H → H in the above work. The result can then be stated for reference as : Corollary 4.3.5 Let (S, B0 (S)) be as in the theorem and β : B0 (S)×B0 (S) → C be a bimeasure. Then there is a Hilbert space H and a bounded linear mapping T : H → H and some x ∈ H such that β can be represented as: β(A, B) = (E(A)T E(B)x, x), A, B ∈ B0 (S), (22) where E(·) : B0 (S) → L(H) is a strongly (same as weakly) σ-additive projection measure. In particular if Z1 (·) = T (E(·))x and Z2 (·) = E(·)x, then the Zi : B0 (S) → H are random measures (when H is realized as L2 (P ) which is possible, since every Hilbert space can be realized as a subspace of L2 (P ) on some probability space (Ω, Σ, P )), we have β(A, B) = (Z1 (A), Z2 (B))H , A, B ∈ B0 (S), and both the Zi (·) are L2,2 -bounded.

(23)

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4 Random Measures in Hilbert Space: Specialized Analysis

Remark 4. It is interesting that from the above Proposition 2 and Corollary 5, we see that the L2,2 -boundedness and Grothendieck’s inequality imply each other. They both have separate extensions and applications of great interest. We shall discuss some of them in what follows. R R Since a bimeasure F : (A, B) 7→ A B dF (u, v) defines a (signed) measure on B0 (S) × B0 (S) if and only if the integral is in the Lebesgue-Stieltjes sense in which case it admits a Jordan type decomposition F = F1 − F2 + i(F3 − F4 )

(24)

where Fj , i = 1, 2, 3, 4 are non negative (regular) bimeasures, hence of finite Vitali variations, whence also of finite Fr´echet variations, it is natural to ask as to when can we present the corresponding decomposition for all bounded bimeasures, which elaborates the MT-integration developed in Section 2.2. We now show that the corresponding Jordan type decomposition for the Fr´echet case depends on the work presented above in this section, and the positivity of the components Fj in (24) is weakened and should be replaced by positive definiteness. We restate the concept for convenience. Definition 4.3.6 A bimeasure β : B0 (S) × B0 (S) → C is positive definite if for ai ∈ C and Ai ∈ B0 (S), we have n X

i,j=1

ai a¯j β(Ai , Aj ) ≥ 0,

or equivalently, if B : B(S) × B(S) → C is the (associated through MTintegration) bilinear form then it is positive definite if for each f ∈ B(S), Z Z B(f, f ) = f (s)f¯(s′ )dβ(s, s′ ) ≥ 0 S

S

Pn where the integral is in the MT-sense. [Taking f = i=1 ai χAi , ai ∈ C, Ai ∈ B0 (S), one obtains the equivalence of both concepts.] The following result is a converse to the last part of Proposition 2.2.6 (see also Theorem 2.2.7) and uses the above work (particularly the Corollary) and is essentially obtained differently by Ylinen (1993). It uses a form of (the easier part of) dilation of a Hilbert space valued vector measure to an orthogonally valued one on a super Hilbert space (cf., e.g.., Corollary 3.4.7. or Corollary 6.2. of Rao (1982)). Theorem 4.3.7 Let (S, B0 (S)) be a measurable space and β : B0 (S)×B0 (S) → C be a bounded bimeasure. Then it is positive definite iff there is a probability space (Ω, Σ, P ) and a random measure Z : B0 (S) → L2 (P ) inducing the bimeasure in the sense that β(A, B) = (Z(A), Z(B))L2 (P ) , A, B ∈ B0 (S).

4.3 Bilinear Forms and Random Measures

149

Moreover, there exists a super space L2 (P˜ ) ⊃ L2 (P ), an orthogonally valued Z˜ : B0 (S) → L2 (P˜ ) and an orthogonal projection Π : L2 (P˜ ) → L2 (P ) such ˜ ˜ ˜ that Z(A) = (Π Z)(A), or equivalently β(A, B) = (Π Z(A), Π Z(B)) L2 (P ) = (E(A)(T E(B))x, x)L2 (P˜ ) = (T E(B)x, E(A)x)L2 (P˜ ) , for some x ∈ L2 (P˜ ), where T : L2 (P˜ ) → L2 (P˜ ) is a positive (bounded) linear operator and E(·) is a spectral measure. [Taking x ‘cyclic’, the Z will be seen to be the same for all such elements x.] Proof. If β : B0 (S) × B0 (S) → C is positive definite, then with the Aronszajn (or Reproducing Kernel Hilbert) space construction one can associate a Hilbert space which can then be realized as a subspace of L2 (P ) on a probability space (Ω, Σ, P ) and a random measure Z : B0 (S) → L2 (P ), of not necessarily orthogonal valued, such that β(A, B) = (Z(A), Z(B))L2 (P ) as shown in an earlier theorem. However, every random measure Z into an L2 (P ) has an orthogonally valued dilation Z˜ onto a super Hilbert space L2 (P˜ ) as shown before on its construction and hence the representation ˜ ˜ β(A, B) = ((Π Z)(A), (Π Z(B)) L2 (P )

(25)

holds, where Π is an orthogonal projection on L2 (P˜ ) → L2 (P ). However, by Corollary 5 above there is a bounded linear operator T : L2 (P˜ ) → L2 (P˜ ) and a spectral measure E : B0 (S) → H = L2 (P˜ ) such that (22) holds. Since β(·, ·) is positive definite and E(·) is a contractive projection operator so that it is also self-adjoint, we conclude that β(A, B) = (T (E(B))x, E(A)x), x ∈ L2 (P˜ ),

(26)

and is positive if A = B ∈ B0 (S). This implies that T is a positive operator. The converse that β given by (24) is positive definite is immediate. Note that in obtaining (24), Grothendieck’s inequality intervenes through Corollary 5, and it is an essential step in that representation. 2 An immediate consequence of this result is the Jordan type decomposition of a bimeasure which is given as follows: Corollary 4.3.8 A bounded bimeasure β : B0 (S) × B0 (S) → C admits a representation as: β = β1 − β2 + i(β3 − β4 ) (27) where each βj (j = 1, . . . , 4) is a (bounded) positive definite bimeasure on B0 (S) × B0 (S). [The decomposition, as in the classical case, is clearly not unique.] Proof. By Corollary 5 above, the β admits a representation (22) with a bounded ∗ T −T ∗ + i( linear operator T : L2 (P ) → L2 (P ). Let T = T +T 2 2i ) = A1 + iB1 (say) so that A∗1 = A1 and B1∗ = B1 . Now we can express the self adjoint (bounded) − 1 ∗ transformations A1 , B1 as A1 = A+ 1 − A1 where A1 = 2 (|A1 | + A1 ) with ± − 1 2 2 |A1 | = A1 , and A1 = 2 (|A1 | − A1 ) so that A1 are positive definite. Similarly

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− + − for β1 a decomposition holds. Hence T = (A+ 1 − A1 ) + i(B1 − B1 ) where each component is a positive definite (bounded) operator. Substituting these for T in (24), and if the resulting bilinear forms are designated as βj , we get the decomposition (27) from the theorem. [Here |Ai | is the positive part of the self adjoint operator Ai .] 2 The preceding ideas and results can be applied with extensions to many problems of interest leading to several new directions. We indicate a couple of these possibilities in the next section.

4.4 Random Measures with Constraints In the preceding work we have been treating random measures on rings without any restrictions except that they are σ-additive into vector spaces based on a probability triple (Ω, Σ, P ), Z : B0 (R) → Lp (P ). The following natural constraint is based on the classical and elementary Poincar´e formula. Namely, if Ai ∈ B0 (R), i = 1, . . . , n, consider for the measure Z(·), the formula: ! n n X X [ Z Z(Ai ) − Z(Ai ∩ Aj ) Ai = i=1

i=1

+

X

1≤i<j≤n

1≤i<j
Z(Ai ∩ Aj ∩ Ak ) − · · · + (−1)n−1 Z(∩ni=1 Ai ). (1)

Suppose that n = 3 and A2 ∩ A3 = ∅ so that (1) becomes with the classical Boole’s equation Z(A1 ∪ A2 ∪ A3 ) = Z(A1 ∪ A2 ) + Z(A1 ∪ A3 ) − Z(A1 ).

(2)

Let ZA (B) = Z(A ∪ B) − Z(A), so that for any C ∈ B0 (R), B ∩ C = ∅, one has ZA (B ∪ C) = ZA (B) + ZA (C) and hence ZA : B0 (R) → Lp (P ) is σ-additive for a fixed A ∈ B0 (R) and p > 0. This “constrained” random measure ZA arises in applications of the following type investigated by I.M. Gel’fand and N.Ya. Vilenkin and others, taking ZA (·) to have independent values on disjoint sets so that it will have, for each given A ∈ B0 (R) a “restrictedly independent” valued measure. Let us elaborate the problem in order to have a better idea, by employing the powerful Fourier transform property. Consider the characteristic functional of ZA (·) : B0 (R) → Lp (P ) defined as: LA (χB1 ∪B2 ) = E(eiZA (B1 ∪B2 ) ), B1 ∩ B2 = ∅, Bj ∈ B0 (R) = E(e

iZA (B1 )

) · E(e

iZA (B2 )

(3)

) = LA (χB1 )LA (χB2 )

by the assumed independence of ZA (Bj ), i = 1, 2. Let ℓA (·) = log LA (·) which is well-defined since LA (·) never vanishes. (Here ℓA (·) denotes the principal

4.4 Random Measures with Constraints

151

value of the logarithm.) It then follows for any bounded Borel functions f1 , f2 with f1 · f2 = 0 that ℓA (f1 + f2 ) = ℓA (f1 ) + ℓA (f2 ). Such a functional is called local which is additive on functions of disjoint supports. Hence the functional LA (·) is given by LA (f ) = eℓA (f ) (4) and a characterization of the additive (not linear) functional ℓA (·) is of importance in applications of generalized random fields. The problem of characterization of the local functionals arose in that context. To appreciate the nontriviality of the problem in characterizing ℓA (·) of (4), let us recall the classical Riesz representation of continuous linear functionals ℓ : C0 (S) → X. Here S is locally compact and C0 (S) is the Banach space (under the uniform norm) of continuous functions vanishing at infinity, and X is a weakly sequentially complete Banach space (e.g., X = Lp (P ), 1 ≤ p < ∞). Then there exists a regular vector measure Z : B0 (S) → X so that Z ℓ(f ) = f (s) dZ(s), f ∈ C0 (S) (5) S

and kℓk = kZk(S), the right side denoting the semi-variation of Z on S and the integral is in the sense of Dunford-Schwartz. If ℓ(·) is only a local functional so that it is non-linear, the representation (5) undergoes a drastic change. When X = R(or C), the above result on Cc (S), the space of continuous scalar functions on S with compact supports, is the classical Riesz-Markov representation which is a relatively deep result. For local functionals we have the following result which is based on the (separate) researches of several authors and will be stated here for reference and to appreciate the problem since it is more involved than the linear case. Theorem 4.4.1 Let ℓ : Cc (S) → R be a local functional satisfying the following pair of continuity conditions: (i) (Sequential) if fn ∈ Cc (S), n ≥ 1 is a bounded convergent sequence (pointwise), then {ℓ(fn ), n ≥ 1} is Cauchy, (ii) (Uniform) for each ǫ > 0, and η > 0, there is a δǫ,η (= δ) > 0 such that for fi ∈ Cc (S), kfi k ≤ η, i ≥ 1, and kf1 − f2 k < δ implies |ℓ(f1 ) − ℓ(f2 )| < ǫ. Then ℓ is representable as: ℓ(f ) =

Z

S

Φ(f (x), x) dµ(x), f ∈ Cc (S),

(6)

where µ is a finite regular Borel measure on S and Φ(·, ·) is a kernel satisfying the following conditions (a)-(c): (a) Φ(0, x) = 0, and Φ(·, x) is continuous for a · a · (x) (in µ-meas.) (b) Φ(α, ·) is measurable for all α ∈ R,

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4 Random Measures in Hilbert Space: Specialized Analysis

(c) Φ(f (x), x) is bounded for a · a · (x), and for {fn , n ≥ 1} as in (i) {(fn (·), ·), n ≥ 1} is Cauchy in L1 (µ). Conversely, if the pair (Φ, µ) satisfy conditions (a)–(c) and ℓ(·) is defined by (b), then ℓ(·) is a local functional with properties (i) and (ii) given above. An (unavoidably) involved detailed demonstration of this result is contained in the author’s Measure Theory book (cf. Rao (2004), pp. 676- 684) and the reader is referred to it where the Riesz-Markov theorem was then deduced from this theorem. Remark 4.1 A significant difference between linear and local functionals on B(S) is that for an integral representation in the linear case the representing measure of the functional is the only unknown and it is determined uniquely under standard conditions. On the other hand in the local case one has to find both the kernel as well as the measure for (6) and uniqueness of representation cannot be expected without further restrictions. If B(S) is replaced by a space such as Lp (µ) on a fixed measure space (S, B0 (S), µ), then one can find conditions for the kernel that goes with the given functional. This important distinction should be noted for applications and integral representation theory. It is also observed that the probabilistic character, arising in random measures given by the Poincar´e relation (2), is transferred into a non-stochastic functional relation such as (3) via Fourier transforms. It is only then the local functional uniquely associated with Z(·) as in (4) becomes a key step in this analysis. The preceding analytical form gives a “L´evy-Khintchine type” representation of the Poincar´e additive random measure Z obeying (2). This is a consequence of the preceding theorem: Corollary 4.4.2 Let ZA : B0 (S) → L0 (P ) be a constrained random measure with independent values for each fixed A ∈ B0 (S) whose local functional ℓA (·) = log LA (·) = log E(eiZA (·) ), satisfies conditions (i) - (iii) of Theorem 1 above. Then the characteristic functional LA (·) of ZA (·) is representable as: Z LA (f ) = exp Φ(f (x), x) dµ(x) , f ∈ Cc (S), (7) S

for a kernel Φ : Cc (S) × (S) → C satisfying Φ(0, x) = 0 and the corresponding conditions (a) - (c) relative to a regular Borel measure µ : B0 (S) → C where S is a locally compact space. The pair (Φ, µ) is determined by LA (·), and is not necessarily unique although LA (·) is unique. [Note that both Φ and µ depend on A.]

In the converse direction, if (Φ(·, ·), µ(·)) are given and LA (·) is defined by (7) then the latter represents a characteristic functional if and only if it is positive definite and this holds when µ is positive, Φ is such that

4.4 Random Measures with Constraints

153

esΦ(f ) is positive definite for all s > 0. This follows from a renowned theorem of I. Schur where Φ(f )(x) = Φ(f (x)), x ∈ S. Thus the work leads to a rich area of stochastic analysis. For instance if S is an LCA group and Z(A) and Z(τn A) have the same distributions (or equivalently Fourier transforms) where τn is the translation operator on G, implying that this theory leads to a study of homogeneous, and, if S = Rn , also to isotropic, random fields; more generally fields with such increments (i.e. local properties). If the random measure Z : B0 (S) → Lp (P ), is not scalar valued, i.e. if Z : B0 (S) × Ω → C considered as Z(A, ω) = Z(A)(ω) is not scalar valued, then the Fourier transform method used in the above work for LA (·) is inapplicable. The purpose is to obtain a (vector) measure induced by the additive vector function or “additive or local operator” in that T : B0 (S) → X, T (A ∪ B) = T (A) + T (B) for A ∩ B = ∅, A, B ∈ B0 (S), or more conveniently T˜ : B(S) → X with T˜(f + g) = T˜(f ) + T˜(g) for f · g = 0, and then to construct a kernel Φ : B(S) × S → X so that Φ(f (x), x) ∈ X. This allows one to define the “integral” suitably so as to have a representation such as: Z T (f ) = Φ(f (x), x) ◦ µ(dx), f ∈ Cc (S). (8) S

The “integral” will be additive only for f1 , f2 having disjoint supports. Although this is quite similar in principle to that of (6), the detailed execution is involved. The problem will simplify somewhat if Cc (S) is replaced by Lp (µ0 ), p ≥ 1, (µ0 , is σ-finite) and X has the “Radon-Nikod´ yn property” so that the vector measure µ will have finite (Vitali) variation and is µ0 -continuous as well as the vector dµ exists. derivative dµ 0 Without assuming that T (0) = 0, an operator T : Cc (S) → X is termed a (Poincar´e) local mapping if for each g ∈ Cc (S), Tg (f ) = T (f + g) − T (g) is local so that Tg : Cc (S) → X is uniformly continuous on bounded sets and Tg (f + h) = Tg (f ) + Tg (h) for f · h = 0. Since T (χA ) = V (A), does not give V (·) as an additive set function, consider for each compact F ⊂ S and h ∈ R, the set P (F, h) = {f ∈ Cc (S) : 0 ≤ f ≤ h if h ≥ 0, and h ≤ f ≤ 0 if h < 0; and f |G = h for G ⊃ F, G open}. Partially order the functions in P (F, α), for fi ∈ P (F, α) as f1 ≺ f2 if supp(f1 ) ⊂ supp(f2 ). Define, for each h, µh (F ) = limf Th (f ) for local Th , the limit in the above ordering. It may be verified (non-trivially) that {µh , h ∈ R} is a family of regular contents (called “height contents”) extending to Borel (vector) measures which in turn determine a (regular) vector measure µ, with finite Vitali variation. These facts are not obvious and have been established by Friedman and Tong (1971). The aim is to obtain a representation such as (8) from which taking X = Lp (P ), 1 < p < ∞, and perhaps P separable, to establish an analog of the representation for Z : B0 (S) → X, as in Theorem 2.3.2 so that Z(·) will have an integral (non linear!) representation with a scalar measure-kernel and a random measure. Thus the open problem here is to obtain a (Poincar´e type) Z : B0 (S) → X, a local measure, an integral representation relative to a better behaving measure Z˜ so that one

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4 Random Measures in Hilbert Space: Specialized Analysis

has: Z(A) =

Z

˜ Φ(χA (x), x) dZ(x),

S

A ∈ B0 (S).

(9)

We leave this as an interesting well-posed problem that opens up a set of new research questions. Finally consider a second order random field {Xt , t ∈ T } with Xt ∈ L20 (P ) ¯ t ) representable as: and covariance r(s, t) = E(Xs X Z Z ∗ g(s, u)¯ g (t, u′ ) dβ(u, u′ ), s, t ∈ T, (10) r(s, t) = S

S

for a class of (complex) functions {g(t, ·), t ∈ T } which are square (strict) M T integrable relative to a bimeasure β : B0 (S) × B0 (S) → C. As we know from Section 3.4, such a process (or field) admits a D–S integral representation of the form (S, T are locally compact spaces): Z g(t, u)dZ(u), t ∈ T, (11) Xt = S

relative to a random measure Z : B0 (S) → L20 (P ). Moreover, every such measure Z can be dilated to an orthogonally valued random measure Z˜ : B0 (S) → L20 (P˜ ) into a larger Hilbert space L20 (P˜ )(⊃ L20 (P )) and Z = V Z˜ where V : L20 (P˜ ) → L20 (P ) is an orthogonal projection onto L20 (P ), (cf. Theorem 3.4.6). Thus the dilated process, analogous to (11), is given by Z ˜ ˜ g(t, u)dZ(u), t ∈ T, (12) Xt = S

˜ provided {g(t, u), t ∈ T } is Z-integrable, in the sense of the D − S concept. This is always possible if g(t, ·) is bounded and satisfies (10), a condition which is valid for (weakly or strongly) harmonizable classes. If µ(·) is a (finite) mea˜ so that sure defined by the orthogonally valued (dilated) random measure Z, ¯ ˜ ˜ ˜ t , t ∈ T } will be a Karhunen µ(A ∩ B) = E(Z(A) Z(B)), A, B ∈ B0 (S), then {X 2 field relative to {g(t, ·), t ∈ T } on L (S, B0 (S), µ) which contains all bounded measurable functions. We state this result for a convenient reference (S, T are as above): Theorem 4.4.3 Let {Xt , t ∈ T } ⊂ L20 (P ) be a process (or a field) and {g(t, ·), t ∈ T } be a family of Borel functions on (S, B0 (S), µ), a σ-finite measure space. If the Xt is a Karhunen process (or a field) relative to this g(t, ·)-family (and measure µ), and if V : L20 (P ) → L20 (P ) is a continuous linear mapping, then {Yt = V Xt , t ∈ T } is a (weak) Cramer ´ process (or field) relative to the same g(t, ·)-family in that Yt ∈ L20 (P ) and Z Z ∗ ¯ g(s, u)¯ g(t, u′ )dβ(u, u′ ) (13) r(s, t) = E(Ys Yt ) = S

S

4.4 Random Measures with Constraints

155

˜ ˜ Z(B)). as an sM T -integral and a bimeasure β(·, ·) : (A, B) 7→ E(V Z(A)V Conversely, if {g(t, ·), t ∈ T } ⊂ L2 (β), satisfying (13) and the g(t, ·) are bounded so that {Xt , t ∈ T } ⊂ L20 (P ) is a (weak) Cram´er process (or field) relative to this g(t, ·)-family and the bimeasure β, then there is an extension space L20 (P˜ )(⊃ L20 (P )) determined by the given X-process, a Karhunen process (or field) {Yt , t ∈ T } relative to the same g(t, ·)-family and a finite Borel measure on (S, B0 (S)) such that Xt = V Yt , t ∈ T where V : L20 (P˜ ) → L20 (P ) is the orthogonal projection with range L20 (P ). Remark 5. This result implies that the class of all (weak) Cram´er fields is closed under bounded linear transformations, and contains all Karhunen fields. The converse implication by dilation apparently is valid only for a large (sub) class of weak Cram´er fields including all weakly harmonizable fields. This is because a (non-bounded) function integrable relative to a general vector measure, in a Hilbert space may fail to have the integrability property relative to the dilated measure on the super space! (See also Exercise 3.5.10 for related information.) An operator representation of a Cram´er process, analogous to that for the Karhunen class given as Theorem 3.4.9, will now be presented as the last item of this chapter. We note that for each Borel function f : R → R and a self-adjoint operator A : H → H (Hilbert space) if fn = f on [|x| ≤ n] and = 0 otherwise (so fn is a truncation of f of absolute height n) let the bounded operator fn (A) be defined through the spectral theorem, and if Df (A) = {x ∈ H : limn fn (A)x exists}, then define f (A) with domain Df (A) as f (A)x = limn fn (A)x, x ∈ Df (A) . This formulation is used in establishing the following: Theorem 4.4.4 Let {Xt , t ∈ R} ⊂ L20 (P ) be a (weak) Cram´er process relative to a class {g(t, ·), t ∈ R} of (scalar) bounded Borel functions, with g(0, u) = 1, u ∈ R. Then there exists a super Hilbert space L20 (P˜ ) ⊃ L20 (P ) and an element Y0 ∈ L20 (P˜ ) a linear (possibly unbounded) operator A : L20 (P˜ ) → L20 (P ) ˜ is symmetric with domain D(A) ˜ dense such that A restricted to L20 (P ), say A, 2 in the subspace sp{Xt , t ∈ R}(⊂ L0 (P )), such that Xt = g(t, A)Y0 , t ∈ R.

(14)

Here g(t, A) is defined, as noted prior to the statement of this theorem. Conversely, if A is a symmetric and densely defined linear operator on L20 (P ) (into itself ), X0 ∈ L2 (P ) and the g(t, ·)-family is as above then the process Yt = g(t, A)X0 , t ∈ R is a (weak) Cram´ R er process relative to the g(t, ·)-family ˜ and a random measure Z so that Yt = g(t, λ)dZ(λ). R

Proof. We present a quick sketch as the result has interesting consequences. So if the Xt -process is of Cram´er type relative to a bounded g(t, ·)-family, then as seen in the preceding result, there is a super space L20 (P˜ )(⊃ L20 (P )) and a Karhunen process {Yt , t ∈ R} in it relative to the same g(t, ·) family and

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4 Random Measures in Hilbert Space: Specialized Analysis

a measure which must be finite since g(0, ·) = 1. Then the Karhunen process admits an operator representation as: ˜ (0), t ∈ R, Yt = g(t, A)Y

(15)

where A˜ is an unbounded self-adjoint operator with dense domain in the subspace spanned by the Yt process in L20 (P˜ ). (See also Getoor (1956) Theorem ˜ is defined by using the spectral theorem for A, ˜ so 3A). Here g(t, A) Z ˜ (0) = ˜ g(t, A)Y g(t, λ)dE(λ)Y (0), t ∈ R, (16) R

˜ ˜ Let Z(·) ˜ where {E(λ), λ ∈ R} is the resolution of the identity for A. = ˜ E(·)Y0 which qualifies to be our random measure (with orthogonal values), to get Z ˜ g(t, λ)(Q · E)(dλ)Y (0) = g(t, A)Y (0), (say). (17) Xt = QYt = R

R

˜ which has positive increments but are not Here A = R λdE(λ), E(λ) = QE(λ), necessarily projections. Such {E(λ), λ ∈ R} is also called a generalized spectral family. Then it is known that A | L20 (P ) (the restriction) is a symmetric densely defined operator of the desired type. Note that g(0, ·) = 1 implies g(0, A) = Q and X(0) = QY (0). The converse depends on Na˘ımark’s dilation argument. Thus if A is symmetric and densely defined, then by the stated Na˘ımark’s theorem it extends to a self-adjoint operator A˜ on a super space L20 (P˜ ) ⊃ L20 (P ) such that A = V A˜ ˜ where V is an orthogonal projection onto L20 (P ), giving g(t, A) = V g(t, A). 2 ˜ ˜ Since {Yt = g(t, A)Y0 , t ∈ R} is a Karhunen process in L0 (P ) by our earlier work (cf. Sec. 3.4) we conclude that {Xt = V Yt , t ∈ R} is a (weak) Cram´er process relative to the (bounded) g(t, ·)-family. 2 We have an immediate consequence of this result as: Corollary 4.4.5 Each vector measure ν : B0 (R) → L20 (P ) is derived from a generalized spectral family. R Proof. Indeed Xt = R eitλ dν(λ), t ∈ R, defines a weakly harmonizable process and by the above theorem there is a super space L20 (P˜ ) ⊃ L20 (P ) and a self-adjoint operator A˜ on it such that with g(t, λ) = eitλ , one has: Z Z ˜ ˜0 , t ∈ R, ˜X ˜0 = eitλ (V E)(dλ) X (18) eitλ dν(λ) = Xt = V g(t, A) R

R

˜ ˜ 0 is an element where {E(λ), λ ∈ R} is the resolution of the identity for A˜ and X ˜ in L20 (P˜ ). If {E(λ) = V E(λ), λ ∈ R}, it is a generalized spectral family and ˜ ν(·) = E(·)X(0), is the desired representation. 2 We end this specialized analysis here and include some complements to the work, in the final section of the chapter.

4.5 Complements and Exercises

157

4.5 Complements and Exercises 1. Extension of a bimeasure from a ring to a σ-ring involves additional argument than that of a signed measure, and is as follows. Let Σi be a ring and β : Σ1 × Σ2 → C, a bimeasure, i.e., β(·, B), β(A, ·) are σ-additive for each A ∈ Σ1 , B ∈ Σ2 . Then it has a unique bimeasure extension β˜ : Σ˜1 × Σ˜2 → C where Σ˜i = σ(Σi ), the σ-rings for i = 1, 2. [Since β need not determine a signed measure on the semi-ring Σ1 × Σ2 , a different argument is needed and here is a sketch. By Corollary 3.5, there is a Hilbert space H and a bounded linear operator T : H → H such that β(A, B) = (E(A)T E(B)x, y) for some x, y ∈ H and strongly σ-additive (orthogonal) projection measures Ei : Σi → H. If E(A × ˜ B) = E1 (A)E2 (B) then it is to be shown that β(A, B) = (E1 (A)E2 (B)x, x) has the properties of a spectral measure. For this consider with A˜ = A × B ∈ Σ1 × Σ2 , 1 1 1 βˆA˜ (x, y) = βA˜ (x + y) − βA˜ ( (x − y)) + iβA˜ ( (x + iy)) − iβA˜ ( (x − iy)) 2 2 2 ˜ and verify that βˆA˜ is a symmetric bilinear functional in x, y for each A. This holds for sets of the form A × B and the collection of sets for which the above equation holds is closed under differences and countable unions. Also |βˆ ˜ (x, x)| = |βˆ ˜ (x, x)| ≤ Kkxk2 , A˜ = A × B ∈ Σ1 × Σ2 . A

A

Next show that this holds if A˜ is a set in Σ1 × Σ2 , and then for each such ˜ and finally verify that E(·) is βˆA˜ there is a bounded hermitian operator E(A) multiplicative by comparing (E(M ∩ N )x, y) and (E(M )E(N )x, y) for which a monotone class argument can be applied. Such methods are often used in Harmonic Analysis (cf. Loomis (1953), pp.93–94). The details are left to be completed by the reader. 2. The above sketch induces a key representation of bilinear forms as an inner product on a Hilbert space which uses Grothendieck’s inequality. This was established in Theorem 3.3. Because of its importance we present an alternative method here which is borrowed from Jameson (1987). It is again enough to establish the result in the discrete case. Consider a bilinear form (in the discrete case) as B = (aij ) on Rn × Rn to be bounded on the unit ball: X aij xi xj | : |xi | ≤ 1, |xj | ≤ 1} ≤ K < ∞, sup{| i,j

so that it is bounded on any (real) Hilbert space H, if we show: XX aij (xi , yj )| : kxi k ≤ 1, kyj k ≤ 1, xi , yj ∈ H} ≤ KG < ∞. sup{| i

j

The result follows from the key estimates obtained in three steps as:

158

4 Random Measures in Hilbert Space: Specialized Analysis x2

(i) If Fn = G1 ∗· · ·∗G1 , is the n-fold convolution of G1 : x 7→ √12π e− 2 , x ∈ R, R so that for x ∈ Rn , a, b ∈ Rn we have Rn ha, xihb, xidFn (x) = ha, bi, and if x, y ∈ Rn of unit length each, then let (x, y) = cos θ, 0 ≤ θ ≤ π so that sin−1 (x, y) = π2 − θ. If x = e1 , (basis vector), y = (cos θ, sin θ, 0, · · · , 0); let A = {u : (x, u), (y, u) ≥ 0}, B = {u : (x, u), (y, u) ≤ 0}, and C = {u : (x, u) R > 0, (y, u) < 0}, D = {u : (x, u) < 0, (y, u) > 0}. Verify that for µ(K) = K dFn (x), K ⊂ Rn , Borel, and with x, y as above, Z sgn (x, u) sgn (y, u) dFn (u) = µ(A) + µ(B) − µ(C) − µ(D), Rn

= 2(µ(A) − µ(C)),

(+)

and also if S ⊂ R2 is taken as the set S = {(r cos θ, r sin θ), r ≥ 0, α ≤ θ ≤ β}, then by Fubini’s theorem, Z 2 2 β−α 1 1 , µ(S × Rn−2 ) = e− 2 (x1 +x2 ) dx1 dx2 = 2π S 2π and hence letting S as {ϕ : θ − π2 ≤ ϕ ≤ π2 } for A, and { π2 ≤ ϕ ≤ θ + π2 } for θ π−2θ C, we get µ(A) = π−θ = π2 (π − θ) = 2π and µ(C) = 2π so that (+) becomes π −1 2 (x, y). This is needed. The evaluation gives ((x, u) being the inner π sin product): X aij sgn (xi , u) sgn (yj , z)| ≤ 1 | i,j

and integrating both sides relative to Fn and using (+) we get X π −1 a sin (x , y ) . ij i j ≤ 2 i,j

(ii) Next let Dn be the collection of the 2n vectors (±1, · · · , ±1) in Rn and if u ∈ Dn , then using the fact that kuk2 = 2n , we get 1 X (x, u)u = x. 2n

Hence

1 X (x, u)(u, y) = (x, y). 2n u∈Dn

If N = 2 , k ≥ 1, then Dnk has 2nk elements, and so RN can be identified P k as RDn with inner product (f, g) = 21n u f (u)g(u). Qk Let wt (x)(u1 , . . . , uk ) = i=1 (x, ui ), which defines a mapping wk : Rn → N R and satisfies (wk (x), wk (y)) = (x, y)k . (iii) The preceding (unmotivated) computations imply for a separable Hilbert space H consisting of vectors with finitely many non-zero terms, having nk

4.5 Complements and Exercises

159

RN (k) , N (k) = 2nk , identified as a subspace , gives with the preceding estimate (involving wk (x)′ s) when sin c(x, y) is expanded (for any c > 0), and the following: sin c(x, y) =

X

(−1)k−1

k

c2k−1 (w2k−1 (x), w2k−1 (y)). (2k − 1)!

P √ P √ Define u(x) = k ck w2k−1 (x), v(y) = k (−1)k−1 ck w2k−1 (y), where ck = P 2k−1 c 2 c2k (kxk2 )2k−1 = sinh(ckxk2 ), and similarly kv(y)k2 = (2k−1)! . Then ku(x)k = sinh(ckyk2 ), and (u(x), v(y)) = sin c(x, y). Choose xi , yi ∈ Rn such that kxk = kyk = 1 ⇒ ku(x)k = kv(y)k = 1. √ Hence the above work simplifies with c = sinh−1 1 = log(1 + 2), to the Grothendieck inequality: X π 1 X aij sin−1 (u(xi ), v(yj ))| ≤ . aij (xi , yj )| = | | c 2c i,j i,j 3. A bimeasure β : Σ1 × Σ2 → C is of finite Vitali variation |β|, if |β|(R, R) = sup{

n n X X i=1 j=1

|β(Ai , Bj )| : Ai ∈ Σi , Bj ∈ Σj , disjoint} < ∞.

Our analysis in Chapter 2 showed that a bimeasure is usually not of finite (Vitali) variation, but is of finite Fr´echet variation. As a consequence we have to use the M T -integration and not the Lebesgue-Stieltjes one. If β is assumed to have finite Vitali variation, then using Lebesgue’s integration and analysis, an extension of the scalar results when β takes values even in a Banach space can be considered and many (not all) product measure results (and Fubini type theorems) using Bochner’s vector integration extend, in contrast with the (nonabsolute) Morse-Transue integrals. Based on some work of Dinculeanu and Muthiah (2000), we present a result of this type in the present exercise to illustrate the distinction. (a) If β : Σ1 × Σ2 → C is of finite Vitali variation and β1 (·) = β(·, S2 ) is the (marginal) signed measure, verify that |β1 |(S1 ) ≤ |β|(S1 , S2 ), whence marginal variations are bounded by the total (Vitali) variation |β|(S1 , S2 ) and for them Lebesgue’s definition of integration suffices. (b) If fi : Si → and either βi -integrable or both β, fi ≥ 0, R R R C is Σi -measurable show that A 7→ A f dβ1 = A S f (ω1 )β(dω1 , dω2 ), defines a (signed) measure on (S1 , Σ1 ), and similarly for f2 and β2 . Show that in this case for any A ∈ Σ1 , and B ∈ Σ2 we have by the classical (L–S) procedure and the obvious notations Z f1 (ω1 )d(f2 β(ω1 , B)) A Z Z Z = f2 (ω2 )d(f1 β(A, ω2 ). = f1 (ω1 )f2 (ω2 ) dβ(ω1 , ω2 ). B

A

B

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4 Random Measures in Hilbert Space: Specialized Analysis

(c) Call a pair (f1 , f2 ), β-integrable, where β is a bimeasure of finite (Vitali) variation if the result of part (b) holds for all (A, B) ∈ Σ1 × Σ2 . Verify that the dominated convergence statement follows, i.e., if (f1n , f2n ) → (f1 , f2 ) a.e. (|β|) and |f1n | ≤ |g1 |, |f2n | ≤ |g2 | where (g1 , g2 ) is β-integrable, then we have Z Z Z Z lim (f1n , f2n )dβ = (f1 , f2 )dβ. n→∞

S1

S2

S1

S2

Many other results (e.g., change of variable formula) can be developed. 4. Here we indicate a self-adjoint dilation of a class of operators as in Theorem 4.4. Let A be a symmetric densely defined operator in a Hilbert space H, and {g(t, ·), t ∈ R} be a family of bounded Borel functions with g(0, ·) = 1. Then {Tt = g(t, A), t ∈ R} defines a set of bounded operators for which there is an extension (or super) Hilbert space K (containing H), a self-adjoint operator A˜ : K → K, extending A, such that Tt = ˜ where V is the orthogonal projection on K with range H. In parV g(t, A) ˜ = Ut is unitary and Tt = V g(t, A) ˜ = if g(t, λ) = eitλ , then g(t, A) Rticular, itλ ˜ e (V E)(dλ), defines a weakly continuous collection of positive definite conR tractions in H = V K. [The extension K is obtained using Na˘imark’s theorem. (cf., Sz.-Nagy (1955).) It implies Theorem IV of the latter paper proved differently and then deduced Na˘imark’s extension theorem. The above result shows that conversely the result is also obtainable from Na˘ımark’s theorem, as seen in Theorem 4.4.4., implying an equivalence of both results. Complete the details of this useful and a key statement following Theorem 4.4.4.] 5. An operator A : H → H (Hilbert space) is symmetric if (x, Ay) = (y, Ax) for all x, y ∈ H and this concept can be extended to Banach spaces X as follows. If A : X → X⋆ , the adjoint (or dual) space of X, then the linear mapping A is called symmetric on X if for any x, y ∈ X we have (Ay)(x) = (Ax)(y) or using the duality pairing (·, ·) from X to X⋆ , so that (x, Ay) = (y, Ax). A symmetric (always linear)operator T : X → X⋆ is called positive if (x, T x) = (T x)(x) ≥ 0 for all x ∈ X. In this definition X and X⋆ may be interchanged. Since |(x, T y)| = |(y, T x)| ≤ kxkkT yk, we note that {kT yk < ∞, y ∈ X}, since X is complete, and T linear, by the uniform boundedness theorem T is bounded, so that a symmetric operator is bounded. We can associate a bilinear form BT : X × X → R for a symmetric operator T : X → X⋆ , by the equation B : (x, y) 7→ (x, T y) which is continuous. Such operators T have the following interesting structure. With every positive symmetric T : X → X⋆ , one can associate a Hilbert space H and a continuous linear mapping A : X → H such that A(X) ⊂ H densely and T = A⋆ A. The factorization of T is essentially unique in the sense that T = A⋆1 A1 , (A1 : X → H1 ) with the same property implies that H and H1 are isometrically isomorphic. The representation allows us to extend many results on bilinear forms on a Hilbert space to the Banach space context. Establish this result using the following sketch: Set N = {x ∈ X : (x, T x) = 0}

4.5 Complements and Exercises

161

which is closed and linear. So H1 = X/N , the quotient space with the bilinear form (˜ x, y˜) = (x + N, y + N ) = (x, T y) is seen to be an inner product on H1 where x ˜ = x + N ∈ H1 and y˜ = y + N ∈ H1 . Let H be the completion of H1 under the above inner product. If Aζ : X → H so that Aζ x = x + N , then (Aζ x, Aζ x)H = (x, T x) ≤ kT k · kxk2 so Aζ is bounded and then (x, T y) = (Aζ x ˜, Aζ y˜) = (˜ x, A⋆ζ Aζ y˜) giving T = A⋆ζ Aζ . If T = A∗1 A1 is another representation then the mapping u : Ax 7→ A1 x for x ∈ X is shown to be kuAxk2H1 = (A1 x, A1 x)H1 = (x, A⋆1 A1 x)H1 = (x, T x) = kAxk2H and so A(X), Aζ (X), and A1 (X) can be identified. This observation on factorization is due to N.N.Vakhania given in his lectures in 1971, and several properties and applications are given in the book by Vakhania, Tarieladze and Chobanyan (1987). 6. We can obtain a form of bimeasures using Aronszajn and Kolmogorov methods of representation of a positive definite bilinear form as follows. Let Λ be an index set, to be taken as a σ-ring Σ below, and consider a positive definite form (or kernel) kP: Λ × Λ → C. Then let H P1n be the linear space of elements n inner product a k(λ ), and g = of the type f = i i,· j=1 bj k(λj,· ) with p Pn Pn i=1¯ (f, g) = i=1 j=1 aj bj k(λi , λj ). Let the norm be kf k = (f, f )H1 , and H the completion of H1 in this norm. Now take Λ = Σ and k(A, B) = β(A, B) where β is a positive definite bimeasure. By the classical Kolmogorov existence theorem there is a random measure Z : Σ → L2 (S, S, P ) on some probability space (S, S, P ) whose covariance bimeasure is β, i.e. β(A, B) = (Z(A), Z(B))L2 (P ) , A, B ∈ Σ. Verify that |β(A, B)|2 ≤ (E(Z(A)2 ) · (E(Z(B)2 ). If β˜ is the vector measure induced by β, i.e. β˜ : A 7→ β(A, ·), A ∈ Σ, and ˜ is its vector space of D − S integrable scalar functions on (Ω, Σ) if L1 (β) where Z : Σ → L20 (P ) is as above, then (by our work in Chapter 2), it follows ˜ = L1 (Z) and if L2 (β) is the completed space of f : Ω → C, where that L1 (β) (f, f ) is β-integrable (i.e. the space for which (f, f ) = kf k2 < ∞, the inner ˜ $ L2 (β). Thus the bounded biproduct in the Aronszajn space) then L1 (β) linear form B(f, g) such that B(χA1 , χA2 ) = β(A1 , A2 ) operates on the larger space L2 (β) on which the Grothendieck inequality is obtained. However, the above argument does not seem to produce all the theory with representation. Note that using Corollary 4.3.8 and a bounded bimeasure β through the representation (25) of Section 3 above one can establish (show this) the relation: β(A, B) =

4 X j=1

(−1)j−1 (Zj (A), Zj (B))L2 (P ) , A, B ∈ Σ,

on a sufficiently augmented probability space (S, S, P ), by the usual adjunction procedure if necessary. 7. Let {Xt , t ∈ T } ⊂ L20 (P ) be a random field where T is a general semi-group with identity. We present here a direct extension of the dilation result of random measures on an LCA group to an orthogonally valued one on a (super) Hilbert space containing the given space L20 (P ) as a subspace. Here we indicate an extension of it to an abelian fairly gen-

162

4 Random Measures in Hilbert Space: Specialized Analysis

eral semi-group. Consider the covariance r(s, t) = (Xs , Xt )L2 (P ) , s, t ∈ T . Then the function r (and the random field Xt itself) is called conservative if r(sh, th) = r(s, t), ∀s, t, h ∈ T , and it is dissipative if ‘equality’ is replaced by ‘≤’, where the (semi) group operation is written multiplicatively. In fact, a conservative field is dissipative, but there exist more of the latter. (a) Suppose that T is abelian and generates an LCA group G. Then by a consequence of a result of T. Ito (1958) (see also its extension in Sz.-Nagy and Foia¸s (1970), p. 22), which extends the Na˘ımark and Sz.-Nagy theoˆ t , t ∈ G}, and rem, there exists a super space L2 (P˜ ), a stationary field {X 2 ˜ ˜ a unitary operator Ut on L (P ) such that its restriction to L2 (P ) is τh , ˜h Xt , h, t ∈ T where τh Xt = Xt+h is the translation in given by τh Xt = U ˆ t , t ∈ G is stationary in the operation of T , here written additively. Thus X 2 ˜ ˜ ˜s X0 )L2 (P ) = r(s, t) L (P ) and coincides with the given one so that (Ut X0 , U where τt X0 = Xt , s, t ∈ T . Using the extension on this super space show that Z ˜ Xt = Ut X0 = ht, λi dZ(λ), t ∈ T, T

and if T = [0, ∞) we get a spectral representation of a conservative process which is a segment of the harmonizable process. Can we generalize this if T is a semi-group (not necessarily abelian) with identity? (b) Consider a specialization as: let {Xn , n ≥ 0} ⊂ L2 (P ) which is conservative so that r(m, n) = (Xm , Xn ) = (Xm+1 , Xn+1 ), m, n ≥ 0. Then by (a), there exists a unitary operator family Um , m ∈ Z = G, such that for 0 ≤ m ≤ n we have r(m, n) = r(n − m, 0) and the collection (r(m, n))m,n≥0 is what is called a Toeplitz matrix. This is now triangular and has many special ˆt, t ∈ G properties. Note also that on the super space L2 (P˜ ) the process X ˆ t = Xt , t ∈ Z+ on is stationary and extends the given field so that X L2 (P ). The interplay between the given conservative field and the stationary one in the super space is of interest in this analysis. Complete now the details. (c) Once again consider a new (not necessarily stationary) process {Xt = Vt X0 , t ≥ 0} ⊂ L20 (P ) where {Vt , t ≥ 0} is a weakly continuous semi-group of normal operators on L20 (P ), and we note that the process has a different representation of importance. This depends on another point of view of the ‘normal operator’ Vt which is expressible as Vt = Rt Ut = Ut Rt where {Rt , t ≥ 0} is a positive self adjoint semi-group collection and {Ut , −∞ < t < +∞} with U−t = Ut∗ is a unitary group and the Rt , Ut families commute so that one has Vs+t = Rt Rs Ut Us = Rs+t Us+t , s, t ≥ 0. Verify that r(s, t) = (Us−t X0 , Rs+t X0 )L20 (P ) = r˜(s − t, s + t) (say), for s, t ≥ 0. Show that r˜ is positive definite in the cone C = {(s, t) ∈ [0, ∞) × (−∞, ∞) : |t| ≤ s2 }. Using the spectral representations of the operators Rt , Ut establish the following:

4.5 Complements and Exercises

Xt = Rt Ut X0 =

Z

[Reλ>0]

163

′ et(log λ+iλ ) dE˜λ,λ′ X0 , t ≥ 0,

˜λ,λ′ = E1λ Eλ′ , the right side operators are respectively the spectral where E families of the classes {Rt , t ≥ 0}, and {Ut , −∞ < t < ∞}. (The problem of positive definiteness in cones is an extension of Bochner’s classical result on Rn as considered by Devinatz (1954) which is invoked here.) 8. Let {Xt ∈ L20 (P ), t ≥ 0} be a dissipative process with covariance denoted by r. Here we indicate a result on the structure of a dissipative covariance which complements the result of the last proposition of Section 3.4 and is of particular interest in the analysis of the following nonstationary classes. (a) Recall that if {Xt , t ≥ 0} ⊂ L20 (P ) is shift invariant then it is representable as Xt = τt X0 where τ : t 7→ τt ∈ B(L20 (P )) is a semi-group of bounded operators, so that r(s, t) = (Xs , Xt )L20 (P ) = (τu Xs , τv Xt )L20 (P ) = r(s + u, t + v), s, t, u, v ≥ 0 where the (semi-) group operation is addition. If the process is moreover dissipative, then the shift semi-group is contractive. Since in this case Xt = τt X0 , t ≥ 0, we have also r(s + u, t + v) = (τu Xs , τv Xt )L20 (P ) [= (τu τs X0 , τv τt X0 )L20 (P ) ]. With the extended Na˘ımark–Sz.-Nagy theorem (by T. Ito for subnormal operators noted above), there is a super Hilbert space, L20 (P˜ ) ⊃ L20 (P ), a unitary group of operators {Ut , t ∈ R} such that there is a stationary process ˆ t , t ∈ R} in the super space with the property that the family restricted {X to L20 (P ) coincides with the given dissipative Xt -process. Thus the covariance can be formally expressed as r(u + s, v + t) = (τu Xs )∗ (τv Xt ) ∈ C. If C is replaced by H, a Hilbert space, and Xt ∈ L20 (P ; H), a Hilbert space valued process, verify that in the above formula for the resulting operator valued covariance one can have r(s + u, t + v) = (τu Xs )∗ (τv Xt ) ∈ B(H) which is meaningful. (b) Here again the above covariance r : R+ × R+ → C is positive definite in the cone {(s, t) : 0 ≤ t < ∞} of the product set, not a group, so that we do not have Bochner’s theorem directly applicable to represent it as a Fourier transform of a positive bounded measure. However using the dilation discussed in (a) above obtain the following form of the covariance r of the dissipative process: Z eitλ dµ(λ), t ≥ 0 r(t)[= r(0, t)] = R

for a positive bounded measure µ on B(R). [Use Stone’s spectral theorem for the integral representation of the (dilated) unitary family {Ut , t ∈ R} and consider its restriction with the initial X0 taken as a constant.] (c) Another special case is the following. Let {Xn , n ≥ 0} be a dissipa¯ tive process Cm,n Pn= E(Xm Xn ), so ¯that for Pn with 2the second Pn moment sequence ¯ ¯k′ E(Xk Xk′ ) satany k i=1 ai Xi k = k,k′ =1 ak a ¯k′ E(Xk Xk′ ) = k,k′ =1 ak a Pn Pn isfies k k=1 ak Xk k2 ≥ k k=1 ak Xk+1 k2 ≥ 0. Suppose that E(|X0 |2 ) > 0

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4 Random Measures in Hilbert Space: Specialized Analysis

and that Xn = τ n X0 where τ is a (contractive) shift operator. If ∆n = det(Ck,l , 0 ≤ k, l ≤ n), then observe that ∆n ≥ 0 and ∆0 > 0 ( as a determinant of the matrix with E(|X0 |2 ) > 0. Assume that n0 is the first index such that ∆n0 = 0, so that n0 ≥ 1, and X0 , . . . , Xn0 are linearly dependent. Verify that τ n0 +k X0 ∈ sp{X0 , τ X0 , . . . , τ n0 X0 ]}. In the case of finite n0 , observe that the structure of τ is directly determined by the moment matrix of the Xn ’s with the linear algebraic methods (eigenvalues etc.). If n0 = ∞, then the structure of τ is more involved. For instance, if kτ n X0 k → 0 for a cyclic vector X0 , then τ n X → 0, ∀X ∈ sp{Xk , 1 ≤ k ≤ N0 }. In this case if 1 1 A = (I − τ ∗ τ ) 2 and B = (I − τ τ ∗ ) 2 are considered, it can be verified that the range of A contains the range of B. This type of analysis leads to operator theory and will be left to the interested readers. [See Mlak (1978) in this connection.]

Bibliographical Notes This chapter contains several special results that use the geometry of Hilbert spaces as well as the analysis with the inner products. Using the Schwartz space, the Itˆ o-Gel’fand theory of generalized random fields, classically given by the latter authors for the stationary case is generalized for harmonizable as well as Cram´er classes. The basic reference is the Gel’fand-Vilenkin (1964) volume for the classical fields, and the corresponding Cram´er type results have been considered by various authors, and the present treatment follows with some tightening of the author’s paper (Rao (1969)). The main motivation here is to get integral representations of various processes and fields relative to suitable random measures. The dichotomy between these measures and the corresponding bimeasures clearly needs the (weaker) Morse-Transue integration. This is essential since there are no natural conditions characterizing bimeasures of finite Vitali variation which has been a repeatedly raised question by Lo`eve (1955, p.477 and in every one of the four editions). A useful and proper concept turns out to be the Bochner V -boundedness criterion. This has been analyzed in detail and used by the author (cf. Rao (1982)). The Bochner condition now appears in almost all the currently available analysis dealing with non-stationary classes. The next stage is to consider processes (and fields) having those properties locally, i.e., only their increments are subject to these restrictions. The basic work, in the context of stationarity, is in Yaglom’s (1957) paper. It forms a fundamental reference, and extensions to the general class (C) types are detailed in Section 2. In their representations random measures have to satisfy interesting supplementary conditions. Many of these results in extended form are given in this section. A key fundamental study involves dichotomy of bilinear forms, Bochner’s L2,2 -boundedness principle and Grothendieck’s very useful inequality (both conceived at about the same time, and in fact independently by these authors) are intimate, and a comparative analysis is included

Bibliographical Notes

165

in Section 3 (perhaps) for the first time. We presented here a simple proof of the inequality due to Blei (1987) in Theorem 3.3, and another longer one by Jameson (1987) as a detailed Exercise 5.2. Both of these are attempts to “simplify” the one given in Lindenstrauss and Pelczy´ nski (1968). Bochner’s Lρ,p -boundedness principles assume the existence of such a dominating measure, but it is shown to be available both in the case of ρ = p = 2 and for many other values, but the existence of a ‘controlling measure’ for random measures is a nontrivial and important (not yet completely settled) problem which was treated in Chapter 3 in great detail. Although Grothendieck’s fundamental result is tied to the Hilbert space geometry at the core, its application is extensive in the “geometry of Banach spaces”. Pisier’s (1986) monograph has a detailed analysis of this subject. Its probabilistic use is considered in a series of papers by Niemi (1975) and slightly later by Ylinen (1978, 1993) which are used here. An interesting application by Ylinen (1993) is to obtain a Jordan-type decomposition of bounded bimeasures which depends on Grothendieck’s important inequality. We presented it in Section 3 and Exercise 5.6 thereafter. See also later on in Section 9.1 for additional results on this interesting topic. Random measures with certain restrictions lead to studies of local functionals and operators (i.e., additive but basically non-linear) on continuous function spaces (including the Schwartz space K) and on Lp (µ)-type classes. We discussed the integral representations in Section 4, but the results are quite incomplete. We tried to explain the many unsolved problems there. One way to focus these in analysis is to assume that they take independent values on disjoint sets. This is a natural assumption for generalized random fields taking independent values on the space K (different from Lp (µ)-types) of disjoint supports. Also representation of processes and fields of Karhunen and Cram´er classes in L2 (P ) use kernels which (unlike the stationary and harmonizable cases) need not be exponentials. They depend on the positive definiteness property of covariance functions. But such mappings also admit factorizations leading to representing certain linear operators as products though intermediate Hilbert spaces. We presented a single result showing how bilinear functionals and operators can be compositions through a Hilbert space starting from (certain) Banach spaces. Exercise 5.5 is culled from the works of Vakhania (1971), and it shows how other extensions can be formulated. Again in this work Pisier’s (1987) volume is of immediate interest. Regarding Problem 5.8 the reader would find further information from the work of Mlak (1978), the dilation of one parameter semi-groups of contractions on a Hilbert space, here L2 (P ), to a one-parameter group of unitary operators on a super Hilbert space (here L2 (P˜ ) ⊃ L2 (P )) which is detailed in Sz.-Nagy and Foia¸s (1970), p. 31). Also a use of Devinatz’s (1954) extension of Bochner’s theorem on positive definite functions on LCA groups is of particular interest in our study. The present application in this form may be found in the author’s (cf. Rao (2008)) article with certain other related representations. The standard reference on Toeplitz forms and its appli-

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cation to stationary processes is the account in the Grenander and Szeg¨ o (1958) monograph. The formal representation of the dissipative covariance in Exercise 5.8 has a resemblance of a factorization, and such forms have interest in the analysis of factorizable covariances and the corresponding measures that represent them. In a particular case this problem has been analyzed in Chang and Rao (1988), and it will be interesting to explore further aspects of this question especially when the dissipative variance actually goes to zero as t → ∞. In fact the covariances here determining a contractive semi-group correspond to the ‘unilateral shift’ operator in the above space. These conditions lead to various interesting decompositions of such operators into ‘unitary and completely non-unitary’ parts. These are contraction operators. The unitary concept is familiar, and a contractive operator V is termed ‘completely nonunitary’ if there is no subspace H1 for V on which it is unitary, so that for no such space V |H1 is unitary. This has a close relation with the classical ‘Wold decomposition’ of stationary processes, into completely or ‘purely’ nondeterministic and ‘deterministic’ parts. This connection will be of interest in a comparative study with Mlak’s (1978) analysis and its extensions. But we have to leave it here for interested readers, and consider our analysis of random measures and integrals. The basic reference is the Sz.-Nagy and Foia¸s (1970) monograph. There are numerous subtle properties of bilateral shift operators that go with the stationary processes and unilateral shifts with dissipative classes, and these are gainfully studied with the analysis in this volume. We next move to the following chapter on representation of random fields relative to special types of operators inspired by the classical Wiener (stochastic) integration. These can have values in different types of vector spaces with metrics that need not be normed so that Fr´echet spaces that are not necessarily locally convex enter. They play a crucial role in the ensuing analysis.

5 More on Random Measures and Integrals

Continuing the analysis of the preceding two chapters, we discuss an essentially new and potentially important operation of the convolution product between pairs of second-order random measures Rn and detail the structure as well as algebraic properties of the resulting collection, i.e., they are Hilbert space-valued. The work proceeds through a consideration of bimeasure algebras associated with such random measures, depending on the Hilbert space geometry. Also multi-linear properties of random product measures are discussed. Next we consider a detailed analysis of integration relative to random measures taking independent values on disjoint sets where the measures are Lp (P, X)-valued with X as a Fr´echet space, the integration being motivated by the Wiener approach and the Hilbert space methods playing a less significant role. Stable random measures and their integrals will get further detail, in order to contrast it with multi (or poly)-measure analysis later.

5.1 Random Measures, Bimeasures and Convolutions If (R, B0 (R)) is a ring of bounded Borel sets and ZiR: B0 (R) → L20 (P ), i = 1, 2 are a pair of random measures, then the integrals R f (λ)dZi (λ), i = 1, 2 are defined for any f ∈ B(R), the space of bounded Borel functions with the uniform norm, as seen in the preceding chapters. Further we note that Z Z ′ ′ B(f1 , f2 ) = f1 (λ)Z1 (λ), f2 (λ )dZ2 (λ ) , fi ∈ B(R) R

R

L2 (P )

defines a bounded bilinear functional which in turn determines a bimeasure β : B0 (R) × B0 (R) → C by the relation β(A, B) = B(χA , χB ) for A, B ∈ B0 (R). Here β(·, ·) in general does not define a (scalar) measure on B0 (R) ⊗ B0 (R). This is a consequence of the fact that the random measures Zi (·) are usually not of finite Vitali variation. To emphasize this point we present

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the following example that must be covered in all our considerations to follow. According to Definition 4.3.1, for a random measure Z(·) which is L2,2 bounded, so that there is a measure µ : B([a, b]) → R+ such that for all f ∈ B(R) one has Z 2 ! Z E f (t)dZ(t) ≤C |f (t)|2 dµ(t) (1) R

R

with some absolute constant C > 0. If {Xt , a ≤ t ≤ b} is Brownian motion, and P if a = t0 < t1 < · · · < tn ≤ b is a partition, then for a simple function n−1 h = i=0 ai χ(ti ,ti+1 ] , if we let Z

b

h(t)dXt =

a

n−1 X i=1

ai (Xti+1 − Xti ),

(2)

the integral on the left is well-defined and since Xti+1 −Xti is normal N (0, σ(ti+1 − ti )) and the increments are independent. So the integral in (2) satisfies (1) with Rb equality, C = σ 2 and µ being the Lebesgue measure. Thus T : h 7→ a h(t)dXt is a bounded linear operator on the simple functions of L2 ([a, b], µ) into L20 (P ) so that it has a unique extension to all of L2 ([a, b], µ) preserving the bounds, and that by the classical Riesz representation theorem for T which is weakly compact by reflexivity of L2 (P ), so that we have a unique representation Z b

Tf =

a

f (t)dZ(t), f ∈ L2 ([a, b], µ)

(3)

for a vector measure Z : B([a, b]) → L20 (P ). Hence this Z is L2,2 -bounded and specializing (3) and (4), we see that Z([ti , ti+1 ]) = Xti+1 − Xti

(4)

[see also Exercise 2.5.4 for a related (slightly different) derivation of this equation]. This correspondence was already noted by Bochner (1955), and was the motivation for his introduction of the L2,2 -boundedness concept (with a generalization toR Lρ,p -boundedness for ρ > 0, p ≥ 1). Hence (2) implies that the integral R f (t)dZ(t) is well-defined for all such random measures obeying an L2,2 -(or even Lρ,p -) bounded condition relative to a measure µ and a constant C > 0. The key observation here is that the random measure Z(·), although is a vector measure, is not of bounded Vitali variation on any non degenerate open set. Thus our development must include this class of random measures. However, as long as the range of Z : B0 (R) → X, is a Banach space [here X = L20 (P ) is included], it has finite Fr´echet variation in the following sense which is also called ‘semi-variation’ in the literature for vector measures:

) ( n

X

ai Z(Ai ) : Ai disjoint, Ai ∈ B(A), |ai | ≤ 1 , (5) kZk(A) = sup

i=1

5.1 Random Measures, Bimeasures and Convolutions

169

where ai ∈ C and B(A) is the trace of B0 on A. Of course here (R, B0 (R)) may be replaced by a general measurable space (S, S) with S as a delta-ring. It was already noted in Chapter 2 that kZk(A) ≤ |Z|(A) ≤ ∞ with strict inequality in the first term when the last one has equality, and the Brownian motion example shows that |Z|(·) need not even be σ-finite. Hence in what follows we have to work only with the Fr´echet variation which is always finite on S. Note that a measure Z : S → L20 (P ), inducing a bimeasure by the relation β(A, B) = (Z(A), Z(B))L20 (P ) , will not necessarily imply that β(·, ·) defines a scalar measure on the product space R2n with S = Rn , n ≥ 1. But there is an important class of problems for which the latter condition holds. Namely if Z : B(Rn ) → L20 (P ) is shift invariant in the sense that (Z(τx A), Z(τx B))L20 (P ) = (Z(A), Z(B))L20 (P ) , A, B ∈ B0 (Rn ), τx A = {x + y : y ∈ A}, x ∈ Rn , τx being ˜ × B) where β˜ : B0 (Rn ) × the shift, then by Theorem 2.3.2, β(A, B) = β(A n B0 (R ) → C is a signed measure. This motivates a method of introducing a convolution operation between pairs of bimeasures and using the work of Chapter 2 and then to transport it back to the random measures themselves. In this connection it is helpful to recall, as already shown in Exercises 2.5.2 and 2.5.3, a use of strict MT-integration (see also Section 3.4) and an elaboration of that method, indicated in Exercise 2.5.5. This will now be explained and extended. We restate the earlier result in a form that is convenient for the present purpose. Theorem 5.1.1 Let (Si , Si ), i = 1, 2 be measurable spaces with Si as(possibly delta) rings and (Ω, Σ, P ) a probability space. If Zi : Si → L20 (P ), i = 1, 2 are random measures, let β : S1 × S2 → C be the (induced) bimeasure defined by β(A1 , A2 ) = (Z1 (A1 ), Z2 (A2 ))L20 (P ) , Ai ∈ Si . Then for fi : Si → C, i = 1, 2 measurable and Zi integrable in the Dunford-Schwartz sense, one has with the (strict) MT-integral Z Z Z Z f2 dZ2 , Ai ∈ Si . (6) (f1 (x), f2 (y)) β(dx, dy) = E f1 dZ1 A1

A2

A1

A2

Moreover, if (S1 , S1 ) = (S2 , S2 ) and Z1 = Z2 then the induced bimeasure β(·, ·) is positive definite. On the other hand if β : S × S → C is a bounded bimeasure, it can be decomposed as : √ (7) β = β1 − β2 + i(β3 − β4 ), (i = −1) where the βj , j = 1, . . . , 4 are positive definite bimeasures on S × S. [An earlier version of (7) was given as Corollary 4.3.8.] Here the first part of the theorem and (6) follow from Proposition 2.2.4, and (7) from Corollary 4.3.8. We now introduce a multiplication operation, termed convolution, between pairs of bimeasures if S is moreover an LCA

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group and then extend the result to random measures using the ideas of Theorem 2.2.5 to obtain an algebra, then use it in analysis. The indirect route to be followed here is somewhat similar to that of introducing convolution in certain “homogeneous Banach spaces”, as in Katznelson [(1968), p.40]. In the present case the detail is different and has also some novel features. To motivate the basic ideas, recall that a bounded bilinear form on a Hilbert space H defined by B : H × H → C is representable, as a consequence of the Riesz representation of bounded linear functionals, in the form: B(x, y) = (x, T y), x, y ∈ H, (8) for a unique bounded linear mapping T : H → H which moreover is positive if B(·, ·) is also positive definite (cf, Riesz-Sz.-Nagy (1955), p.202). If H × H is replaced by H1 × H2 above then T : H2 → H1 is a bounded linear operator and in case Hi = L2 (Ωi , Σi , µi ), i = 1, 2, then (8) becomes Z (T f1 )f2 dµ2 , fi ∈ Hi , i = 1, 2 (9) B(f1 , f2 ) = Ω2

and hence T is uniquely determined by B(·, ·), µ1 and µ2 . This representation is also closely related to Grothendieck’s result discussed in Theorem 4.3.4 (and in fact with the work of Section 4.3, agrees in general). Since this plays an important role here, we restate the proposition in the form that is immediately applicable, after introducing some relevant concepts and terminology. Our interest is to obtain the product of a pair of bilinear functionals (or bimeasures) to be a bimeasure or a bilinear functional on a suitable “product”(or tensor) space with respect to which we intend to introduce a “convolution product” of random measures. This gives a random measure again. The underlying spaces may be locally compact groups. Recall that for a pair {Xi , k · ki }, i = 1, 2 of Banach spaces, Pnthe tensor product space X1 × X2 is the linear span of formal expressions i=1 fi ⊗ gi , satisfying the following conditions for all n ≥ 1 [∼ for equivalence]: Pn Pn fi ⊗ gi ∼ j=1 fij ⊗ gij , (i1 , . . . , in ) is a permutation of (1, . . . , n) (i) i=1 Pn Pn (ii) Pi=1 (ai fi ) ⊗ gi ∼ i=1 i gi ), P Pfni ⊗ (a n n ′ ′ ′′ f ⊗ gi + i=1 fi′′ ⊗ gi , (f + f ) ⊗ g = (iii) i i i=1 i i=1 i and a norm α : X1 ⊗ X2 → R+ is defined to satisfy the conditions Pn Pn 1. α( P if and only if i=1 fi ⊗P gi ∼ 0 ⊗ 0, i=1 fi ⊗ gi ) = 0,P Pn n n n 2. α(a i=1 fi ⊗ gi + i=1 fi′ ⊗ gj′ ) ≤ |a|α( i=1 fi ⊗ gi ) + α( i=1 fi′ ⊗ gi′ ),P Pn Pn Pn n 3. α( i=1 fi ⊗ gi ) = α( i=1 fi′ ⊗ gi′ ), if i=1 fi ⊗ gi ∼ i=1 fi′ ⊗ gi′ . Such an α(·) is called a cross-norm if moreover we have 4. α(f ⊗ g) = kf k1kgk2 , f ∈ X1 , g ∈ X2 (and = 0 if and only if f = 0 or g = 0).

5.1 Random Measures, Bimeasures and Convolutions

171

There exist many cross norms, e.g, each α(·) as Lebesgue’s p-norm, denoted αp (·) for 1 ≤ p ≤ ∞. Two of these are of special interest, called the least cross-norm (l.c.n.) and the greatest cross norm(g.c.n), defined as: (i) Least cross-norm: If X⋆i is the dual Banach space of Xi , i = 1, 2, and fi ∈ X1 , gi ∈ X2 , x⋆i ∈ X⋆i are any elements, then the l.c.n., denoted by λ(·), is defined as: ! n n X o X x⋆1 (fi )x⋆2 (gi ) | : kx⋆1 k ≤ 1, kx⋆2 k ≤ 1 . (10) fi ⊗ gi = sup | λ i=1

It is not difficult to verify that λ(·) is a cross-norm. Pn Pn (ii) Greatest cross-norm: If fi ∈ X1 , gi ∈ X2 and i=1 fi ⊗ gi ∼ i=1 fi′ ⊗ gi′ are equivalent representations, then the g.c.n. denoted γ(·), is defined as: ) ! ( n n n n X X X X ′ ′ ′ ′ fi ⊗ gi . (11) fi ⊗ gi ∼ kfi k1 kgi k2 : fi ⊗ gi = inf γ i=1

i=1

i=1

i=1

It can again be verified that γ(·) is a cross norm, and that one has: ! ! ! n n n X X X fi ⊗ gi fi ⊗ gi ≤ γ fi ⊗ gi ≤ α λ i=1

i=1

(12)

i=1

for all cross-norms α(·). The completions of X1 ⊗ X2 for these norms, are ˘ for ⊗λ , ⊗ ˆ for ⊗γ , one has denoted X1 ⊗α X2 , and using ⊗ ˘ 2 = X1 ⊗λ X2 ⊃ X1 ⊗α X2 ⊃ X1 ⊗γ X2 = X1 ⊗X ˆ 2. X1 ⊗X

(13)

˘ 2 as “injective” tensor product and for X1 ⊗X ˆ 2 as Alternative names for X1 ⊗X “projective” tensor product are also used in the literature. The fundamental ˆ 2 is the space work of Varopoulos (1968) shows that the adjoint space of X1 ⊗X of continuous bilinear forms on the product space X1 × X2 which is of interest in our work. There are two ways of introducing (equivalent) norms here. Namely B ∈ (X1 × X2 )∗ gives a standard method of defining the norm kBk of B: kBk = sup{|B(x, y)| : kxk1 ≤ 1, kyk2 ≤ 1, x ∈ X1 , y ∈ X2 } (14) and using Grothendieck’s inequality one defines |kB|k as : |kB|k = inf{C > 0 : |B(f, g)| ≤ Ckf kX1 kgkX2 , f ∈ X1 , g ∈ X2 }.

(15)

Here X1 = C0 (S1 ), X2 = C0 (S2 ), the continuous scalar function spaces on the locally compact Si , and Xi ⊂ L2 (Si , Si , µi ) are dense where µ1 , µ2

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are probability measures in Grothendieck’s inequality. The work of the latter inequality then implies that k · k and |k · |k are equivalent in the sense that kBk ≤ |kB|k ≤ KG kBk (16)

where KG is Grothendieck’s universal constant. ( π2 < KG < 1.782 the bound is due to J. L. Krivine (1979), the exact value of KG is still not known.) Bilinear mappings considered above can also be identified as eleˆ 2 using the notation of (14) by specialments of the adjoint space of X1 ⊗X izing the spaces Xi , i = 1, 2 as follows. Let Si be locally compact spaces and take Xi = C0 (Si ), the space of continuous scalar functions on Si vanˆ 0 (S2 ))∗ so that ishing at infinity, with the uniform norm. Then B ∈ (C0 (S1 )⊗C B(f, g) = B(f ⊗ g) = hf ⊗ g, Bi using the duality notation. Observe that ˆ 0 (S2 ))∗ ⊃ C0 (S1 )∗ ⊗C ˆ 0 (S2 )∗ . [In this notation the inequality of Sec(C0 (S1 )⊗C tion 4.3 becomes (cf. Theorem 3 there)

|hf ⊗ g, Bi| ≤ Kkf k2kgk2 , f ∈ C0 (S1 ), g ∈ C0 (S2 ) (17) R R where the kf k22 = S1 |f |2 dµ1 and kgk22 = S2 |g|2 dµ2 for some regular probability measures µi on (Si , Si ) and an absolute constant K > 0 which is typically expressed as K = KG kBk where KG is that noted in (16).] Our interest now is in introducing a product operation for pairs of random measures. This will be done through their induced bimeasures. We first consider the result for positive definite bimeasures. The following serves as an initial step based on Aronszajn spaces. Proposition 5.1.2 Let βi : Ti × Ti → C, i = 1, 2 be a pair of positive definite kernels and β = β1 · β2 : (T1 × T1 ) × (T2 × T2 ) → C as the pointwise product. Then β is positive definite. If we let Hβ , Hβ1 , Hβ2 the corresponding reproducing kernel Hilbert (or Aronszajn) spaces, then Hβ = Hβ1 ⊗ Hβ2 , so that Hβ [or H(T1 , T2 ; β)] is a tensor product of Hβ1 and Hβ2 [or H(T1 ; β1 ) and H(T2 ; β2 )]. Proof. First observe that the pointwise product of a pair of positive definite kernels is positive definite, as a consequence of (the well-known) Schur’s lemma which is our context has the following simple proof. Indeed let X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Yn ) be independent centered Gaussian (scalar) random vectors on some probability space. If A and B are their covariance matrices, so ¯ j ) = aij ) and B = (E(Yi Y¯j ) = bij ) then A and B are posthat A = (E(Xi X itive definite matrices. If Z = (X1 Y1 , . . . , Xn Yn ) then clearly Z is a centered random vector with two moments, E(Z) = (E(Xi Yi ) = E(Xi )E(Yi ) = 0, i = 1, · · · , n) and the covariance matrix of the vector Z, denoted ‘cov Z’, is seen to be ¯ j ) · (Yi Y¯j ))) cov Z = (E(Zi Z¯j , 1 ≤ i, j ≤ n) = (E((Xi X ¯ j ) · E(Yi Y¯j ), 1 ≤ i, j ≤ n), by independence, = (E(Xi X = (aij · bij , 1 ≤ i, j ≤ n)

(18)

5.1 Random Measures, Bimeasures and Convolutions

173

which is, being a covariance (matrix), positive definite, as desired. The reproducing kernel Hilbert spaces relative to β, β1 and β2 have already been introduced in Section 2.2 (just prior to Theorem 2.2.5) to establish the stated relations. Observe that the sets {βi (y, ·), y ∈ Ti } generate Hβi , i = 1, 2 and {β1 (y1 , ·) × β2 (y2 , ·), (y1 , y2 ) ∈ T1 × T2 } similarly generates Hβ since β = β1 × β2 : T1 × T2 → C is positive definite by Schur’s lemma. ˜ be generated by the functions f : (y1 , y2 ) 7→ hf, β1 (y1 , ·) × β2 (y2 , ·)i, for Let H ˜ and (y1 , y2 ) ∈ T1 × T2 . If we give the product topology of Hβ1 ⊗ Hβ2 to H, noting that β1 (y1 , ·) × β2 (y2 , ·) determines Hβ1 ⊗ Hβ2 [and (β1 ⊗ β2 )((y1 , y2 ), ·) is the same function as β1 (y1 , ·) ⊗ β2 (y2 , ·) for all yi ∈ Ti ], we can identify the ˜ = Hβ with Hβ ⊗ Hβ . Thus the tensor product here is exactly the spaces H 2 1 desired Aronszajn space. 2 Remark 5.1 If T1 = T2 in the above, then H(T, β) or H(T, β1 × β2 ) where (β1 × β2 )(s, t) = β1 (s, t)β2 (s, t)[= β1 ⊗ β2 |the diagonal of T×T )] may be identified isometrically with the subspace of H(T × T, β1 ⊗ β2 ) = H. In fact let H1 = {f ∈ H(T × T, β1 ⊗ β2 ) : f (t, t) = 0} and H2 = H1⊥ , so that H2 is isomorphic to H(T, β1 × β2 ) where (β1 × β2 )(s, t) = β1 (s, t) · β2 (s, t). It follows also from our work on these spaces that kf k = inf{kgkH : g ∈ H, g|diagonal = f } = kf kH2 . This result extends to any finite number of products. Taking β2 = β¯1 in the above we note that |β|2 gives rise to a reproducing kernel andP determines the ∞ corresponding Hilbert space of functions f of the form f (x) = i=1 fi (x)gi (x) P P 2 2 where kf k < ∞ and kg k < ∞. Thus some of the considerai i i≥1 i≥1 tions above have extensions to product kernels. More details of the nontrivial constructions are discussed in Aronszajn (1950). All this work on products and other relations on positive definite kernels is used to introduce similar (or appropriate) binary relations between random measures to obtain a new measure of the same type and introduce an algebra for these latter objects including analogs of convolutions. The following assertion aids this work. Proposition 5.1.3 Let β : Y × Y → C be a hermitian positive definite kernel. Then there exists a probability space (Ω, Σ, P ) and a random measure Z : Y → L2 (P ) where Y is a δ-ring of the point set T , such that for each pair ¯ 2 )) = (Z(A1 ), Z(A2 )L2 (P ) = β(A1 , A2 ). In fact Ai ∈ Y, i = 1, 2, E(Z(A1 )Z(A we can take P : Σ → [0, 1], to be a Gaussian probability so that β(·, ·) is its covariance bimeasure, and the (Ω, Σ, P ) can always be taken large enough for this representation. Proof. The construction is a specialization of Kolmogorov’s classical existence theorem. Let (Ωi , Σi , Pi ) where Ωi = C, Σi = Bi , the Borel σ-algebra of C R √ |x|2 1 dx2 , x = x1 + −1x2 for x1 , x2 ∈ R. Now take and Pi : Bi → (·) e− 2 dx2π

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N (Ω, Σ, P ) = i∈Y (Ωi , Σi , Pi ), the product probability space determined by the (infinite product) Fubini-Jessen theorem. If we consider the coordinate function Yi (ω) = ω(i), ω ∈ Ω = CY , then E(Yi ) = 0 and E(|Yi |2 ) = 1. Also {Yi , i ∈ Y} is a complete orthonormal basis of L = sp{Y ¯ i , i ∈ I} ⊂ L20 (P ). On the other hand, since β(·, ·) is a Hermitian positive definite function on Y × Y, there is a uniquely defined Aronszajn (or reproducing kernel Hilbert) space Hβ and if {hi , i ∈ Y} is a CON system for it, then the mapping τ : Yi → hi extended linearly sets up an isomorphism of L onto Hβ . Thus the family {Xi , i ∈ Y} ⊂ L where each Xi is a linear span of Y ’s is Gaussian, and the corresponding element in Hβ is the (respective) linear combination of hi ’s so that we have ¯ j ) = hβi , βj i = β(i, j), i, j ∈ Y = B. (19) E(Xi X

If we let Z : A → χA ∈ L ⊂ L20 (P ), A ∈ B, in the above construction, it follows that Z(·) : B → L20 (P ) is a random measure which is moreover Gaussian with β(·, ·) as its covariance bimeasure. 2 One of the purposes of the preceding analysis is to introduce a “multiplication”relation between random measures into a Hilbert space and discuss some aspects of the resulting “measure algebra”. Propositions 2 and 3 lead to the following concept: Definition 5.1.4 As usual let (S, S) be a measurable space and Zi : S → L20 (P ) be a pair (i = 1, 2) of random measures into the Hilbert space of (equivalence classes of) centered (complex) random variables on a probability space (Ω, Σ, P ) with covariance bimeasures βi : S × S → C given by βi (A, B) = (Zi (A), Zi (B)) using the inner product notation. Let β = β1 ⊙ β2 : S × S → C be the product, pointwise as in Proposition 2 above or that given as convolution using (strict) MT-integration, if S = Rn defined in Exercise 2.5.5. If Z : S → L20 (P ) is the induced random measure by β, following Proposition 3 above, then denote Z = Z1 ⊙ Z2 : S → L20 (P ), to be called the product of Z1 and Z2 . (This is unambiguously obtained.)

Here it may be necessary to enlarge the probability space (Ω, Σ, P ) by the standard method of adjunction, familiar in probability theory. Thus the basic space can be assumed rich enough that all these operations are possible to obtain new measures all supported by the same (Ω, Σ, P ). We now include some results on the “random measure algebra” thus obtained for further analysis. Recall that with each bimeasure β : S1 × S2 → C, one can associate a bilinear functional R BR: B0 (S1 , S1 )×B0 (S2 , S2 ) → C through the (strict) MT-integral: g (s2 )dβ(s1 , s2 ), for all (f, g) ∈ B0 (S1 , S)×B0 (S2 , S) → B : (f, g) 7→ S1 S2 f (s1 )¯ C where B0 (Si , Si ) is the space of bounded Si -measurable scalar functions. It then follows by the Grothendieck inequality that

2

|B(f, g)| ≤ Ckf k2 kgk2 ,

(20) 2

where kf k2 is from L (S1 , S1 , µ1 ) and kgk2 refers to L (S2 , S2 , µ2 ) norms, whereas µ1 , µ2 are the corresponding (regular if Si are locally compact) finite

5.1 Random Measures, Bimeasures and Convolutions

175

measures of that theorem. [Here (µ1 , µ2 ) may not be unique but C depends on B only.] It may be noted that the bimeasure β inducing B is not generally of finite (Vitali) variation, and so there is no immediate “controlling measure” to analyze further. But (20) shows that β : S1 × S2 → C is dominated by the pair (µ1 , µ2 ) at once with a universal constant C > 0. This may be contrasted with the fact that β1 (·) = β(·, ) and β2 (·) = β(, ·) are vector measures on S1 and S2 respectively and hence have individual controlling (finite) measures ν1 and ν2 by Theorem 1.3.4 (the Bartle-Dunford-Schwartz controls). This pair (ν1 , ν2 ) thus dominates the bimeasure β(·, ·), but the inequality (20) following from Grothendieck’s theorem is sharper. A similar remark applies to the product spaces coming from Proposition 2. Consequently we need to rely more on the inequality (20) in our work. The following representation will be utilized in the ensuing analysis. Theorem 5.1.5 Let Si , Ti , i = 1, 2 be locally compact spaces with Si , Yi as their δ-rings of bounded Borel sets. If βi : Si × Yi → C, i = 1, 2 are bounded bimeasures, then there exists a bimeasure β on the product delta rings S1 ⊗S2 ×Y1 ⊗Y2 whose control measure pair is µ1 ⊗ µ2 , ν1 ⊗ ν2 where (µi , νi ) is a controlling pair for βi , i = 1, 2. If B1 , B2 and B are the corresponding induced bilinear forms of β1 , β2 and β, then we have the alternative representation for each pair f, g of continuous (complex) functions vanishing at infinity on S1 × S2 and T1 × T2 such that Z h(Q1 ⊗ Q2 )f, gi dν1 dν2 (21) B(f, g) = (B1 ⊗ B2 )(f, g) = T1 ×T2

where Qi : L2 (Si , Si , µi ) → L2 (Ti , Yi , νi ), i = 1, 2 are the bounded linear operators determined by the bounded bilinear forms Bi (similar to (20)). Finally if S1 = S2 = T1 = T2 and the βi are positive definite, then β is also positive definite ; if Z1 , Z2 are the random measures determined by β1 and β2 , as in Proposition 3, then there is a random measure Z corresponding to β and we have the relation Z = Z1 ⊙ Z2 : S → L20 (P ), on a suitably enlarged probability space, where all these (random) measures take their values. Proof. In view of the above Propositions 2 and 3, it is sufficient (and also convenient) to establish the result for the bilinear forms. This can easily be translated to the bimeasures from that work. Thus for f, g as in (21), consider the form, (see (9)) given by Z ((Q1 ⊗ Q2 )(f1 ⊗ f2 ))(g1 ⊗ g2 ) d(ν1 ⊗ ν2 ) B1 ⊗ B2 (f1 ⊗ f2 , g1 ⊗ g2 ) = Z ZT1 ×T2 (Q2 f2 )g2 dν2 (Q1 f1 )g1 dν1 = T1

T2

= B1 (f1 , g1 ) · B2 (f2 , g2 ), by (9).

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5 More on Random Measures and Integrals

But the uniform (or norm) density of C0 (S1 ) ⊗ C0 (S2 ) in C0 (S1 × S2 ), and similarly for the second factor, it follows that B1 ⊗ B2 is well-defined and does not depend on the (non unique) dominating measure pairs (µi , νi ), i = 1, 2; so that B = B1 ⊗ B2 is well-defined on C0 (S1 × S2 ) × C0 (T1 × T2 ). Moreover, (16) implies that kQi k ≤ kBi k ≤ KG kBi k if µi and νi are normalized to be probability measures where KG is the Grothendieck constant. Hence we have for f ∈ C0 (S1 × S2 ), g ∈ C0 (T1 × T2 ), Z ((Q1 ⊗ Q2 )(f ))gdν1 ⊗ ν2 | |B(f, g)| = | T1 ×T2

≤ kQ1 ⊗ Q2 k

Z

S1 ×S2

2

|f | dµ1 ⊗ µ2

(by the CBS-inequality)

12 Z

T1 ×T2

2

|g| dν1 ⊗ ν2

21

,

≤ kQ1 k kQ2 k kf kC0 (S1 ×S2 ) kgkC0(T1 ×T2 )

2 kB1 k kB2 kkf k2,µ1 ⊗µ2 kgkν1 ⊗ν2 . ≤ KG

Next under the additional hypothesis, B1 , B2 and hence B are positive definite and so determine random measures Z1 , Z2 and Z respectively as a consequence of Proposition 3. Here again we may have to enlarge the underlying probability space without changing any conditions. This essentially establishes all the requirements of the theorem and further discussion can be omitted here. 2 Remark 5.2 The point of this result with Grothendieck’s inequality is to be able to obtain the (existence) of the dominating (or controlling) measures µ1 ⊗ µ2 , ν1 ⊗ ν2 which will allow us to consider “products” of random measures to get another random measure. Note that if Si , Ti are different the above method gives Bi (and B) to be bilinear, and not necessarily positive definite functionals. This however is not a problem. As seen in (7), every bounded bilinear form can be decomposed into a finite number (four in fact) of positive definite bounded such forms and consequently the new form Z1 ⊙ Z2 can be expressed as a finite linear combination of the desirable ones and “⊙” is distributive and its binary relation has the same property. Thus this relation can be considered as long as the bimeasure products are pointwise operations and are of convolution type as in Exercise 2.5.5. We shall discuss and develop this aspect further and term it a generalized product. However, unlike the case where randomness does not intervene, if Z1 and Z2 take values in L20 (P ) it does not imply that Z1 (A)Z(B), as a pointwise product, must also belong to L20 (P ) and the product (Z1 ⊙ Z2 )(·) cannot be treated in this simple minded way. Thus Z(·) = (Z1 ⊙ Z2 )(·) defined through bimeasure theory above is considerably different and the reader should look for and avoid such a trap.

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177

5.2 Bilinear Forms and Random Measure Algebras At the outset one notes that the classical convolution products of measures on the underlying groups, even on the real line, do not extend to random measures because of the following new problems. If Z1 , Z2 are a pair of random measures on (R, B), then the standard type concept of convolution of Z1 and Z2 defined as Z (Z1 ∗ Z2 )(A) = Z1 (A − x)dZ2 (x), A ∈ B (1) R

does not necessarily exist, since the integrand in (1) is random and has to satisfy certain “filtration conditions” of the Itˆ o-type for such integrals to make sense. These generally force Z1 or Z2 to be nonrandom measures in general. The subject must therefore be approached differently and with care. A natural idea is to develop the “odot” (⊙) operation indicated in Theorem 1.5 above specializing the spaces (S, S) to locally compact (abelian) groups and study bimeasure algebras to formulate the results for random measures thereafter indirectly. A second approach is to generalize the bilinear inequality due to Grothendieck to a class of multi-linear forms in which we have to consider the range spaces of random measures that can be Lp (P )spaces where p 6= 2 is allowed but multiple integration relative to (product) random measures is permitted. Let us explain this aspect of the subject further. We recall from Theorem 4.3.4 that given a bounded bimeasure β : B(S1 ) × B(S2 ) → C where B(Si ) is a δ-ring on Si , there exist Hilbert spaces Hi , i = 1, 2 and spectral measures Ei : B(Si ) → Hi , i = 1, 2, along with cyclic vectors xi ∈ Hi such that β(A1 , A2 ) = (E1 (A1 )T E2 (A2 )x2 , x1 ) = (T E2 (A2 )x2 , E1 (A1 )x1 )

(2)

for a unique bounded linear operator T : H2 → H1 and Ai ∈ B(Si ), T is also a positive operator if β is positive definite. In this way we take Zi (Ai ) = Ei (Ai )xi and Zi will be random measures when Hi is realized as L2 (Pi ) on some probability spaces. In any case each Zi (·) is orthogonally valued and it remains the same for each cyclic vector which is determined by the bimeasure β. We now consider specializations that each Si is a locally compact abelian (LCA) group and demand that the ⊙-operation be a convolution, starting with the bimeasures β1 , β2 . Let us begin with the regular βj and use some approximations. Recall that, from topological measure theory, the support of a bimeasure β is a closed set F ⊂ S1 × S2 such that β(A, B) = 0 for all sets A × B ∈ B0 (S1 ) × B0 (S2 ) with (A × B) ∩ F = ∅. [Equivalently, F is the complement of the union of all open sets in S1 × S2 on which β vanishes.] Using the dominated convergence criterion, Theorem 2.2.6 for regular bimeasures (with the strict MT integration), we first establish the following:

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5 More on Random Measures and Integrals

Proposition 5.2.1 If β : B0 (S1 ) × B0 (S2 ) → C is a regular bimeasure, then it can be approximated by a similar bimeasure of compact support. Equivalently if B is the bounded bilinear form determined by β, with the MT-integration, then for each ǫ > 0, there is a similar bilinear form Bǫ with a compactly supported bimeasure such that kB − Bǫ k < ǫ, where k · k is the uniform norm on bilinear forms. Proof. This is really a special case of Theorem 2.2.7. Since the details were not given for the earlier result, we sketch the argument for the particular case at handR essentially following Graham and Schreiber (1984). Thus let R g (s2 ) dβ(s1 , s2 ), f ∈ B1 (S1 ), g ∈ B2 (S2 ), bounded B : (f, g) 7→ S1 S2 f (s1 )¯ Borel classes, be the bilinear functional determined by β(·, ·). Let B(f, g) be ˜ = ϕB for ϕ = (ϕ1 , ϕ2 ) ∈ B1 (S1 ) × B2 (S2 ) be expressed as h(f, g), Bi. Let B defined by ˜ = hϕh, Bi, h = h1 = h2 ∈ B1 (S1 ) ⊗ B2 (S2 ) ⊂ B ′ (S1 × S2 ). hh, Bi

(3)

˜ is again a bounded bilinear form for which the Grothendieck inequalThen B ity relative to a regular probability measure pair (µ1 , µ2 ) on (S1 , S2 ) is valid ˜ ≤ kBk kϕ1 k21 µ1 kϕ2 k21 µ2 . By regularity we can find, for any so that kBk ǫ > 0 given, a pair of compact sets Ki ⊂ Si , i = 1, 2, and 0 ≤ ϕi ≤ 1 supported by Ki where µi (Si − Ki ) < ǫ2 so that (k · ku being the uniform norm) ˜ = sup{|hh, B − Bi| ˜ : khku ≤ 1} kB − Bk = sup{hh(1 − ϕ), Bi| : khku ≤ 1} ≤C

sup

{kBk · kh1 k2,µ1 kh2 k2,µ2 k1 − ϕku } < 4Cǫ.

2

khi k21 µi ≤1

We can now introduce the Fourier transform of a bounded bilinear functional B on B1 (S1 ) × B2 (S2 ) when S1 , S2 are LCA groups: Definition 5.2.2 Now let Gi (= Si ) be LCA groups whose character (or dual) ˆ i , i = 1, 2 so that for the group G1 × G2 its dual groups are denoted by G ˆ1 × G ˆ 2 . For each character (λ1 , λ2 ) ∈ G ˆ1 × G ˆ 2 (so λi ∈ G ˆ i , i = 1, 2), is G consider ¯ 2 , Bi, ˆ 1 , λ2 ) = hλ¯1 ⊗ λ B(λ (4) ˆ: G ˆ1 × G ˆ 2 → C is a function, well-defined by the bilinear form B. where B ˆ xi ∈ R and B(e ˆ it1 (·) , eit2 (·) ) [Thus in case Sj = R, λtj = eitj xj , t ∈ R, is an integral defined by (3).] If B(G1 , G2 ) stands for the set of bounded ˆ1, G ˆ 2 ) be the corresponding class of their bimeasures on (G1 , G2 ) let S(G Fourier transforms defined by (4) so that each B ∈ B(G1 , G2 ) has its image ˆ ∈ S(G ˆ 1, G ˆ 2 ). B ˆ1, G ˆ 2 ) will be determined in Next the relations between B(G1 , G2 ) and S(G order to introduce the convolution operation in B(G1 , G2 ) through the algeˆ1, G ˆ 2 ) which is an extension of the case when there is one braic structure of S(G

5.2 Bilinear Forms and Random Measure Algebras

179

ˆ But the structure of an LCA group is also involved. AccordG and hence one G. ing to this theory (cf. e.g., Rudin (1962), p.40) every LCA group G contains an open subgroup G1 so that the quotient (group) G/G1 is discrete and G1 itself is a direct sum of a compact (abelian) group H and a Euclidean space Rn for some n ≥ 0. Thus a direct extension of Fourier transforms to the bimeasure case, through structure theory, is rather round about and we therefore proceed through the “representation theory” of groups which is moreover applicable to the general (nonabelian) case as well. This will be explained first in the context of LCA groups. Motivated by the classical Fourier transform V : t 7→ eit(·) , so that V : R → H the Hilbert space generated by the complex functions x 7→ eitx , t ∈ R, and V (s + t) = V (s)V (t), V (0) = identity, so that V (t) is a homomorphism of R into a vector space X containing the functions {eit(·) , t ∈ R}, consider groups G more general than Rn (n ≥ 1) and spaces in which multiplication is defined. In fact we would like to consider an LCA group G in the first place, and a general locally compact group later in place of Rn . The mapping V : G → B(X) in either case is considered where X is taken as a Hilbert space H (in the special case considered H = C) and B(X) being the algebra of bounded linear operators on X. Such a mapping V on G into B(X), or B(H), is called a representation of G over or H. In the classical case for f ∈ L1 (R, dx), the R R X −itx ˆ the ‘dual’ f (x)dx = R V (t− x)f (x) dx, t ∈ R, Fourier transform fˆ(t) = R e of R (identified as usual with R) we have if R is replaced by an LCA group G, the following: Z Z ˆ ˆ f (x)V (xλ−1 )dx, x ∈ G V (x − λ)f (x) dx = f (λ) = G

G

ˆ is the dual group of the given G, which is also an LCA group and dx and G is the translation invariant (called now (left) Haar) measure on the Borel σˆ The mapping on X = C is one-dimensional and is usually denoted algebra of G. V (x, λ) = hx, λi so that Z ˆ hx, λif (x) dµ(x), f ∈ L1 (G, µ). (5) f (λ) = G

We now use the classical fact that such an invariant measure µ on any locally compact group G exists, called the (left) Haar measure which reduces to a ˆ n (≡ Rn ). In the genˆ = (R) Lebesgue measure when G = Rn , n ≥ 1, and G eral case, therefore, it is necessary to use representation theory fully, and we shall indicate what is needed for our purpose in order to define product random measures, through their bimeasures. There is no direct procedure available. We R first consider the abelian case to use (4). If A 7→ A f (x) dµ(x) is replaced by a measure ν, then (5) becomes Z hx, λi dν(x) = hλ, νi, (6) νˆ(λ) = G

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5 More on Random Measures and Integrals

and for a bimeasure β on B(G) × B(G), the analog of (6) is introduced as: Definition 5.2.3 Let G be an LCA group and u : G → C be a mapping such that M ◦ u : G × G → C is defined as (M ◦ u)(x1 , x2 ) = u(x1 x2 ). For B1 , B2 : C(G1 ) × C(G2 ) → C, a pair of bounded bilinear forms given by a pair of bimeasures β1 , β2 via the (strict) MT-integration. We set using the tensor product notation: (B1 ∗ B2 )(f, g) = (B1 ⊗ B2 )(M f, M g)

(7)

with the convention that f˘(x) = f (x−1 ) = f ∗ (x) = f¯(x−1 )(= f˘(x)), and ˘ g) = B(f˘, g˘), B(f, ¯ g) = B(f ∗ , g ∗ ), for f ∈ C0 (G1 ), g ∈ C0 (G2 ) for the LCA B(f, groups G1 , G2 where in (7) we can extend B1 ⊗B2 : C0 (G1 ×G1 )×C0 (G2 ×G2 ) → C to the closure Cb (G1 × G1 ) × Cb (G2 × G2 ) of bounded continuous functions on the LCA groups G1 and G2 . It is to be shown that the product bilinear functional (7) is the correct definition of the product bimeasures and hence will define the convolution product for these objects. Later we see that this extends to the nonabelian case as well allowing us to form products of random measures making up an algebra for further analysis. Note that (7) as well as the above definition did not really use the property that G is abelian. We first verify that Theorem 1.5 takes the following form: Proposition 5.2.4 If B1 , B2 are bounded bilinear forms on C0 (G1 , G2 ) with the associated Grothendieck probability pairs µi , νi i = 1, 2, and Q1 , Q2 are their induced operators Qi : L2 (G1 , µi ) → L2 (G2 , νi ), i = 1, 2 then B = B1 × B2 has Q1 × Q2 as its measure on L2 (G1 × G1 , µ1 ⊗ µ2 ) → L2 (G2 × G2 , ν1 ⊗ ν2 ) and is defined by Z (M −1 P (Q1 ⊗ Q2 )M (f ))M (g) dν1 ∗ ν2 (8) B(f, g) = (B1 ∗ B2 )(f, g) = G2

for f ∈ C0 (G1 ), g ∈ C0 (G2 ) and P : L2 (G2 × G2 , ν1 × ν2 ) → M (L2 (G2 , ν1 ∗ ν2 )) is a projection operator of norm at most one so that B is well-defined where M (·) is given in the preceding definition. Proof. This is an immediate consequence of the previous work. In fact Z ((Q1 × Q2 )M (f ))M (g) dν1 × dν2 (B1 ∗ B2 )(f, g) = G1 ×G2 Z (M −1 P (Q1 ⊗ Q2 )M (f ))(g) d(ν1 ∗ ν2 ) = G2

which is (8).

2

5.2 Bilinear Forms and Random Measure Algebras

181

Corresponding to V (t, ·)(= eit(·) ) of the real case, we consider the representation V : G → B(H) to have the property that V (g, ·) : f → V (g, ·)f having the same norm, and be a homomorphism together with its adjoint having similar properties. Thus we take the representing family to be unitary operators, and then V : G → B(H) must satisfy: (i) V (g1 g2 ) = V (g1 )V (g2 ), (ii) kV (g)f k = kf k, g ∈ G, and (iii) V (g −1 ) = V (g)∗ , kV (g)f − V (h)f k = kV (g −1 h)f − f k → 0 as h → g in the topology of G. When (i)-(iii) hold, V : G → B(H) is termed a unitary representation of G on B(H). For each representation V of G, we can restrict attention to V (G) ⊂ B(H), which is a group of unitary operators. The dimension of V (G) is called the degree of the representation V , the space H being the representation space of V , denoted V (H) (instead of V (G)). Thus if G is an LCA group, the degree of the representation is one, and we recover the classical case. As examples, for the right and left translations of G on L2 (G, µ) = H where µ is a left Haar measure, we have (Rg f )(x) = f (xg) and (Lg f )(x) = f (g −1 x). Then one shows that kRg f1 − Rh f2 k2 = kRgh−1 f1 − f2 k2 → 0 as kf1 − f2 k2 → 0. Other properties can be verified with standard results from measure theory. If G = R, then these are familiar. In the case of groups, if f ∈ L1 (G, µ) with µ as the normalized Haar measure on a compact group G (so µ(G) = 1), and if V : G → B(H), H = L2 (G, µ) is a unitary representation, then the Fourier transform of f is defined as an operator valued function fˆ by the equation (with 1, f ∈ L2 (G, µ) so that Vλ f1 , Vλ 1 are in L2 (G, µ)) : Z fˆ(λ) = f (x)Vλ (x) dµ(x) = hVλ 1, f i (9) [fˆ(λ)h =

Z

G

G

f (x)Vλ (x)h(x) dµ(x), h ∈ L2 (G, µ) ∩ L1 (G, µ), G is LCA]

and fˆ(λ) is a (finite order) matrix function, the dimension of the matrix d(λ) is called degree of representation. This fact is not obvious and in fact is a deep theorem due to F. Peter and H. Weyl obtained in 1927. If G is abelian then d(λ) = 1 and the earlier case is included. With this motivational discussion we consider the case for bimeasures in this context for LCA groups G1 , G2 . Definition 5.2.5 If G1 , G2 are LCA groups, V1 (·), V2 (·) are strongly continuous unitary representations of G1 , G2 on a Hilbert space H, let S(G1 , G2 ) = {α : G1 × G2 → C | hV1 (g1 )ξ, V2 (g2 )ηi = α(g1 , g2 )}

(10)

for some ξ, µ ∈ H. Here ξ, η are arbitrarily fixed. [If G1 , G2 are compact, these can be replaced by 1 each. In general one takes ξ (and η) to be cyclic meaning {V (g1 )ξ, g1 ∈ G1 } linearly spans an H(V ), called the Hilbert space of the representation V of G1 .] The point of this long-winded introduction is that it will exhaust the Fourier transforms of bounded bimeasures extending the measure case. More

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5 More on Random Measures and Integrals

precisely we have the following result, from Graham and Schreiber (1984) and it plays a key role for us in introducing convolution operations for random measures: ˆ 1 , Gˆ2 as their dual groups. Theorem 5.2.6 Let G1 , G2 be LCA groups with G ˜ 1 , G2 ) be the space of all bounded bilinear forms [obtained from bounded Let B(G ˆ1, G ˆ 2 ) be the corbimeasures through the MT-integration as before] and S(G responding function space of Definition 2.5. Then the following conclusions obtain: ˜ 1 , G2 ), its transform exists, B ˆ ∈ S(G ˆ1, G ˆ 2 ), and (i) For each B ∈ B(G ˆ1, G ˆ 2 ) there is a unique B ∈ B(G ˜ 1 , G2 ) satisfying for each α ∈ S(G ˆ α = B. ˆ 1, G ˆ 2 ) and B ∈ B(G ˜ 1 , G2 ) of (i) so that α = B, ˆ we (ii) For each α ∈ S(G have kBk ≤ kξkkηk, ξ, η thereby defining α in (10). (11) Proof. Instead of working with bimeasures, translating the problem to bilinear functionals facilitates a direct (and simply convenient) application of Grothendieck’s inequality relative to which we can associate a pair of finite (probability by normalization) measures µ1 , µ2 . Using these we associate two Hilbert spaces L2 (Gi , µi ), i = 1, 2 and then their direct sum H = L2 (G1 , µ1 ) ⊕ L2 (G2 , µ2 ) in which we can assume that µ1 and µ2 are 2 which only changes the uniform constant in the the same (or replace by µ1 +µ 2 Grothendieck inequality). Note that in this case, C ⊂ H (i.e., complex and real constants are in H) and it can be used. This is detailed in three steps for ease and simlicity. ˜ 1 , G2 ), i = 1, 2, the bounded bilinear forms, we 1. Thus for Bi ∈ B(G have (cf. Proposition 4 and expression (8)) a pair of bounded linear operators Qi : H1 → H2 such that, using the notations of (4), Z (Qi f )¯ g dµ2 , f ∈ H1 , g ∈ H2 . (12) Bi (f, g) = hf ⊗ g, Bi = G2

If V1 , V2 are the strongly continuous representations of G1 , G2 into H1 , H2 so ¯ 1 f1 , V2 (λ2 )g = λ ¯ 2 g, we have that for the characters λi (∈ Gˆi ), V1 (λ1 )f = λ (since 1 ∈ Hi ) ¯1 ⊗ λ ¯2 , B1 i = hQ1 λ1 , λ2 i = hQ1 V1 (λ1 ), |V2 (λ2 )|i. Bˆ1 (λ1 , λ2 ) = hλ

(13)

Consider Q1 as a bounded operator on H = L2 (G1 , µ1 ) ⊕ L2 (G2 , µ2 ) by idenˆ : H → H where Q˜ ˜ x = Q1 x1 by tifying it as a block matrix onH, i. e. let Q x1 ˜ = 0, 0 ˜ x = Q1 x1 . writing Q and x ˜= ∈ H(or x ˜ = x1 ⊗ x2 ∈ H) so Q˜ Q, 0 x2 Then ˜ = sup{kQ˜ ˜ xk : k˜ 0 < kQk xk ≤ 1} = sup{kQ1 x1 k : kx1 k ≤ 1} = kQ1 k = c < ∞.

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183

˜=Q ¯ is a contraction on H. Hence by the known Hilbert space results, Thus 1c Q ˜ Q is dilated to a unitary operator U on a larger Hilbert space K ⊃ H, (cf. SzNagy and Foia¸s (1970), p. 22). We let K = H⊕H⊥ = L2 (G1 , µ1 )⊕L2 (G2 , µ2 )⊕ H⊥ . Define V˜1 = V1 ⊕I⊕I, V˜2 = U ∗ (I⊕V2 ⊕I)W, x˜ = (c·1, 0, 0), y˜ = U ∗ (0, 1, 0), where I is the identity and 1 ∈ L2 (G1 , µ1 ) as well as 1 ∈ L2 (G2 , µ2 ). ˜ ≤ KG kB1 k; k˜ Thus x ˜, y˜ ∈ K and k˜ xkK ≤ kQk ykK ≤ kU ∗ k ≤ 1, Moreover, ˆ 1 , λ2 ) = hcU (V1 (λ1 )1, 0, 0), (0, V2 (λ2 )1, 0)i B(λ = hcV1 (λ1 )1, 0, 0), U ∗(0, V2 (λ2 )1, 0)i = hV˜1 (λ1 )˜ x, V˜2 (λ2 )˜ y i.

(14)

ˆ ∈ S(G ˆ 1, G ˆ 2 ) as asserted in (i). Only uniqueness and (ii) remain to be Hence B shown. ˆ1, G ˆ 2 ) corresponds uniquely to a bi2. We now assert that each α ∈ S(G 1 1 ˆ to complete the linear form B on L (G1 , µ1 ) × L (G2 , µ2 ) and that α = B 1 ∧ 1 argument. Thus let f ∈ (L (G1 , µ1 )) and g ∈ (L (G2 , µ2 ))∧ so that by the ˆi, µ standard (Fourier) analysis, with (G ˆi ) as the dual Haar measure spaces of ˆi, µ (Gi , µi ), i = 1, 2, there exist uniquely ϕi ∈ L1 (G ˆi ), i = 1, 2, such that ϕˆ1 = f and ϕˆ2 = g (cf.(9)). We then can introduce a bilinear form by the equation Z Z hf ⊗ g, Bi = ϕ1 (−λ1 )ϕ2 (−λ2 )α(λ1 , λ2 ) dˆ µ1 (λ1 )dˆ µ2 (λ2 ). ˆ2 G

ˆ1 G

Since B(·, ·) is clearly bilinear, let us verify its boundedness. Consider Z Z |hf ⊗ g, Bi| = ϕ1 (−λ1 )ϕ2 (−λ2 )hV1 (λ1 )x, V2 (λ2 )yi dˆ µ1 (λ1 )dˆ µ2 (λ2 ) ˆ ˆ ZG2 G1 Z ϕ2 (−λ2 )V2 (λ2 )y dˆ µ2 (λ2 )i = h ϕ1 (−λ1 )V1 (λ1 )x dˆ µ1 (λ1 ), ˆ2 G

G1

=

|hV1 (ϕ¯∗1 )x,

≤

kV1 (ϕ¯∗1 )k

V2 (ϕ¯∗2 )yi| ,

kV2 (ϕ¯∗2 )k

Vi (ϕ∗i )

being the integral transforms of ϕi shown,

kxk · kyk.

(15)

ˆ ∞ (G2 , µ2 ) preThus B is a bounded bilinear form. It extends to L∞ (G1 , µ1 )⊗L serving the bounds, again as a consequence of Grothendieck’s inequality seen as follows. By that inequality we have |hf ⊗ g, Bi| ≤ Ckf k2 kgk2 , f ∈ C0 (G1 ), g ∈ C0 (G2 ), for a pair (µ′1 , µ′2 ) of Grothendieck’s measures. Then there exist fn , gn such that fn → f, gn → g in L2 (G1 , µ′1 ) and L2 (G2 , µ′2 ), by the density of C0 (Gi ) in L2 (Gi , µi ), and so the above inequality persists. It now follows that the result ˆ 2 (G2 , µ2 ). holds on L2 (G1 , µ1 )⊗L

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5 More on Random Measures and Integrals

¯ 2 , Bi, λi ∈ G ˆ or that α(λ1 , λ2 ) = hλ¯1 ⊗ λ ˆi, 3. We next show that α = B, verified by approximating with suitable functions contracting to the characters λi , i = 1, 2. This needs some special attention. Note that for each neighborhood ˆ i , there is a function 1 ≥ ϕNi ≥ 0 supported in Ni of the identity ei ∈ G R µi (x) = 1, (ϕ∗Ni = ϕNi , being nonnegative). This Ni such that Ni ϕNi (x) dˆ ˆ i ) for each σ ∈ G ˆ i , and x in choice is possible (due to regularity of µ ˆ i on the G compacts of Gi , we have if τσ is the shift so (τσ f )(x) = f (x − σ), and for the Ni chosen (τσ (ϕNi ))∧ (x) = (x, σ)ϕˆNi (x) → (x, σ), x ∈ Ci , (Ni ↓ e)

(16)

uniformly on compacts Ci ⊂ Gi . Since the Grothendieck measures µi are regular the above limit also holds in L2 (Gi , µi ), i = 1, 2. Thus for x, y ∈ H determined by the Grothendieck pair µi , constructed prior to Step 1 above, we have, using the norm continuity of the representations V1 , V2 (µ1 , µ2 being probability measures), Z Z | µ1 (λ1 )dµˆ2 − hx, yi| ϕN1 (λ1 )ϕN2 (λ2 )hV1 (λ1 )x, V2 (λ2 )yi dˆ ˆ2 G ˆ1 G Z Z ≤ µ1 (λ1 )dˆ µ2 (λ2 ) ϕN1 (λ1 )ϕN2 (λ2 )|hV1 (λ1 )x, V2 (λ2 )yi − hx, yi| dˆ N2

≤ sup

N1

sup |hV1 (λ1 )x, V2 (λ2 )yi − hx, yi|

λ1 ∈N1 λ2 ∈N2

ˆ i. → 0 as Ni ↓ ei ∈ G

(17)

Letting N = (N1 , N2 ) ↓ (e1 , e2 ), we have ¯⊗λ ¯2 , Bi hλ = hlim((τλ1 ϕ¯N1 )∧ ⊗ (τλ2 ϕ¯N2 )∧ , B)i N↓

= limh(τλ1 ϕ¯N1 )∧ ⊗ (τλ2 ϕ¯N2 )∧ , Bi N↓ Z Z ˆ 2 (λ′ ) = lim µ1 (λ′1 ) dµ (τλ1 ϕ¯N1 )(λ′1 )(τλ2 ϕ¯N2 )(λ′2 )α(λ′1 , λ′2 ) dˆ 2 N↓ G ˆ2 G ˆ1 Z Z = lim µ1 (λ′1 ) dˆ µ2 (λ′2 ) ϕN1 (λ′1 − λ1 )ϕN2 (λ′2 − λ2 )hV1 (λ′1 )x, V2 (λ′2 )yi dˆ N↓ G ˆ2 G ˆ1 Z Z = lim µ1 (λ′1 ) dˆ µ2 (λ′2 ) ϕN1 (λ1 − λ′2 )ϕN2 (λ2 − λ′2 )hV1 (λ′1 )x, V2 (λ′2 )yi dˆ N↓

ˆ2 G

ˆ1 G

since the neighbourhoods can be taken symmetric, Z Z = lim µ1 (λ′1 ) dˆ µ2 (λ′2 ) ϕN1 (λ′1 )ϕN2 (λ′2 )hV1 (λ′1 )V1 (λ1 )x, V2 (λ′2 )V2 (λ2 )yi dˆ Ni

ˆ2 G

ˆ1 G

= hV1 (λ1 )x, V2 (λ2 )yi = α(λ1 , λ2 ).

(18)

ˆ For uniqueness, let B1 , B2 be a pair of bounded This establishes α = B. ˆ1 = B ˆ2 (= α). We claim that B1 = B2 on bilinear functions such that B

5.2 Bilinear Forms and Random Measure Algebras

185

ˆ 2 (G2 , µ2 ), which needs an additional but easy argument as L2 (G1 , µ1 )⊗L follows. Let (µi1 , µi2 ) be the Grothendieck pair associated with Bi , i = 1, 2. Taking µ′i = 12 (µi1 + µi2 ), i = 1, 2, both of which having C0 (Gi ), as dense subsets, we can consider L2 (Gi , µ′i ) = L2 (Gi , µ1i ) ∩ L2 (Gi , µ′2i ), i = 1, 2, and see that B1 and B2 agree on trigonometric polynomials of L2 (Gi , µ′1 ) ⊗ L2 (G2 , µ′2 ), which ˆ L2 (G2 , µ2 ) are dense. So B1 = B2 also holds by continuity on L2 (G1 , µ′1 ) ⊗ ˆ ˆ and hence B1 = B2 = α implies that B1 = B2 and the theorem follows. 2 This detailed argument is given as the resulting structure is needed to introduce convolution of bimeasures and of random measures. The desired concept can now be stated as follows. Definition 5.2.7 Let B1 , B2 be a pair of bounded bilinear functionals on ˆ2 be their Fourier transforms as given in Theorem 6, and are G1 , G2 and Bˆ1 , B ˆ1, G ˆ 2 ). [As shown below it is an algebra so that the pointwise elements of S(G ˆ1 · B ˆ2 ∈ S(G ˆ1, G ˆ 2 ).] We define the convolution product has the property that B ˜ ˆ1 · B ˆ2 which B1 ∗ B2 as an element of B(G1 , G2 ) by the equation (B1 ∗ B2 )∧ = B ˆ ˆ is unambiguously defined in the (complex-valued) space S(G1 , G2 ). [It is seen below to be an algebra.] The desired algebra property is inferred from: ˆ1, G ˆ 2 ) of Definition 2.5 is closed under pointwise Lemma 5.2.8 The set S(G products, sums and complex conjugation so that it is an algebra. ˆ 1, G ˆ 2 ) and V1 , V2 be unitary representations, and x, y ∈ H Proof. Let α, β ∈ S(G as in Definition 2.5 above for α. Similarly if V1′ , V2′ and (x′ , y ′ ) are for β on H′ , another Hilbert space, then on the direct sum H ⊕ H′ , the representations V1 ⊕ V1′ , V2 ⊕ V2′ are unitary and α(λ1 , λ2 ) + β(λ1 , λ2 ) = hV1 (λ1 )x, V2 (λ2 )yiH + hV1′ (λ1 )x′ , V2′ (λ2 )y ′ iH′ = h(V1 (λ1 )x, V1′ (λ1 )x′ ), (V2 (λ2 )x, V2′ (λ2 )y ′ )iH⊕H′

= h(V1 ⊕ V1′ )(λ1 )(x, x′ ), (V2 ⊕ V2′ )(λ2 )(y, y ′ )iH⊕H′ ,

ˆ1, G ˆ 2 ). Regarding multiplication, consider their tensor product so α + β ∈ S(G ′ ˜ space H ⊗ H (in view of the direct sum) and observe that the representations ˜ ′ . Also using the definition of tensor V1 ⊗ V1′ and V2 ⊗ V2′ are unitary on H⊗H product operations we have α(λ1 , λ2 )β(λ1 λ2 ) = hV1 (λ1 )x, V2 (λ2 )yiH hV1′ (λ1 )x′ , V2′ (λ′2 )y ′ iH′ = hV1 (λ1 )x ⊗ V1′ (λ1 )x′ , V2 (λ2 )y ⊗ V2′ (λ′2 )y ′ iH⊗H ˜ ′

= h(V1 ⊗ V1′ )(λ1 )(x ⊗ x′ ), (V2 ⊗ V2′ )(λ2 )(y ⊗ y ′ )iH⊗H ˜ ′

ˆ 1, G ˆ 2 ) as asserted. Since complex conjugation and which shows that αβ ∈ S(G multiplication by scalars are evident the result follows. 2

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5 More on Random Measures and Integrals

We now observe that the composition operation of Definition 2.7 above, which is well-defined by the preceding lemma (and Theorem 2.6) is also continuous. More precisely the following modified Banach-algebra inequality holds: Proposition 5.2.9 If B1 , B2 are bounded bilinear forms on C0 (G1 ) × C0 (G2 ) then their composition B1 ∗ B2 , of Definition 2.7 is continuous in the sense that 2 kB1 ∗ B2 k ≤ KG kB1 k kB2 k, (19) where KG is the Grothendieck constant. Proof. Since the convolution of B1 and B2 is defined in terms of their (Fourier) ˆ1 and B ˆ2 the desired bound will be deduced through the latter, transforms B with the dilation procedure used in Theorem 6. This type of round about argument seems unavoidable since the bimeasures βi determined by Bi need not have finite (Vitali) variations and since the MT-integral behaves well only if the integrand is a product of single parts and hence the integrand β1 (A − x, B − y) to be integrable for β2 (·, ·) will not be well-defined. [Compare with Exercise 5 of ˜ be Chapter 2.] Thus using the dilations in the proof of Theorem 6 above, let Q the operator determined by B1 (cf. (14) there) along with x ˜, y˜ in the extended space K, so that ˜ ≤ KG kBk, and k˜ k˜ xkK ≤ kQk ykK ≤ kU1∗ k = 1.

(20)

Consider the associated unitary representations that go with B1 namely ViB1 (on ˜B1 , y˜B1 (∈ K′ B1 ) such that the space K′ B1 ), and vectors x ˆ1 (λ1 , λ2 ) = hV B1 (λ1 )˜ ˆ i , i = 1, 2. B xB1 , V2B1 (λ2 )˜ yB1 i, λi ∈ G 1 We then get the bound (cf.(11)) as (KG being Grothendieck’s constant) yB1 k ≤ KG kB1 k · 1·, by (20). kB1 k ≤ k˜ xB1 k k˜

(21)

Similarly we obtain a bound for B2 . To consider B1 ∗ B2 we need to recall the tensor product (to use Bˆ1 · Bˆ2 ). Thus let K′ Bi , i = 1, 2 be the enlarged Hilbert spaces, and consider K = ˜B2 , y = y˜B1 ⊗ y˜B2 and use the bounded (new) ˜B1 ⊗ x K′ B1 ⊗ K′ B2 , x = x representations Vi = ViB1 ⊗ ViB2 , i = 1, 2. Then α(λ1 , λ2 ) = hV1 (λ)x, V2 (λ2 )yi, by definition, and this with (21) simplifies, as the new bilinear functional given by α(·, ·), to the following: 2 kB1 k kB2 k. yB2 k ≤ KG yB1 k˜ xB2 kk˜ kB1 ∗ B2 k ≤ kxkK kykK = k˜ xB1 k˜

(22)

This gives the asserted bound, after such a long and round about route at last! 2

5.2 Bilinear Forms and Random Measure Algebras

187

Remark 5.3 With this procedure we are able to show that there is a uniquely defined bilinear functional B = B1 ∗ B2 as a convolution product. Also in (22), KG > π2 so that the convolution algebra of Bi ’s is not a Banach algebra. Unlike in the one dimensional case (cf. Rudin (1962), p.22) we do not yet have an inversion formula for bilinear functionals and direct computations, without assuming Vitali variations, are not available. None-theless, the convolution is well-defined for bimeasures and we can use it to define the same operation for random measures as we are going to show. The special nature (and difficulties) of this new operation should be recognized. The following result then shows how random measure convolutions on LCA groups can be defined and analyzed by extending the ideas of the earlier Proposition 5.2.4. Theorem 5.2.10 Let (Gi , Gi ), i = 1, 2 be a pair of LCA groups with Gi as the δ-rings of bounded Baire sets of Gi . Suppose Zi : Gi → L20 (P ), i = 1, 2 are random measures whose covariance bimeasures are regular, determining bounded bilinear forms Bi on C0 (Gi ) so that their composition B1 ∗ B2 is welldefined (whence (19) holds). Then there exists a unique random measure Z on G1 ⊗ G2 with values in L20 (P ) such that its covariance bimeasure determines a bounded bilinear form B which is precisely B1 ∗ B2 and Z is then denoted as Z = Z1 ∗ Z2 : G1 × G2 → L20 (P ) where the probability space (Ω, Σ, P ) can be taken rich enough (by the adjunction procedure if necessary) to support all this structure. Proof. Let Z1 and Z2 be random measures, as given on (G1 , G1 ) and (G2 , G2 ), with induced bilinear forms obtained by their covariance bimeasures β1 , β2 which are positive definite. Then by Lemma 8 and Proposition 9 above it is quickly seen that the convolution B = B1 ∗ B2 is positive definite where B1 , B2 are the bilinear forms induced by β1 , β2 using the (strict) MT-integrals. The positive definite bilinear functional B then determines on a probability space, which can (and will) be taken as (Ω, Σ, P ) and hence on L2 (P ) itself. But then by Proposition 1.3 there exists a random measure Z (based on G1 × G2 ) into L20 (P ) with covariance bimeasure determined by the bilinear form B. We denote this new random measure Z as Z2 ∗ Z2 , and it is termed the convolution operation. The previous work implies that Z(·) is σ-additive in L20 (P ), and is uniquely determined by Z1 and Z2 . [Here we may have to consider tensor products of Z1 and Z2 on G1 × G2 but the essentials are clear.] The method being similar to that of the above Proposition 2.9 we can omit further detailed discussion. 2 Remark 5.4 1. The preceding work opens the way for a new study of random measure algebras, and one can investigate properties motivated by the classical (measure) deterministic algebras keeping in mind that now we need to use the

188

5 More on Random Measures and Integrals

MT-integration and the related bilinear forms which avoid the analogs of integrals of functions that are not of the “product form”, i.e., β(A − x, B − x) be replaced by f (A − x) · g(B − x) which need not hold in this context. Bimeasures on sub and quotient groups and their consequences for random measures are possible. 2. There are several new problems and opportunities to explore in this context. For instance if Z1 , Z2 are a pair of independent random measures on S → L20 (P ), then Z = Z1 · Z2 , the pointwise product presents an interesting ¯ problem since β1 (A, B) = E(Z(A)Z(B)) = E(Z1 (A)Z¯2 (B)) · E(Z1 (A)Z¯2 (B)) = β1 (A, B) · β2 (A, B) gives a positive definite bimeasure (by Schur’s lemma), and the resulting special analysis will be interesting. All these particular questions can be pursued, but we shall leave them at present as interesting new problems and continue on other aspects of the subject. 3. In the classical studies, the R convolution of the (vector) measures Z1 , Z2 is defined as (Z1 ∗ Z2 )(A) = R Z1 (A − x) dZ2 (x). This formula for genuine random cases presents really serious problems. Here Z1 (A − x) is a random integrand, and hence the integral has to be in the sense of Itˆ o or one of its extensions. However, the Lebesgue-Stieltjes method should be replaced by the stochastic integration procedure, which at the very start depends on a “filtration” of a family of σ-algebras Si ⊂ Σ, Si ⊂ Si′ , for i ≤ i′ and Si = ∩j>i Sj . This is automatic in the classical (non-stochastic) case since the integrands are deterministic (i.e, nonrandom) so that Si = {∅, Ω} for all i. But the filtration is needed in the stochastic case. For random measures which should include Brownian motion, orthogonally valued and “martingale types” the filtration condition appears to be crucial. A generalized (weaker) integral due to Stratonovich also depends on the filtration condition, and is usable for classes of random integrators called “semi-martingales” and “quasi-martingales”. However, our measures Z(·) need not satisfy any of these conditions. In fact Z(A − x) dZ(x) in the above has as variables both A and x which are not tied to the filtration conditions. [We study some martingale measures and integrals in Chapter 6.] It does not seem likely that, even with a useful connection established by Itˆ o (1975) between the martingales integrals and the Stratonovich formulation, one can use a direct approach and define Z1 ∗ Z2 . There is another variant of the Itˆ o-calculus, extending the classical Stieltjes approach due to L.C.Young (1970, 1974) tailored to ‘martingale type’ methods. As yet these methods have not been successfully extended to generalize the convolution algebras. Combining these ideas with the formal operations on stochastic differentials, introduced by K. Itˆ o (1975) forming an abelian ring will be interesting to pursue in studying random measure algebras. This is an open area for a future exploration and for an extended investigation of “random convolutions”.

5.3 Vector Integrands and Integrals with Stable

Random Measures

189

5.3 Vector Integrands and Integrals with Stable Random Measures In this section we restrict random measures to be of stable class, thus excluding several Hilbert space valued measures, but including Brownian motion classes. We now allow general X (Banach and/or Fr´echet)-valued integrands. If (S, S) is a measurable space where S is locally compact and S its δ-ring of bounded Borel (or Baire) sets, then a mapping Z : S → Lp (Ω, Σ, P, X) which is additive on disjoint elements of S, will be called a (vector valued) random measure, P∞ ∞ if An ∈ S disjoint, ∪∞ A ∈ S implies Z(∪ A ) = n=1 n n=1 n n=1 Z(An ) in the topology of Lp (P, X), p ≥ 0. Since P (Ω) = 1 implies Lp1 (P, X) ⊃ Lp2 (P, X) if 0 ≤ p1 ≤ p2 ≤ ∞, for the present analysis we make the following assumptions: (i) X is a Banach (or a Fr´echet) space and (ii) Z(A) has p-moments, p ≥ 0, and Z(A), Z(B) are independent if A ∩ B = ∅, so that the σ-additivity is in P -measure which is equivalent when p ≥ 0 to the Lp (P, X)-metric, where the R p integral E(|Z(A)|X ) = Ω |Z(A)|pX dP is in Bochner’s sense. It is thus clear that one has to make assumptions on X (to include the multivariate case) and from the preceding analysis (of Chapters 1–4) with special geometry of the Hilbert space, when p = 2, one needs to concentrate essentially for 0 ≤ p ≤ 2. Thus we consider random measures in this space first in order to study in some depth and detail. The general class considered is the stable family, of which the Gaussian collection (including Brownian motion) is the most prominent one. This study supplements the earlier work in Sections 3.2 and 3.3 on the subject. Let us recall the concept due to P. L´evy for a convenient reference and use. Definition 5.3.1 Let (S, S) be a measurable space with S as a δ-ring, so that if µ : S → R+ is a measure it is finite on S. A random measure Z : S → Lp (P ), on some probability space (Ω, Σ, P ) is p-stable if (a) for disjoint A1 , . . . , An ∈ S, {Z(A1 ), . . . , Z(An )}, n ≥ 2 is a vector of mutually independent random variables, (b) each Z(A), A ∈ S, has a stable distribution FA so that its characteristic function φA (·) is given by φZ(A) (t) = E(eitZ(A) ), t ∈ R,

= exp{iα(A)t − c(A)|t|p [1 − β(A) sgn t · ψ(p, t)]

where |β| ≤ 1 and ψ(p, t) =

tan πp 2 , 0 < p 6= 1 ≤ 2 2 log |t|, p = 1. π

(1)

(2)

Here α(A) ∈ R, c(A) ≥ 0; and (c) if An ∈ S, disjoint, ∪n≥1 An ∈ S, then ! ∞ X [ Z Z(An ) (3) An = n=1

n=1

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5 More on Random Measures and Integrals

the series converging in probability hence (by independence) also with probability one. The functions α, β and c are called parameters of p-stability. It follows from the additivity of Z(·), the same is inherited by α(·) and c(·) so that they are signed and positive measures respectively. But β(·) only has the weaker property: ∞ X

n=1

c(An )β(An ) = c

∞ [

n=1

∞ [ An β An

(4)

n=1

and hence the σ-additivity of β obtains only in the very special case that c(·) ≡ 1. Note that if S is a locally compact space, S can be the δ-ring of bounded Borel sets so that c : S → R+ . In the abstract case, of a measure µ(·) given on S, µ(A) must be finite for each A ∈ S, and should dominate (= control) Z. In particular µ(·) can be a positive multiple of c(·), although the measure µ need have no relation to α(·), c(·) or β(·), in the formulation of the problem. It is however useful to observe that a random measure Z(·) as in the above ˆ µ) is definition always exists on any given δ-ring, and in particular if (S, S, ˆ a measure space and S = {A ∈ S : µ(A) < ∞} so that S is a δ-ring then we can have the concept of a p-stable random measure. One says that Z(·) is symmetric if Z(A), A ∈ S is a symmetric (real) random variable. For a better appreciation we present an existence result, based on a given set of parameters (α, β, c) and measure µ : S → R+ , for reference in the following form: [It is known that each p-stable random measure Z : S → Lp (P ) determines a unique set of parameters (α, β, c), and the converse is asserted by the next proposition, so that (α, β, c) and µ determine a p-stable Z(·) for some 0 < p ≤ 2.] The following relatively “simple result” will play a conceptually important part later on. ˜ µ) be a measure space with S ⊂ S ˜ as the δProposition 5.3.2 Let (S, S, ring of finite µ sets, and α ∈ R, c ≥ 0, |β| ≤ 1 be set functions defined on S so that (α, β, c) are possible parameters, satisfying conditions (1)(4) above. Then for any 0 < p 6= 1 ≤ 2, there exists a probability space, (Ω, Σ, P ) and a random p-stable measure Z :R S → L0 (P ) with (α, β, c) as its parameters for each µ : S → R+ , so that S f (s) dZ(s) is defined for all f ∈ Lp (µ). Proof. By the uniqueness theorem for characteristic functions we start with a functional of the form (1) using the parameters (α, β, c) and verify that it is positive definite and continuous on (S, µ), and then associate a random measure on a probability space, showing that the latter is (distributionally) unique. Thus consider a function space Xp = L1 (µ) ∩ Lp (µ) with the metric

5.3 Vector Integrands and Integrals with Stable 1 1∧ p

kf kp = kf k1 + kf kp

Random Measures

191

0 < p < 1 or 0 < p ≤ 2, f ∈ Xp ,

,

and define φp (f ) = iα

Z

S

f dµ − c

Z

S

|f |p (s){1 − iβ sgn f (s)ψ(p)} dµ(s),

(5)

1 where ψ(p) = tan π2 p and for p = 1 let f ∈ X1 = L1 (µ) ∩ L1+ e (µ). Now set Z Z 2 φ(f ) = iα f (s)dµ(s) − c |f |(s){1 − iβ sgn f (s) log |f (s)|} dµ(s). (6) π S S Thus (5) and (6) have the corresponding formula for f ∈ Xp and f ∈ X′1 . This difference for p = 1 and p ∈ (0, 2] − {1} is often termed a “disagreeable feature” for stable families and this is a fact of life. It has to be treated separately. Since from the classical Probability Theory the exponential function (1) is a characteristic function (ch.f.) as a consequence of an important theorem due to I.J. Schoenberg (1938), and it is also true that for each λ ≥ 0 the expression (φZ(A) (t))λ is a ch.f., and from this fact, approximating f by simple functions in (6), we conclude that exp φ(f ) is a characteristic functional, i.e., is continuous and positive definite (f being integrable) for f ∈ Xp . Then an extended Bochner theorem is applicable for an integral representation of exp φ(f ) (cf. Bochner (1955), Theorem 5.4.4, p.136) according to which there is a probability space (Ω, Σ, P ), Ω = Xap the algebraic dual of Xp (no topology) and a bilinear form < ·, · > : Xp × Xap → R with Σ as the cylinder σ-algebra such that < f, · > : Xap → R is measurable for each f ∈ Xp (hence for all cylinder sets) such that Z ei dP (g), f ∈ Xp . (7) exp{φ(f )} = Ω

[The space (Ω, Σ) is thus constructed using the classical Kolmogorov existence theorem and the fact that exp{φ(f )}’s form a consistent family.] This is the key point for the existence result. Now let Y : Xp → L0 (P ) be a mapping such that for each f ∈ Xp , Y (f ) = < f, · > (the equivalence class) in L0 (P ) as usual. Then Y is a well-defined random variable, and we put Z : A 7→ Y (χA ), A ∈ S(χA ∈ Xp ). It is now asserted that Z(·) is the desired random measure. Replacing χA by tχA , t ∈ R in the definition we get by (5) and (7) that Z Z i eitZ(A)(ω) dP (ω) = eφ(tχA ) . (8) e dP (g) = Ω

Pn

Ω

Taking f = i=1 ai χAi , Ai ∈ S disjoint, ai ∈ R, so f ∈ Xp , we see from (5) and (8) the following:

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5 More on Random Measures and Integrals

Z

eit

Pn

j=1

aj Z(Aj )(ω)

dP (ω) =

Ω

Z

eit dP (ω)

Ω

= exp{iαt

n X j=1

= =

ai µ(Aj ) − c

n X j=1

|taj |p µ(Aj )

×{1 − iβ sgn (taj ) tan

n Z Y

πp } 2

eitaj Z(Aj )(ω) dP (ω)

Ω

j=1 n Y

exp{φ(t, gj χAj )}.

j=1

Hence Z(A1 ), . . . , Z(An ) are independent p-stable random variables. To show that Z(·) is σ-additive in probability, considering for ε > 0, we use the so-called truncation inequality (cf., e.g. Lo`eve (1963), p.196): P {ω ∈ Ω : |

n X

j=m

Z(Aj )(ω)| ≥ ε} ≤ 7ε = 7ε

Z

1 ε

0

Z

0

−c

1 ε

Re(1 − eiφ(tf ) ) dt Re(1 − exp[iαt

n X

j=m

n X

µ(Aj )

j=m

µ(Ai )|t|p {1 − iβ sgn t tan

πp }] dt. 2

P∞ S∞ Since A = j=1 Aj ∈ S, so that µ(A) < ∞ and µ(A) = j=1 µ(Aj ) < Pn ∞, the integral converges to zero since µ(A ) → 0, as m, n → ∞. j j=m Hence Z(·) is σ-additive, and we conclude the existence as asserted in the statement. If p = 1, a similar argument works when |f | log |f | is shown to be integrable for f ∈ X′p , the rest of the argument being the same. This is concluded by the following elementary estimates for 0 < t < 1. We have −

1 1 ≤ t log t ≤ t1+ e e

so that for f ∈ X′p and 0 < δ < 1 Z Z |f (s)| log |f (s)|dµ(s) ≤ S

[|f |<δ]

+

Z

|f log |f || dµ +

[log |f |≥1]∪[|f |<1]

Z

e−1 dµ

[δ≤|f |<1] 1

|f |1+ e dµ < ∞.

5.3 Vector Integrands and Integrals with Stable

Random Measures

193

Thus eφ(f ) is a ch.f. in both cases, and hence Z(A) = Y (χA ) gives a random measure by the procedure above since eφ(t) is a ch.f. Hence the result follows. 2 Remark 5.5 The parameter set (α, β, c, µ) of a p-stable random measure on (S, S) has the following significance for β. If β = 1 or (−1), then the random measure Z(·) with these parameters is positive or (respectively) negative. If |β| < 1, then the values of Z(S) is the set R itself. The distributions are not continuous in the parameters. [For details, see Zolotarev (1986).] It was already noted in Section 3.1 that every random measure Z : B0 (R) → L0 (P ) has a dominating or controlling measure ν whenever Z(A) is infinitely divisible for each A ∈ B0 (R), the δ-ring of bounded Borel sets. For a p-stable measure with parameter set (α, β, c, µ) we have noted that ν = |α|(·) + c(·) can serve as a controlling measure and Z(A) is infinitely divisible. Hence the mapping Z f dZ, f ∈ C0 (R). (9) T : f 7→ R

is well defined from Lp (µ) ∩ C0 (R) → Lp (P ). We can now present a more detailed information about T than before. The next key result, sharpening Theorem 3.1.1., is due to Okazaki (1979).

Theorem 5.3.3 Let (S, S, µ) be a measure space with S as a δ-ring and µ : S → R+ , a (finite) measure. If Z : S → Lp (P ) is a p-stable (0 < p ≤ 2) random measure with (α, β, c, µ) as its parameter set, and Xp (X′p ) are the metric function spaces as in Proposition 2, so that ν can be taken as µ + c + |α| and T : Xp (X′p ) → Lp (P ) is uniquely defined and it has the following structural description: 1. T : Xp → Lp (P ) is one-to-one and its range consists of p-stable random variables whose ch.f. is given in detailed form as: Z eitT (f )(ω) dP (ω), f ∈ Xp , (10) eφ(f,t) = Ω

where the functional φ(f, t) is described as (with f = f + − f − ) Z Z πp } φ(f, t) = iαt f dµ − c|t|p (f + )p dµ{1 − iβ sgn t tan 2 S S Z πp −c|t|p (f − )p dµ{1 + i sgn t tan }, 2 S and for f ∈ X′p it is given by (with f = f + − f − )

(11)

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Z Z 2 f log |f |dµ t φ(f, t) = i α f dµ + cβ π S ZS 2 −c f + dµ |t|{1 − iβ sgn t log |t|} π S Z 2 −c f − dµ |t| 1 + iβ sgn t log |t| . π S

(12)

Moreover for 0 ≤ q < p, T : Xp → Lq (P ) is also continuous (and linear). 2. If either α = 0 and p 6= 1, or p = 1 and β = 0 = α, the mapping T (is well-defined and) takes Lp (µ) → L0 (P ). It is an isomorphism of Lp (µ) into Lq (P ) for 0 ≤ q < p ≤ 2. 3. If moreover Z(·) is symmetric (with parameters just (0, c, 0, µ)), then T : Lp (µ) → Lq (P ), 0 ≤ q < p, whose range consists of symmetric p-stable random variables and its characteristic functional given by: Z Z eitT (f )(ω) dP (ω) = exp −c|t|p |f |p dµ , f ∈ Lp (µ). (13) Ω

S

p

Proof. Since Z : S → L (P ) is a random measure for the given parameter set (α, β, c, µ), it follows from Proposition 2 (and even otherwise on using the fact that Z is infinitely divisible) and Theorem 3.1.1 that the integral T f given by (9) is defined for all f ∈ Xp (or f ∈ X′p ) and because of the availability of a controlling measure for Z(·), we conclude that as fn → 0 in Xp (or X′p ), T fn → 0 in probability so that T : Xp → L0 (P ) is a continuous linear mapping and that (10) for (T f ) is a ch.f., and the same holds if X′p is used instead as the domain. We now establish formulas (11) and (12). By Proposition 2 we have the ch.f. E(ei(T f ) ) given by the following (cf. (6)), where f is replaced by tf , i.e., φ(f ) is written as φ(tf ) = φ(f, t) so that we have Z Z πp p }dµ. (14) φ(f, t) = iαt f (s)dµ(s) − c|t| |f (s)|p {1 − β sgn (tf (s)) · tan 2 S S Here sgn (tf (·)) can be simplified as sgn t(f + − f − ) = ( sgn t)χ[f >0] − ( sgn (t)) χ[f ≤0] , so that the above formula reduces to (11). R If T f = 0, then (10) shows that | Ω ei(T f )(ω) dP (ω)| = 1 and hence R R p (11) (and (12)) imply that e−c S |f | dµ = 1. So S |f |p dµ = 0 or f = 0 a.e. [µ]. Consequently T is one-to-one. Since (T f ) is a p-stable random variable, p > 0, then for 0 < q < p, it is clear from elementary integration that E(|T f |q ) < ∞. Hence with α = 0, by the closed graph theorem T : Xp → Lq (P ) is continuous. This implies all the assertions of part 1. For 2., we take β = 0 (and p = 1), so that using the same argument, T is seen to be continuous. The preceding analysis shows that T is

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an isomorphism for all 0 ≤ q < p, i.e., T is an isomorphism of Lp (µ) into Lq (P ). Finally 3. follows immediately from 1. and 2. 2 In the above work we focused on mappings T : Xp → Lp (P ) defined by a p-stable random measure Z : S → Lp (P ), as integrals. A natural and immediate question is to replace Xp (= Lp (µ)) by a space of vector valued µ-integrable functions so that Lp (µ) of scalar functions is replaced by Lp (µ, Y)(= Lp (S, S, µ, Y)) of Y-valued functions, integrable relative to µ in the classical Bochner (or Pettis) sense when Y is a suitable Fr´echet or a Banach space or even a space of (vector valued) random variables. Thus a vast area of the subject opens up, and we indicate here a few of the consequences leaving the possibility of replacing the random measure Z(·) itself with other classes, such as martingale measures considered in the following chapters. There are new (and many) possibilities. A basic reference here is L. Schwartz (1973, Part II) on general cylindrical measures. A simplified account for the present purposes, also outlined in Okazaki (1979), will suffice for our work below. [See pp. 145–6, Complements of Chap. 8, on control measures.] First recall the space Lp (µ; Y), p ≥ 0 of Y-valued functions on (S, S, µ), integrable for pth power. Thus if f : S → Y is a mapping, suppose that it is a limit of simple functions fn : S → Y so that, letting d(·) for the invariant metric in Y relative to which it is complete (the pair (Y, d(·)) is a Fr´echet space, Y = Lp (µ), p ≥ 0 being examples), we have d(f (s) − fn (s)) → 0 as n → ∞ for a.a.(s) (or µ, a.e.) such f are called strongly - µ measurable. This implies that f is (almost) separably valued. To simplify matters we assume that (Y, d(·)) itself is separable so that it will have a dense denumerable subset. For p ≥ 0, R d(f ) R and if S [d(f )(s)]p dµ(s) < ∞, p > 0, and ||f ||0 = Ω 1+d(f ) (s)d(µ(s) 0, S

and kf k0 defined above, that (Lp (µ; Y), k · kp ) is a complete metric space when equivalence functions are identified as usual. This reduces to Lp (µ) considered earlier if Y = R (or C) as the reader can easily verify. If Z : S → Lp (P ) is a random measure having a control measure Pkn ν, and if fn is a simple function as above fn : S → Lp (µ; Y), fn = k=1 yk χAk define the elementary integral Z

fn (s)dZ(s) = A

kn X

k=1

yk Z(Ak ),

y ∈ Y, Ak ∈ S,

(15)

where the Ak are disjoint. It is verified immediately that the expression in (15) is a uniquely defined element of Lp (P ; Y) the corresponding space of Y-valued pth power integrable functions given by the left side (of (15)). We then say that f is D-S integrable if (i) there is a sequence of simple functions fn : S → Y, fn → f

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R pointwise as n → ∞, and (ii) the set { A fn dZ, n ≥ 1} is Cauchy in the metric space Lp (P ; Y) for each A ∈ S, the δ-ring of µ-finite sets of S. The unique limit, denoted as Z Z fn dZ, A ∈ S, f ∈ Lp (µ, Y), (16) f dZ = lim A

n→∞

A

is called the integral of the T-valued strongly measurable f relative to the random measure Z. It depends on two basic results: (i) the uniqueness of the integral in (16) uses the general Vitali-Hahn-Saks theorem and the Egorov theorem, both of which hold for Fr´echet spaces. This is proved in DunfordSchwartz (1958, IV.10.8) for Y = R, but the proofs hold for any (separable) Banach or Fr´echet spaces as needed here. (ii) The other essential property of Z(·) is the existence of a controlling measure ν : S → R+ , as assumed here. The existence of such a measure was proved in Theorem 3.1.1 which was noted to be (possibly) σ-finite. However each σ-finite measure on a σ-algebra is easily seen to be equivalent to a finite measure, and so we take ν to be such. But here we are interested in p-stable random measure Z(·) which is infinitely divisible, and this controlling property is available. Consequently, Z f dZ, f ∈ Lp (µ; Y), (17) T : f 7→ S

is well-defined, and T f is again a p-stable variable having the property that for f1 , f2 with disjoint supports, T f1 and T f2 are independently distributed as seen already before. So far the p-stable measure is arbitrary. But for a given set of parameters (α, β, c, µ) and the (metric or) Fr´echet space Y, we have constructed (possi˜ Σ, ˜ P˜ )) for which bly a different or enlarged, underlying probability space (Ω, the above method can be applied. For this we need to employ the construction of Proposition 2 through cylindrical measures which we now describe, as it is a basic item here. A good reference is the book by L. Schwartz (1973) and the relevant part was also included in the form directly applicable here in the author’s volume (of Rao (1995), Section 1.4). It is an extension of the fundamental Kolmogorov existence theorem and is restated as follows. Let Y be a locally convex Fr´echet space. It is desired to define the Fourier transform of f ∈ Lp (µ, Y), and this is nontrivial when Y is infinite dimensional, which is the case here. Thus let F be the collection of all closed subspaces F of Y of finite deficiency in the sense that the quotient (or factor space Y/F (often denoted F ⊥ )) is identifiable as a finite dimensional subspace of Y. If F1 , F2 ∈ F, then we set F1 ≺ F2 iff F1 ⊃ F2 so that {F1 , ≺} is a directed set, i.e., for each pair F1 , F2 in F either F1 ≺ F2 (or F2 ≺ F1 ) or there is an element F3 ∈ F such that Fi ≺ F3 , i=1, 2. Let YF = Y/F and πF : Y → YF be the quotient (or canonical) map which is a linear projection operator. Similarly if E, F ∈ F and E ≺ F , then πEF : YF → YE can be defined and the family {πE , πEF , (E, F ) ∈ F} satisfies πEF ◦ πF G = πEG (πEE = identity) for E ≺ F ≺ G and the composition rule πEF ◦ πG = πE also holds. The

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collection {(YE , ΣE , µE , πEF )E≺F , E, F ∈ F} is called a projective system of (finite) measure spaces when µE is a finite measure on the Borel σ-algebra ΣE −1 of YE , and the measures µE satisfy the compatibility relations: µE = µF ◦ πEF −1 for all E, F ∈ F with E ≺ F . Let Σ0 = ∪F ∈F πF (ΣF ) which is an algebra. It can then be verified that there is a unique additive µ ˜ : Σ0 → R+ such −1 that µ ˜|πF (ΣF ) = µF . When the system of measures {µF , E ∈ F} is a regular Borel system (i.e., µE (A) = sup{µE (B) : B ∈ ΣE , B ⊂ A ∈ ΣE }, then µ ˜ is also σ-additive on the σ-algebra Σ = σ(Σ0 ), the one generated by Σ0 . This nontrivial theorem is an abstract form of Kolmogorov’s existence result recalled above, a proof of which is in Schwartz’s monograph as well as in the author’s volume (Rao (1995), pp. 20–24). The limit measure µ ˜ need not be Borel regular however although some weaker form (that suffices here) holds. The space on which µ ˜ operates is denoted Ω = lim(YF , πEF ) and is usually ←− isomorphic to Y∗ ′ , containing the second adjoint of Y where Y∗ ′ is the algebraic dual of Y∗ , the latter being the topological dual of the locally convex Fr´echet space Y. [It is for this purpose we assumed that Y is locally convex so that hy, y ∗ i = 0 for all y ∗ ∈ Y∗ implies y = 0 i.e., Y∗ separates points of Y, and this property is not always true for general Fr´echet spaces such as Lp (µ), 0 ≤ p < 1. The purpose of this long-winded, but necessary, discussion is to conclude that we can define the Fourier transform of µ ˜ on Y by the equation Z ∗ ˆ eihy,y i d˜ µ(y), y ∗ ∈ Y∗ (18) µ ˜(y ∗ ) = Y

where hy, y ∗ i is the duality pairing and y ∗ → h·, y ∗ i is a measurable function of Σ. Taking each µE to be a (regular) probability, we get ˆ˜(y ∗ ) Φ(y ∗ ) = µ

(19)

to define Φ(·) as a positive definite function on Y∗ , which is continuous at ‘0’ and hence everywhere. Then by the Bochner-Schoenberg theorem there is a probability space (Ω, A, P ) such that for every strongly measurable f : Ω → Y the mapping T : f → L0 (Ω, A, P ; Y) defined as T (f )(ω) = hf, ·i(ω, ·) : Y∗ → R is a random variable in ω, and moreover Z ∗ eihf (ω),y i dP (ω) = Φ(y ∗ ), y ∗ ∈ Y∗ (20) Ω

is positive definite and continuous in the weak∗ -topology of Y∗ so that it defines a ch.f. on Y∗ . Taking f˜ = χA y, A ∈ S (i.e., f˜ ∈ L0 (S, S, µ, Y) and Z(A) : S → L0 (P ) as a random measure, we have (20) as Z eitZ(A)(ω) dP (ω) = ΦZ (t, A), (21) Ω

a ch.f. of Z(A) with parameters (α, β, c, µ) when it is p-stable. For each f ∈ L0 (S, S, µ, Y), and Z : S → L0 (Ω, A, P ) a p-stable random measure (hence

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has a controlling positive finite measure µ) we can define the stochastic integral Z T : f 7→ f dZ, f ∈ Xp = L1 (S, S, µ, Y) ∩ Lp (S, S, µ, Y) (22) Ω

and T f ∈ L0 (Ω, A, P, Y). Its ch.f. and properties are as follows. Regarding the scalar p-stable random measure Z(·) with parameters (α, β, c, µ), it is known from the theory of p-stable classes that if β = +1(−1) then Z(A) ≥ 0 (≤ 0) i.e. so that it is a nonnegative (nonpositive) independently valued (infinitely divisible) random measure, i.e., with values in (L0 (P ))+ ((L0 (P ))− ). Since we already noted that T f of (22) is well-defined for Z(·) having a controlling measure, its properties in the case that f : Ω → Y (and strongly measurable) can be obtained with analogous conclusions as in the preceding theorem. So we have, using the above analysis, the following result: Theorem 5.3.4 Let (S, S, µ) be a finite measure space, S being the δ-ring of µfinite sets of S, and Z : S → L0 (P ) a p-stable random measure with parameters (α, 1, c, µ), implying that it is positive valued. If Y is a locally convex Fr´echet space and Xp = L1 (S, µ; Y) ∩ Lp (s, µ, Y) with metric defined earlier as (cf.,(22) above) kf kp =

Z

S

d(f )(s)dµ(s) +

Z

S

1∧(1/p) d(f ) (s)dµ(s) , 0 < p < 1; 1 < p ≤ 2 p

for strongly measurable f : S → Y relative to µ then the mapping T : f → Lp (P ; Y) by (22) for f ∈ Xp is well defined, linear, continuous and one-to-one. In fact for 0 < q < p < 1 the mapping T : Xp → Lq (P, Y) is continuous and its ch.f. is given by Z Z ∗ eith(T f )(ω),y i dP (ω) = exp{iα hf (s), y ∗ idµ(s) Ω S h Z πp i p −c |t| (hf (s), y ∗ i+ )p dµ(s) 1 − i sgn t tan 2 S h Z πp i −c |t|p (hf (s), y ∗ i− )p dµ(s) 1 + i sgn t tan }, 2 S (23) where hf (s), y ∗ i is written as hf (s), y ∗ i+ − hf (s), y ∗ i− , y ∗ ∈ Y∗ , each of the terms being non-negative. If α = 0, then T : Lp (µ; Y) → L0 (P, Y) is also well defined and for 0 < q < p < 1, T is one-to-one and continuous, with T (Lp (S, µ; Y)) ⊂ Lq (P ; Y). The proof is a simple modification of the one given for Theorem 3 above and will be omitted. Note that in the one-to-one case we need to use the separability of Y [or equivalently the strong µ measurability of f ∈ L0 (µ; Y)]. In fact

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when R T f = 0, its ch.f. is identically unity and hence, taking absolute values, we get S |hf (s), y ∗ i|p dµ(s) = 0, y ∗ ∈ Y∗ implying that |hf (s), y ∗ i| = 0, and now the separability of Y implies the existence of a countable set of yn∗ ∈ Y∗ that determines the norm of Y (from the general theory of such spaces) and hence kf k(s) = supn |hf (s), yn∗ i| = 0 so f (s) = 0 a.e., giving the one-to-one property of T , (T f = 0 ⇒ f = 0, a.e.). The other conclusions are analogous to that of Theorem 3 above. Positive random measures will be discussed separately in Section 4 below. If Y is an infinite dimensional Fr´echet space that is not locally convex (e.g., Y = Lp (µ), 0 ≤ p < 1), then it need not have a dual space Y∗ of continuous linear functionals that separate points of Y. In this case we have only a weaker conclusion, the one-to-one part will not hold, and measurability problems also have to be reconsidered. Note that Y still includes not only Banach spaces, but also Schwartz spaces such as C ∞ (R) and the “distribution spaces” as well as “countably normed” vector spaces all of which are in the locally convex Fr´echet class. We have already discussed between (15)–(17) that for a random measure Z : S → Lp (P ) one can define the mapping T : Lp (µ, Y) → L0 (P, Y) using the D − S integration if Z(·) is infinitely divisible for all separable Fr´echet spaces. Here we show that T can be analyzed if Z(·) is p-stable and Y = Lp (µ), 0 < p ≤ 2 in various forms. We now present the existence of the continuous linear R mapping T : f 7→ S f dZ for f ∈ Lp (µ, Y), when 0 < p < 1 is included, as follows: Theorem 5.3.5 Let (S, S, µ) be a finite measure space (S a δ-ring) and Z : S → L0 (P ) be a p-stable random measure with (α, β, c, µ) as its parameter set. If Yq R= Lq (W, A, P ) where P (W ) = 1, with metric (sometimes norm) as kf kq = [ W |f R|q dP1 ]1∧(1/q) , f ∈ Yq , then T is a continuous linear mapping, where T : f 7→ S f dZ, defined as in the following parts: [Part (a)]: 1. 1 ≤ q < p ≤ 2, and T : L1 (µ; Yp ) ∩ Lp (µ; Yp ) → Lq (P ; Yq ) 2. 0 < q < 1 < p ≤ 2, and T : L1 (µ; Yp ) ∩ Lp (µ; Yp ) → L1 (P, Yq ) 3. 0 < q < p < 1, and T : L1 (µ, Y1 ) ∩ Lp (µ; Y1 ) → L1 (P ; Yq ) 4. 0 < q < p = 1, and 1

T : L1 (µ, Y1+ 1e ) ∩ L1+ e (µ; Y1+ 1e ) → L1 (P ; Yq ).

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[Part (b)]: If 1 ≤ q < p ≤ 2 then the ch.f. of T f can be given as, Z Z n ith(T f )(ω),y ∗ i e P (dω) = exp itα( hf (s), y ∗ i dµ(s)) Ω S Z π p ∗ + p − c|t| (hf (s), y i ) dµ(s) 1 − iβ sgn t tan p 2 ZS p π o 1 + iβ sgn t tan p , hf (s), y ∗ i− dµ(s) − c|t|p 2 S (24) where y ∗ ∈ Y∗q ⊂ Y∗p , and T is injective as well as continuous. The proof of this result is easily obtainable with the D-S integration as discussed earlier and in our discussion: for (b), Y∗p exists nontrivially since Yp is now a Banach space. The injectivity follows as in Theorem 4. We leave the details as an exercise. [This result is given establishing the existence of T f with a separate proof in each case, reducing it in order to apply the ‘truncation inequality’ by Okazaki (1979), and it is somewhat round about since he never used the D–S method. However his work is self-contained.] The following result is a corollary of the above theorem and is stated for a convenient reference. Proposition 5.3.6 Let (S, S, µ) and Yp = Lp (W, A, P ) be as in the above theorem and Z : S → L0 (Ω, Σ, P ) = L0 (P ) be a p-stable random measure with parameters (α, β, c, µ). Let R 0 ≤ q ≤ max(1, p); 1 < p < r ≤ 2 with p = 2 implying r = 2. Then T f = S f dZ is a well-defined continuous linear mapping for f ∈ L1 (µ, Yr ) ∩ Lp (µ; Yr ), with T f ∈ Lq (P ; Yr ). It is one-to-one and into. Its ch.f. is given [as in (24) by: Z Z n ∗ eith(T f )(ω),y i dP (ω) = exp iαt hf (s), y ∗ i dµ(s) Ω S Z p πt −c|t|p (hf (s), y ∗ i+ dµ(s)[1 − iβ sgn t tan ] 2 S Z p πp o −c|t|p (hf (s), y ∗ i− dµ(s)[1 + iβ sgn t tan ] 2 S (25) for y ∗ ∈ Y∗r . In particular if α = β = 0, then the range of T is of symmetric p-stable Yr -valued random variables. The key role played by the vector (Fr´echet-) valued integrand in the D − S integration together with its dominated convergence theorem, discussed earlier, is exemplified here. We shall include some related applications in the Complements Section. In the next section we consider random measures, motivated

5.4 Positive and Other Special Classes of

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201

by the class of p-stable ones with its parameter β = 1, which are non negative. These also arise in “point-processes” with immediate practical applications, but again they have interest in certain embedding properties of spaces, preserving some order relations. It will show moreover that, even when random measures are vector valued measures, they are also important for certain applications.

5.4 Positive and Other Special Classes of Random Measures As seen in the preceding section there are p-stable random measures Z : S → Lp (P ), 0 < p ≤ 2, whose parameter set (α, β, c, µ) with β = 1, give rise to Z(A) ≥ 0 for all A ∈ S. There exist other random measures Z : S → Lp (P ), p ≥ 0, which need not be stable. An important example of the latter is a conditional probability function relative to a given σ-subalgebra. There are still others called “completely random measures” cutting into both the above types. We now discuss these classes briefly to better understand and appreciate the subject. As a simple consequence of the last proposition of the preceding section we have (with Y∗ as the bounded linear measurable set): Proposition 5.4.1 Let Y be a locally convex Fr´echet space, B(Y) the Borel σalgebra of Y, and R µ : B(Y) → [0, 1], a probability measure having one moment κ1 , i.e., κ1 = Y d(y)dµ(y) < ∞ where d(·) is the metric of Y. Then there is a p-stable (BorelR ) probability Q : B(Y) → [0, 1], 0 < p < 1 such that for every 0 < q < p, Y d(y)q dQ(y) < ∞. If κ1 = 0 (i.e., the first moment of µ is zero) and the functionals yn∗ ∈ Y∗ are measurable as well as yn∗ → 0 in µ-measure, then yn∗ → 0 in Q-measure, and conversely yn∗ → 0 in Q-measure implies the same conclusion for µ, so that both induce the same topology on Y∗ . Proof. Let (α, 1, c, µ) be a set of parameters for a (possible) p-stable random measure on the space (Y, B(Y), µ). Then by Proposition 3.2 above, there is a probability space (Ω, A, Q) and (since β = 1) a positive p-stable random R measure Z : B(Y) → Lq (Q) such that T : f 7→ Y f dZ is uniquely defined, for each f ∈ L1 ((Y, B(Y), µ); Y). Also the variable T f is p-stable by Theorem 3.4. Since µ(F ) = 1, so that the (identity) mapping I : Y → Y is an element of L1 (B(Y), µ; Y), we have T I ∈ Lq (Q; Y). If ν is the distribution of T I, then ν is a p-stable measure on (B(Y), Y), having qth moment 0 < q < p < 1. So its ch.f. admits a representation as in (25) of Section 3 above. If α = 0 in the parameter set, i.e. (0, 1, c, µ) is the parameter vector, then ν is centered and its ch.f. has only the parameters c and µ. The separability of Y implies that each y ∗ ∈ Y∗ is ν-measurable (as also for µ). If yn∗ (·) → 0 in

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R ihy,y∗ i n implies ν-measure then its Fourier transform R | Y e∗ p dν(y)| → 0. But this in (25) of Section 3, that exp{−c Y |hy, yn i| dµ(y)} → 1 so that yn∗ → 0 in µ-measure and also conversely. Thus the µ and ν convergences of Y∗ , induced by Y, for 0 ≤ q < p are equivalent. 2 So far, given the numbers α, β ∈ R, |β| ≤ 1 and c ≥ 0, we have been able to associate a random (p-stable) measure on a δ-ring (S, S) [e.g., S is a topological space with S as the class of its bounded Borel sets], into a family of random variables defined on some probability space (Ω, Σ, P ) i.e. Z : S → L0 (P ). We then analyzed various properties when 0 < p ≤ 2 through the L´evy form. If β = 1, then the random measure Z(·) is non negative valued. But the latter can occur without regard to the p-stable property. We now introduce such a class and analyze some aspects of its structure. Let (Ω,RΣ, P ) be a probability space X : Ω → R be a random variable such that Ω |X| dP < ∞. If B ⊂ RΣ is a σ-subalgebra then the set function and νX (·) νX : B → R given by νX (A) = A X dP, A ∈ B, is well-defined R is (clearly) PB (= P |B)-continuous. The variation |νX |(Ω) = Ω |X| dP < ∞ dνX exand by the classical Radon-Nikod´ ym theorem for scalar measures, dP B B ists PB -uniquely and is a B-measurable function, denoted as E (X). This satisfies, Z Z Z dνX E B (X) dPB , A ∈ B. (1) dPB = XdP = νX (A) = A A dPB A Taking X = χB , B ∈ Σ, (1) implies on writing E B (χA ) = Q(A|B), Z Z Q(B|B) dPB , A ∈ B, B ∈ Σ. χB dP = P (A ∩ B) = A

(2)

A

Here Q(B|B) is a.e. unique (by the uniqueness of the Radon Nikod´ ym theorem) and satisfies 0 ≤ Q(B|B) ≤ 1, B ∈ Σ. It is clear that Q(·|B)(ω) : Σ → [0, 1] is a B-measurable nonnegative bounded function. Also (1) and (2) imply that Q(·|B)(ω) is σ-additive on disjoint sets for almost all ω ∈ Ω. It then follows that Z(·) = Q(·|B) : Σ → L0 (Ω, B, PB ) is a nonnegative bounded random measure, also denoted QB (·). It is usually called (with some qualifications) a conditional probability function on (Ω, Σ, P ), given a σ-algebra B. This abstract formulation was introduced by A. Kolmogorov in 1933 and it plays a fundamental role in Markov processes and in many places in stochastic analysis. Thus Z(·)(·) = QB (·)(·) : Σ → L0 (P ) defines a nonnegative (bounded) random measure which we now analyze as it need not necessarily be of p-stable class. From (2) it is clear that QB (·)(·) is an equivalence class member (just as elements of Lp (P ), p ≥ 0), and one may wonder whether it is possible to select a function, of two variables, denoted V (A, ω) = QB (A)(ω) for all A ∈ Σ and ω ∈ Ω, for a fixed B, so that (i) V (·, ω) : Σ → R+ ,

ω ∈ Ω, σ-additive, and

5.4 Positive and Other Special Classes of

Random Measures

203

(ii) V (A, ·) : Ω → R+ , A ∈ Σ, B-measurable. Hence the analysis can be reduced to a function of two variables. Since the equivalence class changes with A and the collection of each such class {V (A, ·), A ∈ Σ} can be “large”, the problem of selection becomes nontrivial. This selection problem is called a lifting and has the following solution. [See A. Ionescu Tulcea and C. Ionescu Tulcea (1969), Chapter IV, for details.] Lifting Theorem. For a complete measure space (Ω, A, µ) where µ is finite (or σ-finite or even more generally “localizable”) measure and Lp (µ), 1 ≤ p ≤ ∞, there exists a selection mapping (called a lifting) ρ : Lp (µ) → Lp (µ) if p = ∞, but for 1 ≤ p < ∞ there does not exist such a mapping if there is at least one set A0 ∈ A such that µ(A0 ) > 0 and µ|A(A0 ) is diffuse, i.e., is nonatomic. Here ρ must satisfy: (i) ρ(f ) ∼ f , (ii) f ∼ g ⇒ ρ(f ) = ρ(g), (iii) f ≥ 0 ⇒ ρ(f ) ≥ 0 and (iv) ρ(·) is linear as well as multiplicative, i.e., ρ(f g) = ρ(f )ρ(g). [f ∼ g means f, g are a.e. equal.] Remark 5.6 Since for a p-stable class, 0 ≤ p ≤ 2, the random variables are infinitely divisible, it follows that no such mapping generally exists by this theorem, unless the probability space (Ω, A, P ) is degenerate (or discrete). By the last part, since conditional probability functions are bounded (and non negative) such a lifting exists and V (A, ω) = ρ((QB (A))(ω) can be treated as a function of the two variables (A, ω). However, the existence proof of ρ depends on the ‘Axiom of Choice’ and so there is no method available in the literature for the ρ-construction. In view of this difficulty, we must proceed in our theory (as we did so far) without depending on individual choices of measures Z(·) to be scalar functions and try to apply the Real Analysis methodology. Because of the use and interest in working with conditional measures as ‘ordinary measures’, for each ω, there is a great deal of effort to find conditions for the validity of this property under the name ‘regular conditional probability’ measures. [For a detailed analysis of this topic, see Rao (1993, 2005).] We assume the regularity in the work below. Nonnegative bounded random measures, under the regularity assumption, have applications of the following type. Thus let (S, S) be a measurable space and Z : S → L0 (P ), a positive random measure in the sense that for A, B ∈ S, A ⊂ B ⇒ Z(A) ≤ Z(B) a.e.. Now regard L0 (P ) as a vector lattice under the order 0 ≤ f, g ∈ L0 (P )S⇒ min(f, g) ≤ f, g ≤ n max(f, g) so that for An ∈ S, disjoint, since k=1 Ak ↑ as n ↑, we obtain

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Z

∞ [

n=1

n [ [ An = Z Ak + Z Ak k=1

=

n X

Z(Ak ) + Z

k=1

k>n

[

k>n

Ak .

If the σ-additivity condition is replaced by Z(Bn ) ↓ 0 for Bn ↓ ∅, as n → ∞, Bn ∈ S, Z(·) is called order continuous to use the lattice structure. ThusPZ(·) becomes σ-additive in the order topology, and one can define for n f = i=1 ai χAi , Z n X ai Z(A ∩ Ai ). (3) f dZ = A

i=1

This is well-defined and J.D.M. Wright (1969) has introduced a vector integral for such Z(·) using the lattice structure of L0 (P ), regarded as real random variables. He then showed that Ra monotone convergence theorem is valid for this integral, and that T : f 7→ A f dZ, A ∈ S for f ∈ B(S), bounded (real) Borel functions on (S, S), is a linear operator. Much of the work in the preceding section extends. It is clear that no probabilistic ideas have been essential here. However, this type of treatment is important in some applications of probability, especially “sufficiency theory” in the undominated case, (cf., e.g., Rao (2005), Sec. 6.5). It is also of interest when Z : S → L0 (P ) and, for A, B ∈ S disjoint, Z(A), Z(B) are independent, an assumption which is important in applications of problems dealing with “point process”. Now suppose there is a σ-finite (or just finite) measure µ : S → R+ that dominates or controls the random measure Z(·) which is independently valued on disjoint sets. Such a Z(·) has a very interesting probabilistic structure, analyzed by Kingman (1967) and Garling (1974). We now discuss this as it exemplifies the analysis of such measures. A positive random measure Z : S → L0 (P ) which takes independent values on disjoint sets is termed completely random S∞ (by Kingman (1967)) if there is also a disjoint sequence, An ∈ S, S = n=1 An , satisfying the condition that P (Z(An ) > 0) > 0 for each n. Then the Laplace transform LA : t 7→ E(e−tZ(A) ) exists for all t ≥ 0 and {µt (A) = − log LA (t), A ∈ S} defines a measure for each A ∈ S. It was shown that Z(·) admits a Lebesgue type decomposition: Z(A) = Za (A) + Zd (A), A ∈ S, (4) where Za (·) is P a nonatomic random measure and Zd (·) is “purely atomic” so ∞ that Zd (A) = n=1 Xn δn (A), where the Xn are mutually independent (real) random variables and δn (·), is the Dirac point measure. Moreover Za , Zd are independent and Za is infinitely divisible since for each A ∈ S, there exist disjoint Ajn ∈ S, A = ∪nj=1 Ajn , Z(Ajn ) > 0 are independent and E(e−tZ(Ajn ) ) =

e−

µt (Ajn ) n

. Taking t = 1, for any c > 0 we have

5.4 Positive and Other Special Classes of

Random Measures

205

i h P [Za (Ajn ) ≥ c] = P 1 − e−Za (Ajn ) ≥ 1 − e−c ≤

E(1 − e−Za (Ajn )) ) , 1 − e−c

Markov’s inequality,

−µ1 (A)

=

1−e n 1 − e−c

,

(5)

so that limn→∞ max1≤j≤n P [Za (Ajn ) ≥ c] = 0, implying also that Za (Ajn ), 1 ≤ j ≤ n, are ‘infinitesimal’. This shows that Za (·) is infinitely divisible. R Consequently there is a controlling measure ν : S → R so that f 7→ S f dZa is defined for all 0 ≤ f ∈ Lp (ν), as discussed at the early part of Section R 3.1. Since Zd (·) is countably supported, so we can define T : f 7→ S f dZ for bounded Borel functions f , as before, although Z(·) need not be a pstable measure 0 < p ≤ 2. In this connection the range of a linear map T : B(S) → L0 (P ) has an interesting (special) characterization, established by Garling (1975) which sharpens some early work due to Urbanik (1967). The point of this discussion is to obtain the analogs of the last part of the preceding section by showing that the mapping T induced by a positive independently valued random measure again gives (order preserving) embedding results. Here one has certain Fr´echet spaces of functions mapped into L0 (P ), determined by (and determines) a Z(·). We now utilize the infinite divisibility instead of stability used earlier, the underlying idea however being the same. The following representation which is a modification of a classical result, called the L´evy-Khintchine representation, for Laplace transforms of infinitely divisible (i.d.) positive random variables (random measures here), is available (see e.g., Feller (1966), p.426) and it will be utilized: Theorem 5.4.2 (Representation of i.d. Laplace Transform) Let (S, S) be a measurable space and Z : S → (L0 (P ))+ a (positive) random measure, on a probability space (Ω, Σ, P ), where Z is infinitely divisible. Then + there exist measures α, µ : S → R and a “measure kernel” k : S × + + R → R , meaning that (a) limu→0 k(s, u) = 0, limu→∞ k(s, u) = 1, for all s ∈ S and (b) k(·, u) is S-measurable, u > 0, in terms of which one has ϕZ(A) (u) = E(e−uZ(A) ), A ∈ S, Z Z = exp −α(A)u − A

0

∞

h(u, v)k(x, dv) dµ(x) ,

(6)

where h(u, v) = (1 − e−uv )(1 − e−v )−1 , u, v > 0. Conversely if α, µ and k(·, ·) are functions satisfying (a) and (b) above, then there exists a probability space (Ω, Σ, P ) and a random measure Z : S → L0 (P ) which is i.d. and whose Laplace transform {ϕZ(A) (u), u ≥ 0, A ∈ S} has the unique representation (6).

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5 More on Random Measures and Integrals

+ Now using the explicit form of R ∞h(·, ·) we see that the function M (·, ·) : S ×R → + R , defined as M (s, u) = 0 h(u, x)k(s, dν), s ∈ S, has the property that ↓ so that M (s, ·) is a concave M (s, ·) is continuous and increasing but M(·,u) u function like k → |k|p , 0 < p < 1. Analogous to Lp (µ), but somewhat more generally we can define a generalized Orlicz space, for each s ∈ S, but in this case it is called the Musielak-Orlicz space. [See Rao and Ren (1993), who called it generalized Orlicz space when using M (1, ·), and for the general case called a modular space; see Musielak (1983) for details and applications used below.] If we define the set LM (µ) = LM (S, S, R µ) as the equivalence class of measurable f : S → R such that ρM (af ) = S M (s, |f (s)|af )dµ(s) < ∞ for some af > 0, then LM (µ) is a (real) vector space, and if d : LM (µ) → R is defined as |f | ≤k , dM (f ) = kf kM = inf k > 0 : ρµ k

it is not difficult to verify that {LM (µ), dM (·)} is a complete metric (or Fr´echet) space. If M (s, u) = |u|p , 0 < p < 1, this reduces to Lp (µ). [It needs some, not entirely trivial, computation to verify the above statements. The details are also found in the two references given above.] An important fact to remember here is that (LM (µ), dµ(·)) is not locally convex although it is a vector lattice under the usual ordering ‘≤’, (f ≤ g means f (s) ≤ g(s), a.e.) and so additional computations are needed. In the representation result of Theorem 2 above, the kernel k(s, du) will be independent of s if Z(·) is (strictly) stationary which means that its finite dimensional distributions are shift invariant, i.e. (Z(A1 ), . . . , Z(An )) and (Z(τx A1 ), . . . , Z(τx An ) have the same n-dimensional distribution for Ai ∈ S, n ≥ 1. As usual τx Ai = {x + y : y ∈ Ai } and S is a Borel ring of Rn or more generally a semi-group. When this condition holds M (·) becomes an increasing (but not necessarily convex) function, and LM (µ) will be a generalized Orlicz space which still is only a Fr´echet space, generalizing Lp (µ), 0 < p < 1. In either case, the function M (·, ·) is said to satisfy a (local) growth condition (such as ∆2 in the classical Orlicz space theory) to be called a generalized ∆2 condition if there are fixed constants K > 0 and λ0 > 0 such that for some 0 < γ < 1, M (s, λu) ≤ Kλγ M (s, u),

u > 0, λ ≥ λ0 , a.a(s).

(7)

If λ0 = 0, the condition is said to hold globally. We now present the interesting counterpart of Theorem 4.5 for positive R random measures Z : S → L0 (P ), so that T : f 7→ S f dZ defines an embedding T : LM (µ) → L0 (P ) when M satisfies the growth condition (7). Theorem 5.4.3 (Garling) Let (S, S, µ) be a measure space and M : S × R+ → R+ be a Musielak-Orlicz function satisfying the generalized ∆2 -condition (cf.(7)) and M(x,u) ↑ for x ∈ S as u ↑, for some 0 < α < 1. Then there exists uα

5.5 Complements and Exercises

207

a probability space (Ω, Σ, P ), a positive independently valued random measure Z : S → L0 (P ) and a generalized Orlicz space Lφ (P ) where φ is a (generalized) R φ(u) linear Young function such that φ(u u ↓ as u ↓ but R+ u1+α du < ∞, and that the R mapping T : LM (µ) → Lφ (P ) defined by the (random) integral T f = S f dZ, for f ∈ LM (µ), is continuous, order preserving, and one-to-one. So T is an isomorphism of LM (µ) into Lφ (P ), which otherwise said, is an embedding and a positive mapping. A proof of this interesting result is given in complete detail in Garling’s paper, along with some supplementary results. We omit it for space reasons and ask the reader to consult the original memoir. However some related results will be given in the complements section to follow.

5.5 Complements and Exercises 1. Let (S, S) be a measurable space and β : S× S → C some bounded bimeasure. Using the M T -integration associate a bilinear functional B : B0 (S) × B0 (S) → C such that kBk = sup{|B(f, g)| : kf kB0 (S) ≤ 1, kgkB0 (S) ≤ 1} < ∞, where B0 (S) is the space of scalar S-measurable bounded functions, k · kB0 (S) being the uniform norm. Then verify that there exist a pair of probability measures µ, ν on (S, S) and a constant KG (Grothendieck constant) such that (writing k · k2,µ for the norm in L2 (S, S, µ)) |B(f, g)| ≤ kBk · KG kf k2,µ kgk2,ν , and B(·, ·) extends uniquely (by density) to all of L2 (µ) × L2 (ν) preserving the norm. Let H be the Hilbert space determined by this product space and use the same symbol B(·, ·) for its unique extension to H so that it is bilinear on it. By the classical Riesz representation theorem we can have B(f, g) = (f, Ag)H for a unique bounded linear operator A : H → H. It was shown in Corollary 4.3.5, with a similar representation that B can be expressed as a sum of (four) positive definite bilinear forms. There are also other types of representations and the following is one such. Express A = V R(= RV ) where V : H → H is partially isometric and R : H → H is a positive definite operator (both commute). Verify that B(f, g) = (Rf, V ∗ g) where V ∗ is also partially isometric with domain as the closure of the range of V ∗ which coincides with the range of the above noted positive definite operator R. 2. Let (S, S) be a measurable space and Zi : S → L2 (P ) be a random measure with βi (·, ·) as its bimeasure soR that R βi (A, B) = (Zi (A), Zi (B))L2 (P ) , A, B ∈ g (y)dβi is an M T -integral, so that Si , i = 1, 2. If Bi : (f, g) 7→ S S f (x)¯ Bi : Bi (S) × Bi (S) → C is a bilinear functional on the space of bounded measurable scalar functions on (S, S), suppose Bi (·, ·) is bounded when Bi (S) is given the (usual) uniform norm. Using Grothendieck’s representation, observe

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5 More on Random Measures and Integrals

that there exist Hilbert spaces Hi and bounded linear operators Ai : Hi → Hi such that Bi (f, g) = (f, Ai g)Hi , f, g ∈ Hi , i = 1, 2. Show that we can define a bounded bilinear functional B as B = B1 ⊗ B2 , a kind of tensor product on H1 ⊗ H2 such that (expanding the work of Exercise 1 above) B(f1 ⊗ f2 , g1 ⊗ g2 ) = B1 (f1 , g1 )B2 (f2 , g2 ), and moreover we have for fi , gi ∈ Hi , i = 1, 2, B(f1 ⊗ f2 , g1 ⊗ g2 ) = (f1 ⊗ f2 , (A1 ⊗ A2 )(g1 ⊗ g2 ))H1 ⊗H2 satisfying 2 kBk = kB1 ⊗ B2 k ≤ KG kB1 kkB2 k.

(+)

Give the corresponding description in terms of Grothendieck’s measures (µi , νi ), i = 1, 2. 3. We can present an analog of the Lebesgue type decomposition of a bimeasure into continuous and discrete (or atomic) parts. Let (Si , Si ), i = 1, 2 be measurable spaces and β : S1 × S2 → C be a bimeasure. We say that β is continuous if for all finite subsets Ai ∈ Si , the bimeasure β|S1 (A1 )×S2 (A2 ) = 0, i.e. β(A1 , A2 ) ≡ 0. If on the other hand, there is an increasing finite sequence Ani ↑ in Si such that kβ − β|An1 ×An2 k → 0 as n → ∞, then β is termed discrete. If T = {A1 × A2 : Ai ∈ Si , finite, i = 1, 2}, directed by inclusion, i.e., (Ai , Bi ) ∈ T, i = 1, 2 then A1 × A2 ≺ B1 × B2 if and only if A1 ⊂ B1 and A2 ⊂ B2 . Let Bc and Bd be the classes of continuous and discrete bimeasures on S1 × S2 . Verify that Bc and Bd are closed linear subspaces of all bounded bimeasures B on S1 × S2 and that T : B → Bd is a projection of norm at most one, where B = Bc ⊕ Bd . [Note that the norms are uniform and not of L1 (µ)-type so that B1 ∈ Bc and B2 ∈ Bd , B = Bc + Bd does not imply kBk = kBc k + kBd k the right side is usually larger and only triangle inequality is assured.] [This result is from Graham and Schreiber (1984). The fact that Bc and Bd are closed needs a little care, but the reader can find the detail in the preceding reference if necessary.] 4. This problem explains the necessity of multidimensional random measures to represent some important sets of stochastic processes. Let {Xt , t ∈ T ⊂ R} ⊂ L20 (P ) be a class of random variables indexed by T such that they are left continuous in t with right limits with probability one, i.e., P [limh→0 Xt+h = Xt ] = 1, t ∈ R and P [Xt = Xt−0 ] = 1. Also let Ht be the linear span of 2 Xs for s ≤ t in L2 (P ), i.e., Ht (X) = sp{X ¯ s , s ≤ t} where Ht (X) ⊂ L0 (P ) and suppose that the ‘remote past’ is trivial, i.e. ∩t Ht (X) = {0}. The process is then called “purely nondeterministic”. Show that such a process admits a representation of the following description. There exist a set of random measures Zn : B(R) → L20 (P ), n = 1, . . . , N which are mutually orthogonal (i.e., (Zm (A), Zn (B)) = δm,n for all A, B ∈ B(R)), each orthogonally valued,

5.5 Complements and Exercises

209

and for a corresponding set of measurable kernels gn (t, ·), integrable relative to Zn , n ≥ 1, such that Xt =

N Z X

n=1

R

gn (t, λ) dZn (λ)[=

Z

R

(g(t, λ) · dZ(λ))]

(+)

where g(t, ·) = (g1 (t, ·), . . . , gN (t, ·)) and Z(·) = (Z1 (·), . . . , ZN (·)) with PN R 2 2 n=1 R |gn (t, λ)| dFn (λ) < ∞, Fn (A) = E(|Zn (A)| ). The number N satisfies 1 ≤ N ≤ ∞, which is uniquely defined, by the X-process, and is termed the multiplicity. [In (+) we used the “dot product” g(t, ·) · dZ(·) as a short notation for the sum.] ¯ t ), Use the following sketch in establishing this result: Let K(s, t) = E(Xs X the covariance of the process. First obtain a suitable series representation of K from which one may deduce (+). For this it is best to use the Aronszajn (or reproducing kernel Hilbert) space technique introduced already in Section 2.2 based on the positive definite kernel such as K. Thus let HK be the RKHS generated by {K(·, t), t ∈ T } as in Section 2.2 and let Ht = sp{K(·, ¯ s), s ≤ t} ⊂ HK . Our assumptions on the process (continuous on the left with right limits) imply that H 0 = ∩t∈T Ht = {0}, Ht = ∪s
T

There is an isometric onto mapping τ : Ht → Ht and a random measure Zn such that τ (Zn ([a, b])) = fn ([a, b]), a < b, a, b ∈ T, and kZn (A)k22 = kfn (b) − fn (a)k22 if A = [a, b]. Letting gn (t, ·) = ψn (t, ·) ∈ L2 (T, µn ), one gets the Xt representation (+) above for 1 ≤ n ≤ N, the integer 1 ≤ N ≤ ∞ which is then the multiplicity of the process. This detailed structure depending on the classical Hellinger-Hahn theorem is a deep one. An early recognition and use of it is due to Hida (1960). It is later considered by several authors. [For instance Kallianpur and Mandrekar (1965), and Hida’s

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5 More on Random Measures and Integrals

procedure which works with more general cases, (not assuming left continuity and right limits) is also included in the present author’s book (cf. Rao (2000), Sec.5.2). It is given here to show how nontrivial applications may have to employ different types of tools from different parts of analysis.] 5. In Definition 4.3.1 we introduced the L2,2 -boundedness concept and observed that Brownian motion induces a random measure Z that is L2,2 -bounded. Thus if I = [a, b] ⊂ R is an interval and f : I → RR is an element of L1 [I, dλ], where dλ is the Lebesgue measure then τ : f 7→ I f dZ is well defined and τ : L2 (I, dλ) → L2 (P ) is a bounded linear operator. Show that τ f can also be expressed as vector integrals in two ways Z Z (∗) τ (f ) = f dZ(t) = f (t)Y (t) dt, f ∈ L2 (I, dt), I

I

for a unique Bochner integrable Y : I → L2 (I, dt) although both integrals in (∗) are defined differently. Moreover the left integral in (∗) can be treated in the Lebesgue-Stieltjes sense (if and) only if Z(·) has finite Vitali variation (which for the B.M. it does not hold). This is why in general we cannot always assume that a random measure be of finite Pnvariation. [Main ideas of proof: If f = i=0 ai χ(ti ,ti+1 ] , a < ta < t1 < · · · < tn+1 ≤ b then by definition of L2,2 -boundedness 2

E(|τ f | ) ≤ C

Z

a

b

|f (t)|2 dt

(+)

and if Xt is B.M., determining Z, then there is equality in (+) with C = σ 2 the variance parameter. In any case τ : L2 (I) → L2 (P ) is a bounded linear operator defined on a dense set of L2 (I), and has a unique bound preserving extension to all of L2 (I) which is weakly compact by reflexivity of L2 (I). Now by the classical Riesz representation theorem, there is a vector measure Z : B(I) → L2 (P ) R such that τ (f ) = I f dZ, and Z is induced by the Xt -process in the B.M. case. Moreover, Z Z (∗+) E(|τ (f )|2 ) ≤ C |f |2 dt = C |fˆ|2 dtˆ, I

Iˆ

where I is an LCA group, with addition, mod (b − a), as group operation ˆ dtˆ) with and (∗+) is simply the Plancherel equation on the dual group (I, ˆ dt as its (normalized) Haar measure, which here is a constant multiple of the Lebesgue measure and fˆ is the Fourier transform of f relative to dtˆ. So fˆ = F (f ) and let T = τ ◦ F −1 , to get T (fˆ) = τ (f ). Once again by the ˆ → L2 (P ) such Riesz theorem there is a (unique) vector measure Z˜ : B(I) that Z ˆ ˜ ˆ dˆ fˆ ∈ L2 (I, u), T (f ) = fˆ(u)dZ(u), Iˆ

and by a suitable application of Fubini’s theorem, if e(u, t) = eiut , we have

5.5 Complements and Exercises

Z

R

ˆ = f (t) τ (f ) = T (f) I Z = f (t)Y (t)dt

Z

Iˆ

211

˜ e(u, t)dZ(u) dt

I

˜ ˆ is a Bochner and C1 dtˆ = dt. Thus {Y (t), t ∈ I} where Y (t) = C1 I e(u, t)Z(du), integrable function on I giving (∗). But dXt = Y (t)dt is not valid, as the integrals are not in Lebesgue’s sense. To treat it as a Leb-Stieltjes type, let πn : a ≤ t0 < t1 < . . . < tn−1 = b be a partition and f ∈ B(I), a bounded Borel function. Then we have Z n X f (ti )(Z(ti+1 ) − Z(ti )) ∈ L2 (P ) ⊂ L1 (P ) Sn (f ) = fπn dZ = I

Pn

i=0

where fπn (t) = i=0 f (ti )χ(ti ,ti+1 ] and Z((a, t]) = Z(t). If one uses the Stieltjes definition then Sn (f ) → τ (f ) as |πn | → 0, by refinement, and kSn (f )kL2 (P ) ≤ kf k∞ kZk(I). Hence supn kSn (f )kL2 (P ) < ∞, for each f ∈ B(I). By the uniform if we Pn boundedness theorem supn kSn k < ∞. Hence ω ∈ B(I), sgn (Z(t ) −Z(t ))(ω)χ (u), then h consider hω (u) = i+1 i (ti ,ti+1 ] n n i=0 ω and |Sn (hω n )| ≤ kSn kkhn k∞ < ∞. Integrating over ω ∈ Ω, one gets the equation that Z : B(I) → L1 (P ) has finite variation, so only then the integral can be in the L − S sense!] (This useful observation on variation of Z is due to P. A. Meyer. See also Chapter 7, Sec. 7.1 for some other extensions.) 6. In Section 2.2 (see Definition 2.2.3 and discussion preceding it) we noted a class, called “harmonizable” processes, and their integral representations. Briefly, let {Xt , t ∈ R} ⊂ L20 (P ) be a (centered) class of second order pro¯ t ). It is cesses with covariance r, given by r(s, t) = (Xs , Xt )L20 (P ) = E(Xs X (weakly) stationary if there is a shift operator τs : Xt → Xt+s for s, t ∈ R and kXt k2 = kXt+s k2 or equivalently r(t + u, s + u) = r(t, s) for all s, t, u ∈ R and τs is linear. This definition implies that {τs , s ∈ R} is actually a unitary class with τs+t = τs τt and τ−s = (τs )∗ , the adjoint so that {τs , s ∈ R} is in2 ¯ deed a unitary group on L20 (P ), extended from sp{X t , t ∈ R} to all of L0 (P ), defining it for instance, as identity on the orthogonal complement of this closed span. If we only consider right shifts, so that r(s + h, t + h) = r(s, t) for h ≥ 0, then the process is called conservative and need no longer be stationary. However it is (weakly) harmonizable and {τs , s ≥ 0} now becomes a semi-group of bounded operators. Integral representations of such processes {Yt = τt X0 , t ≥ 0}, relative to random measures is possible. A closely related concept is that the process is dissipative if the variance Pn r(s + h, s + h) ≥ r(s, s) for h ≥ 0, s > 0. Or equivalently if Yn (h) = j=1 aj Xtj +h , for each finite linear combination, the variance σn2 (s) = E(|Yn (s)|2 ) ≥ σn2 (s + h), n > 0, s > 0, h > 0. This time the result demands even more effort in getting integral representations, but again the process is (weakly) harmonizable. We sketch some ideas in obtaining integral representation relations to

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random measures in these cases, showing the versatility of the theory. Let ¯ t ) de¯ t ) = E(Xs X {Xt , t ∈ R+ } ⊂ L20 (P ) be a process and r(s, t) = (Xs , X fine its covariance function r. If Hr is the Aronszajn space Pn determined by r, then it is generated by linear combinations Yn = j=1 aj r(sj , ·), with Pn Pn norm kYn k2 = (Yn , Yn ) = i,j=1 ai a ¯j r(si , sj ) = i,j=1 ai a ¯j (r(si , ·), r(sj , ·)) = Pn Pn k j=1 aj r(sj , ·)k2 = k j=1 aj Xi k22 (since r(·, ·) is positive definite) the mapping τ : rs → Xs from H → L2 (P ) is an isometric isomorphism. It is also R linear and by the Riesz representation we get Xs = τ (rs ) = R+ r(s, λ) dZ(λ), where Z : B(R+ ) → L20 (P ) is a random measure which may not be orthogonally Rvalued. Specializing r(·, ·) one gets various representations, such as Xt = R+ eitλ dZ(λ), t ≥ 0, where Z(·) does not have orthogonal values, and gives a representation of Xt as a harmonizable process. The integral is in the Dunford-Schwartz sense (cf. Section 3 above). Show that the process is dissipative in this set up. Other extensions and specializations are inviting and possible. 7. Let 0 < q < p < 1 ≤ r ≤ 2 with (S, S, λ) as a measure space and S0 = {A ∈ S, λ(A) < ∞}, the δ-ring, Y = Lr (S, S, λ). Then for any cylindrical Radon probability on (Y, B(Y)), which has one moment finite, there exists a p-stable probability measure ν on (Y, B(Y)) with parameters (α, β, c) such that its ch.f. is given by: (a) Z Z ∗ eithy,y i dν(y) = exp{ iαt hy, y ∗ i dµ(y) Y Z Y πp p ∗ + p } −c|t| (hy, y i ) dµ(y){1 − iβ sgn t tan 2 ZY πp p ∗ − p } }, −c|t| (hy, y i ) dµ(y){1 + iβ sgn t tan 2 Y R (b) ν satisfies Y kykq dν(y) < ∞, 0 < q < p, and finally (c) if α = 0, then Lp (Y, B(Y), µ) and Lq (Y, B(Y), ν) induce the same topology on Y∗ in the sense of Theorem 3.4. [The proof uses the random measure Z : B(Y) → Lq (P ; Y), although the measure Z does not appear in the statement. Follow the analysis in Theorems 3 and 4 of Section 3 in the text. This result is also due to Okazaki (1979).] 8. Let X and Y be normed linear spaces and V : X → Y be a continuous mapping. Then V is called “p-stable” if there exist a number 0 < q < p ≤ 2 and a sequence {fn , n ≥ 1} of independent p-stable symmetric random variables with the parameter set (0, 0, 1), on a probability space (Ω, Σ, P ) such that for an absolute constant K and that every finite set (x1 , . . . , xn ) ⊂ X, "Z # q1 ! p1 n n X X k u(xi )fi (ω)kq dP (ω) ≤ K kxi kp , Ω

l=1

i=1

5.5 Complements and Exercises

213

where 0 < K < ∞ is as noted. It is likewise called “cotype p-stable” if

q ! q1 # P1 " n Z n

X X

u(xi )fi (ω) dP (ω) ≤ K′ kxi kp .

Ω i=1

i=1

{In case q = 1 and the fn are replaced by the Rademacher sequence εn taking values ±1 with equal probability, then the p-stable type is called simply “p-type” and similarly the other called “cotype p”, but now εn ’s are no longer stable!} We have the following consequences for our “p-stable types” to be established by the reader using the sketch provided below: 1. Let (S, S, µ) be a finite measure space when S as a δ-ring and Z : S → L0 (P ) a stable random measure with parameters (0, 0, c, µ). If X, Y are a pair of normed linear spaces and u : X → Y is a linear p-stable operator, then the mapping Tu : Lp (µ; X) → Lq (P, Y), 0 < q < p Pn defined as usual, for instance for simple f = i=1 xi χAi , xi ∈ X, as Tu (f ) =

n X

u(xi )Z(Ai ),

i=1

is continuous. 2. In the converse direction, if µ is such that L1 (µ) is infinite dimensional, and u : X → Y is continuous, Tu defined above is continuous for some 0 < q < p, then µ is of p-stable type. [A Sketch: Note that Z(·) is p-stable of parameters (0, 0, c = µ(A), µ) which ˜ shows that Z(A) = Z(A)/µ(A)p , A ∈ S, is also p-stable with (0, 0, 1, µ) as its parameter set and hence Z

Ω

k

n X

! 1q

q

u(xi )Z(Ai )k (ω)dP (ω)

i=1

=

Z

Ω

≤K =K

k

n X i=1

n X i=1

Z

S

! q1

1 p

˜ i )(µ(Ai )) kq (ω)dP (ω) µ(xi )Z(A p

! p1

kxi k µ(Ai )

k

n X i=1

! p1

xi χAi kp (s)dµ(s)

,

implying that the map Tu is continuous, giving 1. above. The converse (part 2) is similar.]

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5 More on Random Measures and Integrals

[The case for p-type (and p-cotype) uses slightly different argument (not depending on p-stability) and is detailed with applications in Pisier (1986).] 9. The preceding problem has the following counterpart for the operators T without reference to the p-stable mapping u : X → Y, but the class of spaces X (and Y) are somewhat restricted. Namely a Banach space X is pstable if the identity mapping I : X → X is p-stable in the sense of the above problem. Thus if (S, S, µ) is a measure space, X is a type p-stable complete normed linear space, and RZ : S → L0 (P ) is a symmetric p-stable random measure define T : f 7→ S f (s)dZ(s), first for simple functions which suffices and then extend, so that T : Lp (µ, X) → Lq (Ω, Σ, P, X) = Lq (P, X) is defined (as above with u as identity) then T is continuous, for all 0 ≤ q < p ≤ 2, and if 0 < p < 2, then T is an isomorphism into (i.e., an embedding). [Proof Sketch. That T is well-defined for the dense set of simple functionsPis as in the preceding problem. With u = I (identity) we get for n f = i=1 xi χAi , xi ∈ X, Ai ∈ S disjoint, Z

S

p1 p = kf (s)k dµ(s)

n X

≤K =K

i=1

kxi µ(Ai )kp

Z

Ω

Z

k

n X

! p1

1 p

q

! q1

xi (µ(Ai ) f (ω)k dP (ω)

i=1

q1 k(T f )(ω)k dP (ω) . q

Ω

So T f = 0 implies f = 0 a.e. and T is continuous as well as one-to-one. From this one shows that T −1 is also continuous, and the rest follows easily.] 10. We have seen in Section 4 above that a positive random measure Z(·) on (R, B(R), µ) with independent values is completely random so that it is infinitely divisible. This is based on a Lebesgue decomposition of Z. A somewhat more general case is as follows. Let (S, S) be a measurable space, Z : S → Lp (P ), p ≥ 1, a random measure such that for each disjoint An ∈ S, Z(An ) → 0 in Lp (P ) as n → ∞, (Z(·) is then called (strongly- or) s-bounded). Let ν : S → Lr (µ) be a measure, r ≥ 1, with σ-finite (Vitali) variation, i.e., |ν|(·) is σ-finite on S. Then Z = Z1 + Z2 uniquely where Zi is Lp (P )-valued (i = 1, 2), Z1 is absolutely continuous relative to |ν|, written Z1 << ν and Z2 ⊥ ν, in the sense that on every set of positive |ν|(·)measure Z2 (·)-vanishes and there is a set A0 ∈ Σ, |ν|(A0 ) = 0 but Z2 (A) 6= 0. [If Lr (µ) = R, and then ν is a signed measure with the translation invariance property so that ν = aλ where a ∈ C and λ is Lebesgue measure, then the random measure Z1 (·) will be infinitely divisible (assuming Z ≥ 0). Thus we have a generalization of Lebesgue’s decomposition in this more gen-

Bibliographical Notes

215

eral case, although we cannot assert the infinite divisibility now. This is still a particular case of a decomposition in Banach spaces given by the author (cf., Rao (1967), pp.112–113) which will be discussed further in Chapters 8 and 9.]

Bibliographical Notes This chapter is devoted to some deeper aspects of bimeasures and random measures in a general context. First we considered second order random measures and their bimeasures, then obtained a Jordan type decomposition into (four) positive definite (bounded) bimeasures for general (bounded) bimeasures. K. Ylinen (1993) had obtained such a result for bimeasures even on C ∗ -algebras and our treatment slightly simplified his work for the particular case of scalar continuous function space C0 (S), functions vanishing at infinity on a locally compact set S, wherein we used the M T -integration developed earlier. More particularly our aim is to take S = R and define a convolution operation for random measures through their associated bimeasure algebras, investigated by G.C. Graham and B.M. Schreiber (1984). We included the relevant portions of their work (see also Gilbert, Ito and Schreiber (1987), generalizing the earlier work to all locally compact groups and we again utilized some of their ideas). Here it should be noted that the classical convolution method does not seem to extend directly even with the weaker Stratonovitch’s generalized Riemann-Stieltjes method. All this work is restricted to random measures with values in a Hilbert space, and an extension to the L1 -context (as in the classical work) is of interest, but is not yet available. It will also be useful to consider convolution algebras in this contest, and analyze them for further properties. This will be discussed once again in a general context in Chapter 9. If random measures take values in L0 (P ), then we have to leave bimeasures and turn to more inherently probabilistic treatment. So we assume that they take independent values on disjoint sets and are p-stable 0 < p ≤ 2, to invoke the L´evy-Khintchine formula and develop integration. Here the investigation starts with the classical or non-stochastic integrands, both scalar and vector valued cases are studied. The general detailed and deeper analysis of Y. Okazaki (1979) is utilized in our treatment of the subject. Now the random integrator takes values in L0 (P ), a complete non-locally convex linear metric space, or a Fr`echet space, as the range of our random measures. The integration of such integrands is fully amenable to the (Bartle-) Dunford-Schwartz method, given in their monumental work (1958), and we have exploited it fully. Okazaki has established the desired integrals each time afresh, and we are able to simplify some of his computations utilizing the methods and results of D–S integration. The work for Lp , 0 < p ≤ 2 is satisfying and the analysis is nontrivial. We have used the characteristic functionals, and Fourier transform on (algebraic) duals of Fr´echet spaces. Most of the interesting available work, essen-

216

5 More on Random Measures and Integrals

tially based on Okazaki’s (1979) important analysis which in part also depends on cylindrical measures, is included. Regularity of these measures, so that their (projective) limits exist, plays a crucial role. The long Section 3 includes all this work. To go beyond the p-stable class, we particularize to nonnegative random measures. These already appear as conditional measures which are bounded, nonnegative, and not necessarily stable. On the other hand if the nonnegative random measures take independent values, then they are called “completely random” and studied by J.F.C. Kingman (1967). Using his structure theory of such measures, which need not be p-stable, both stochastic integration (easy part), and the deeper embedding theory of the resulting integrals were developed by D.J.H. Garling (1975). We have included some salient features of both (bounded and not necessarily so) in our treatment. It is an interesting fact, already pointed out by Garling, that the range spaces of the associated linear operators turn out to be certain “generalized Orlicz spaces” (if the random measures are strongly stationary) and otherwise “Musielak-Orlicz spaces” both of which are subspaces of the L0 (P ) space. All of this work gives the special character of random measures which are vector valued, but have their own nontrivial individualistic analyses. These ideas will be analyzed further in the following chapters. We have also included several additional results as complements with sketches of some arguments asking the reader to fill in the details. These include some results on tensor products of bimeasures in (tensor) Hilbert spaces, providing some motivation for multidimensional random measures and specialized properties of stochastic integrals. Also we included a result to show why one cannot use Lebesgue-Stieltjes analysis here, and discussed “p-stable” types (and cotypes) of the geometric theory of Banach spaces, but in the Fr´echet space context. This was also considered by Okazaki and we followed his lead. Thus we have covered large parts and raised interesting problems to proceed from here on to other properties of random measures and also contrasting these with classical vector measures. One of the important aspects of stochastic integration that we have not touched in this chapter is to study the problems if the integrands are also stochastic. A general analysis as in Lebesgue’s theory presents numerous difficulties. They can however be studied for large classes of stochastic integrands with somewhat restricted random measures. We call them martingale type measures, and these will be taken up in the following chapters and their analysis will be presented in some detail. It may also be noted that this chapter acts as a bridge between the classical stochastic integrals of the preceding chapters that have shown the beginnings of the new theory, away from the restriction of being absolute integrals, and the forthcoming general classes that are inclusive of ‘bilinear integration.’

6 Martingale Type Measures and Their Integrals

In the preceding chapters we have considered random measures and defined their integrals mostly for nonrandom scalar or vector integrands. But the study of convolution of random measures in a Hilbert space which is an L2 (P ) here, leads to random integrands of a general nature and their integrals invite a study in new directions. This is considered in this chapter for measures that again generalize Brownian motion, namely martingales and some extensions allowing large classes of random integrands. All these ideas depend on a new concept, called a “filtration” of underlying σ-algebras as a key feature. R Let us analyze the definitions of integrals I f (t) dZ(t) given in the preceding chapter where Z(A) ∈ L0 (P ) and f (t) ∈ C so that f (t) ∈ L0 (P ) also. Here Z(A) is measurable for the σ-algebra B = σ(Z(A), A ∈ S) ⊂ Σ and f (t) is measurable for σ(∅, Ω) the trivial field ⊂ Σ. These two properties were noted by Itˆ o, as f (t, .) is measurable for all σ-algebras Σ since the trivial field is contained in all of them. The other σ-field B is generated by Z(·), the B.M., which is independently valued. He then made the fundamental observation that f can simply be a random function which is independent of the algebra generated by the B.M. for values of t ≥ s, where s goes with f (s). Therefore the ‘constant’ f (t) can be replaced by f (t, ω) and Z(A), by Z((t, t′ ]), so that f (t) and Z((t, t′ ]) are mutually independent random variables. With this the integration techniques of the last chapter extend. Thereafter Doob working with martingales made the key observation that Itˆ o’s idea extends to all random measures that are induced by martingales, and since B.M. itself is a martingale the proposed general integration for such induced measures is possible and then he made extensions. The following section is devoted to this point of view for integrators that are more general, called semi-martingales. It will be seen that the procedure succeeds if the measures can be decomposed (additively) such that one is a martingale part, and the other having finite variation on bounded sets with features of the very classical type of a Stieltjes measure.

218

6 Martingale Type Measures and Their Integrals

In Section 2 we proceed to consider “two parameter” functions and processes, and see that the extension is somewhat deeper. It can thereafter be considered for multi-parameter cases. The work is detailed with a view to use it for some important problems. In both cases, even in L2 (P ) spaces, the filtration obscures the geometry and the analysis has to take this difficulty into consideration. Section 3 is devoted to some aspects of analysis leading to certain types of (random or stochastic) differential equations which are an important motivation for this extension. Unlike the Lebesgue (-Stieltjes) case the stochastic integrals, in all but simple problems, are not absolute. We then consider some generalizations in Section 4 including a brief account of ‘McShane’s stochastic calculus’ which is gaining interest in contemporary applications. In the last section on complements and exercises, we consider several adjuncts to the main analysis, usually with sketches of the main ideas and arguments. Some of this work will later be simplified under additional restrictions and contrasted with classical (Banach valued) vector measures and integrals [Another one using Kakutani-Klee theorem, cf. p. 445, is possible.].

6.1 Random Measures and Deterministic Integrands Before the study of integration with stochastic integrands starts, we consider briefly the integrals relative to Z(·), with values in L0 (P ) which was not yet discussed in detail. We just treated the case of Z : S → Lp (P ), 0 < p ≤ 2 when Z(·) is p-stable, or infinitely divisible and a special treatment when Z(·) is positive in L0 (P ), as well as independently valued. It is thus appropriate to consider the case of Z(S) ⊂ L0 (P ) also. R Observe that T : f 7→ S f dZ for f ∈ Lp (µ) is analyzed as a mapping on Lp (µ) → Lp (P ) and in various related forms. The properties of T depend both ¯ + depending on the probabilistic behavior of on the existence of a µ : S → R Z(·) and particularly that µ dominates or controls the variation of Z(·). The existence (or availability) of such a measure µ is essential for a proper definition R of S f dZ. The following result on the existence of a controlling measure for Z(·) with values in L0 (P ), without further restrictions was established by M. Talagrand (1981), and independently also by N.J. Kalton, N.T.Peck and J.W. Roberts (1982), which plays an important role. We now present this aspect of the (Wiener-type) integration before embarking on the analysis of integrals with stochastic integrands which is the main focus of this chapter. Recall that L0 (P ) is a complete linear (invariant) metric space, under the metric d : L0 (P ) → R+ , given by convergence in probability or equivalently d(f, g) = d0 (f − g) where Z |f − g| dP, (1) d(f, g) = Ω 1 + |f − g|

6.1 Random Measures and Deterministic Integrands

219

and (L0 (P ), d0 (·)) then becomes a Fr´echet space. If (S, S) is a measurable space and ν : S → X is a vector measure where (X, d(·)) is a Fr´echet space with invariant metric d(·), then it is not known whether there exists a measure µ : S → R+ that dominates (or is “controlling”) ν : S → X in the sense that limµ(A)→0 ν(A) = 0 in the topology of X. However, if X = L0 (P ) above, then a positive answer is obtained as follows. We shall outline the ideas since the proof of the result is fairly involved and the details are found in the above noted readily available sources. [Hereafter d0 (·) is written simply as d(·) for convenience.] Theorem 6.1.1 Let (S, S) be a measurable space and Z : S → L0 (P ) be a random measure, i.e., σ-additive in the topology of L0 (P ) as defined by the metric given by (1). Then there exists a positive random variable g, i.e., 0 ≤ g ∈ L0 (P ), and a vector measure Z1 : S → L2 (P ) such that Z(·) = gZ1 (·) : S → L0 (P ), and since Z1 is a vector measure into the Hilbert space L2 (P ) it has a controlling (probability) measure µ (so Z1 ≪ µ), [by Theorem 1.3.4 even when L2 (P ) is replaced by a Banach space] which therefore controls Z as well. Sketch of Proof. In the proof of Theorem 1.3.4 if ν : S → X, a Banach space, then x∗ ◦ ν : S → R is a signed measure, hence bounded for each x∗ ∈ X∗ , and so we deduce that {kνk(A), A ∈ S} is bounded. But in a Fr´echet metric, X∗ need not separate points of X and can even be trivial (as in the case of Lp ([0, 1])), 0 ≤ p < 1, with Lebesgue measure so that (Lp ([0, 1]))∗ = {0}). So we first suppose that {ν(A) | A ∈ S} ⊂ L0 (P ) is bounded. Now the classical results in functional analysis show that the closed convex hull of this set, denoted E, is also bounded. When X = L0 (P ), a function space, then E is identifiable as a set of bounded measurable functions on S, a subspace of B(S), and under uniform norm, the elements of E can be identified as a subspace which in turn can be considered isometrically isomorphic ˜ the space of continuous functions on a compact Hausdorff space S. ˜ to C(S), [This is done by the Gel’fand-Na˘ımark theorem.] Then a result of Maurey ˜ is a ‘factor space’ of L2 (˜ (1981) shows that C(S) µ) where µ ˜ is a probability measure. Hence we finally have E/L2 (˜ µ), the factor space to be one dimensional. Now denote it as {g}, for g ≥ 0 so that one can identify Z and a vector measure ν into L2 (˜ µ) and g ∈ L0 (P ). We can enlarge the measure 2 space (Ω, Σ, P ) so that L (˜ µ) ⊂ L0 (P ) and one gets the result as announced. Note that all of this depends on the fact that Z(S) ⊂ L0 (P ) is a bounded set. This along with its consequence is really at the root of our argument. 2 Theorem 6.1.2 Let (S, S) be a measurable space and Z : S → L0 (P ) be a σadditive function, i.e., a random measure. Then the range {Z(A), A ∈ S} ⊂ L0 (P ) is contained in a ball, i.e., it has a bounded range. The proof is indirect, as in Theorem 1.3.4. Assuming that it is not bounded, we make several reductions, starting with the simplification that P may be as-

220

6 Martingale Type Measures and Their Integrals

sumed not atomic, i.e. has a nonatomic part S1 , and showing that {Z(A), A ∈ S} must be unbounded, and S1 may be taken to be separable in the Fr´echet metric d of S1 , where d(A, B) = µ(A ∆ B) in which S1 ⊂ Σ can be assumed etc. Eventually this leads to a contradiction so that the range Z(S) ⊂ L0 (P ) must be bounded. There are several details to be filled in, and we refer the reader for the two sources (of papers) noted for the preceding result. [More on this is discussed at the end of Bibliographical Notes of Chap. 8] Remark 6.1 The problem originally raised by D. Maharam in the late 1940’s is this: Does every vector measure ν : S → X, Fr´echet space, has a dom¯ where ν ≪ µ? This problem is still unsolved. inating measure µ : S → R p If X = L (P ), 0 ≤ p ≤ ∞, a positive answer is furnished by the above result and that of Theorem 1.3.4. The significance of a random measure, and its specialized analysis are thus made explicit. The next result is motivational. If no special properties are assumed for the vector or random measure Z: S → R L0 (P ), then we can only associate for the mapping T : f 7→ S f dZ, f ∈ B(S) ∩ Lp (µ), p ≥ 0 and µ the dominating measure of Z, very few properties, such as its continuity and linearity. Further conditions related to independent values, positivity or p-stability will give more interesting properties. The following is one such result. So its essential details are included. Theorem 6.1.3 Let (S, S) be a measurable space and Z : S → L0 (P ), a random measure into a space of scalar random variables on (Ω, Σ, P ). (a) If µ is the controlling measure of Z, guaranteed by Theorem 1, then p 0 p the R mapping T : L (S, S, µ) → L (P ) is linear. Also if f ∈ L (µ), T f = ˜ ˜ S f (s) dZ(s) is defined (i.e., f is Z-integrable) then T fn → T f for all fn → f in µ-measure when |fn | ≤ f < ∞ so that T is continuous in this (weak) sense. (b) If in the above Z is p-stable and symmetric with parameters (0, 0, 1, µ), then for every 0 ≤ q < p < 2, the mapping T given in (a), i.e., R (2) T : Lp (µ) → Lq (P ), (T f = S f dZ), is actually an isomorphism.

Proof. (a) In this part we did not assume much on Z, and so only a weak Rconclusion obtains and this is established as follows. It is clear that T : f 7→ S f dZ is linear and defined for all simple f . By Theorem 1, there exists some finite measure µ on (S, S), denoted (S, S, µ) such that Z(·) is controlled by it. If f is a function in Lp (µ), p ≥ 0 and is (D–S) integrable for Z, then by the well-known vector integration of scalar functions on (S, S, µ) each |g| ∈ Lp (µ), |g| ≤ |f | when f is Z-integrable, (cf. Dunford-Schwartz (1958), IV.10.9), and a Vitali convergence theorem holds for it. Hence (a) follows immediately.

6.2 Random Measures and Stochastic Integrands

221

(b) If additionally we assume Z to be p-stable (0 0, and the p-stable condition on Z(·) implies |f |p dµ S Pn that there is a constant K > 0 such that for each simple function f = i=1 ai χAi and 0≤q
i=1

holds by Theorem 5.4.5 (with Y = C here). It then follows from the fact that T −1 exists and is also continuous, whence T is an isomorphism, since simple functions are dense in Lq (µ). Here we also use the results with Fourier transforms in the form of characteristic functions for p-stable random variables. This gives the conclusion. 2 The result can be extended to vector valued functions if the range space Y, of Lp (µ, Y), and Lq (P, Y) is a Banach space of “p-stable type” as discussed in the preceding chapter (or a projective limit of such spaces which will be a locally convex Fr´echet space). But these extensions will not be treated here. Without further conditions not much more can be said about Z : S → L0 (P ). On the other hand one can drop the hypothesis of “p-stable” condition and assume that Z(·) is infinitely divisible (without being positive), and the reader could consider an interesting extension of Garling’s theorem (Theorem 5.4.3) to this case. We proceed to the Itˆ o-type integrals [Another one using Kakutani-Klee theorem, cf. p. 445, is possible.]

6.2 Random Measures and Stochastic Integrands R The basic problem is to generalize Wiener’s integral I f (t) dXt where {Xt , t ∈ I ⊂ R} is B.M., but allowing f (t, .) to be a random function. The procedure becomes the following one naturally. Thus let BI be the Borel σ-algebra of I and (Ω, Σ, P ) a probability space. Now consider a σ-subalgebra O ⊂ BI ⊗ Σ of thePproduct. An O-simple function f : I × Ω → R (or C) is of the form n f = i=1 fi χAi , Ai ∈ BI disjoint, and fi χAi is O-measurable with fi as a bounded random variable. Let X : I × Ω → R (or C) be BI ⊗ Σ measurable so that X = {Xt , t ∈ I} is a process and Xt ∈ Lp (P ), t ∈ I. Now define a mapping, slightly modified from Wiener’s original definition, τ : f → Lq (P ) as (the Ai need not be intervals): Z n X fi Xsup(Ai ) − Xinf(Ai ) , (1) τ (f ) = f (t) dXt = I

i=1

where if Ai = (ti , ti+1 ] ⊂ I then we will have the increment of X, in (1). Thus τ is clearly well-defined and is additive on such functions f . The problem is to extend τ to a larger collection of (jointly measurable) f ’s for which

222

6 Martingale Type Measures and Their Integrals

we need to evaluate E(|τ f |q ) and obtain a suitable bound when f ∈ Lp (P ). Even for p = q = 2 the calculation of these bounds is somewhat involved as we illustrate it for the B.M. which already explains the difficulties. Let I = (a, b] and a = t0 < t1 < . . . < tn ≤ b with Ai = (ti−1 , ti ]. Then (1) becomes n X fi (·)(Xti − Xti−1 ), τ (f ) = i=1

and (take fi real for simplicity) E(|τ f |2 ) =

n X i=1

E(fi2 (Xti − Xti−1 )2 )

+2

X

1≤i<j≤n

E (fi fj )(Xτi − Xti−1 )(Xtj − Xtj−1 ) .

(2)

To evaluate this, we use the identity E(Y ) = E(E A (Y )) for any integrable Y on (Ω, Σ, P ) with E A (·) as the conditional expectation operator for any σ-algebra A ⊂ Σ. In order to utilize the independence of increments of the B.M. (so they are necessarily conditionally independent), the fi , fj must be assumed A-measurable. The appropriate condition is to take A to be a σalgebra Ai generated by X0 , X1 , . . . , Xi and suppose moreover that each fi is measurable for Ai (⊂ Ai+1 ). With this condition, (2) can be simplified. Thus the second term in (2) becomes on using some familiar properties of conditioning (found in any basic graduate text book on Probability) and 1 ≤ i < j ≤ n: E(E Ai (fi (Xti − Xti−1 )fj (Xtj − Xtj−1 ))

= E fi (Xti − Xti−1 )E Ai fj (Xtj − Xtj−1 )

= E fi (Xti − Xti−1 )E Ai E Aj−1 fj (Xtj − Xtj−1 ) = E fi (Xti − Xtj−1 )E Ai fj E Aj−1 (Xtj − Xtj−1 )

= 0,

since E Aj−1 (Xtj − Xtj−1 ) = 0 because Aj−1 is

independent of Xtj − Xtj−1 implying

E Aj−1 (Xtj − Xtj−1 ) = E(Xtj − Xtj−1 ) = 0.

(3)

With this (2) simplifies on using E Ai−1 (Xti − Xti−1 )2 = σ 2 (ti − ti−1 ), also the independence of the B.M. increments and its variance relations in (2) and (3), to get: Z n X E(fi2 )(ti − ti−1 ) = σ 2 E(f 2 (t)) dt. (4) E(|τ f |2 ) = σ 2 i=1

I

Since such f ’s span a dense subset of L2 (dt × dP ), we see that τ extends to all L2 (dt × dP ) valued in L2 (P ), and satisfies for such (simple) f and then for all f with the measurability conditions in the former space:

6.2 Random Measures and Stochastic Integrands

E

Z 2 ! Z Z 2 2 f (t) dXt ≤ σ |f (t, ω)| dt dP. I

I

223

(5)

Ω

R Hence τ : f 7→ I f (t) dXt ∈ L2 (P ) is a bounded linear operator on a subspace of L2 (dt × dP ) into L2 (P ). These restrictions were discovered by K. Itˆ o for the first time in early 1940’s and then extended the Wiener type integral for deterministic integrands that we discussed in the last section as well as in Chapter 5. It was then noted by Doob that only the conditional expectations of the increments should be zero for Itˆ o’s analysis, and that property is restated as (6) E Aj−1 (Xtj ) = Xtj−1 , j ≥ 1. But this is the defining property of martingales, already used by J. Ville and P. L´evy in the late 1930’s. Comparing (4) and (5), it is seen that the inequality E Aj−1 (Xtj − Xtj−1 ) ≥ 0, j ≥ 1, suffices and this makes (6) valid if we replace ‘=’ by ‘≥’, the resulting class is called a submartingale. Thus the stochastic integration can be established for random measures determined by (sub) martingales extending the B.M. case substantially. But one must also remember that the integrands, f (t, ω), satisfy the corresponding order restrictions involved in the families of σ-subalgebras. Thus we have not only the random measures, all of which considered above include B.M. as a fundamental building block, and hence need not be of finite variation, but the integrands now must satisfy the corresponding measurability hypothesis relative to increasing families of σ-subalgebras of Σ in the basic triple (Ω, Σ, P ). Note also that the inequality (5) which resulted here is none other than the Bochner L2,2 boundedness of Definition 4.3.1. We shall generalize this for many integrals with random integrands and measures, influenced just by the classical Real Analysis results. We summarize the preceding discussion, for reference, as follows: Theorem 6.2.1 Let {Xt , a < t ≤ b} be a B.M. on (Ω, Σ, P ) and Z(A) = Xt − Xa for A = (a, t]. Then Z : R → L2 (P ) is an additive set function on the semi-ring R of half-open intervals (left open, right closed). The function Z extends to a random measure on the Borel σ-algebra B([a, b]) on which the measure Z is defined, Z : B([a, b]) → L2 (P ). Let At be the σ-algebra generated by {Xr , a < r ≤ t} and f : (a, b] × Ω R→ RC be a bounded function such that f (t, ·) is At measurable, (and especially Ω [a,b] |f (t, ω)|2 dt dP (ω) < ∞.) Then the mapping Z b τ : f 7→ f (t) dZ(t) ∈ L2 (P ), f ∈ L2 (dt dP ), (7) a

is well-defined (by (1) for simple functions, and satisfies the fundamental inR Rb equality (5), i.e., E(|τ f |2 ) ≤ σ 2 Ω a |f |2 dt dP , where σ 2 is the variance parameter of the B.M. and in fact there is equality here. Further τ : L2 ((a, b] × Ω, B((a, b]) ⊗ Σ, dtdP ) → L2 (Ω, Σ, P ) is a bounded linear operator.

224

6 Martingale Type Measures and Their Integrals

We now extend this result, clarify the inequality (5), and analyze its underlying conditions which incorporate randomness and show how it applies to many more situations. This is the Bochner L2,2 -boundedness principle which includes the Itˆ o integral, where the measure Z(·) never has a finite (Vitali) variation (cf., e.g., Exercise 5 of Chapter 5 where a detailed argument is sketched about this fact). It is useful to note that the σ-algebras At ↑⊂ Σ, can be taken complete, and f (t, ·) is measurable relative to At for each a < t ≤ b and thus the t-space has an order structure. Also the restriction Z|At is a measure generated by {Xt′ , a < t′ ≤ t}, inheriting some valuable properties. These are given abstractly in the following form for the process Xt that determines a random measure Z on ∪t At → L2 (P ). We state it with some new terminology as: Definition 6.2.2 Let (Ω, Σ, P ) be a complete probability space and {Ft , t ∈ R+ = [0, ∞)} be a family of σ-subalgebras of Σ such that for 0 ≤ t < t′ < ∞, Ft ⊂ FT t′ ⊂ Σ and each Ft is complete. It is often called a filtration. If also Ft = s>t Fs , t ≥ 0, then the filtration {Ft , t ≥ 0} is moreover termed standard. A family of random variables {Xt , t ∈ R+ } is said to be adapted to a filtration {Ft , t ∈ R+ } if each Xt : Ω → R (or C) is Ft -measurable, i.e., Xt−1 (B(R)) ⊂ Ft , t ∈ R+ , written symbolically as {Xt , Ft , t ≥ 0}. An example is Ft = σ{Xs , s ≤ t} ⊂ Σ. If moreover, {Xt , t ≥ 0} ⊂ L1 (P ), and adapted to a filtration Ft , then it is a (sub-)martingale if E Fs (Xt ) = (≥)Xs , a.e. for 0 ≤ s < t < ∞, and if Ft = σ(Xs , s ≤ t) it is also written as, (8) E Fs (Xt ) or E(Xt |Fs ) or E(Xt |Xn , 0 ≤ u ≤ s), which by the Radon-Nikod´ ym theorem is equivalent to: Z Z Xs dPFs , ∀A ∈ Fs , 0 ≤ s < t. Xt dPFs = (≥) A

(9)

A

Thus the work uses the conditional probability analysis fully and the general theory of the last chapter undergoes a specialized treatment. This new ‘technology’ is restricted to get refined results. All these considerations are necessary and interesting for applications in (stochastic) partial or ordinary differential equations among others. We now present a brief motivation of how a process {Xt , Ft , t ≥ 0} may determine (and is determined by) a random measure. Suppose that X : t 7→ Xt ∈ Lp (P ), p ≥ 1 is right continuous, in that P [limt↑t0 Xt = Xt0 ] = 1, t0 > 0, then the set function Z(·) defined as Z(A) = Xsup(A) − Xinf(A) , A ⊂ R+ (or more simply if A = (a, t] let Z(A) = (Xt − Xa )) define an additive function on the semi-ring based on such intervals which then can be uniquely extended, using standard methods, to the ring generated by these intervals, to be additive. If moreover the process is of finite variation on all compact subsets of R+ , then the variation of Z(·) is additive and is of finite variation on

6.2 Random Measures and Stochastic Integrands

225

compact sets, which is also left continuous so that it has a σ-additive (and σ-finite) extension that dominates Z(·). It follows that Z(·) is a σ-additive vector measure on L1 (P ) and is of finite (Vitali) variation |Z| on bounded sets. If 1 < p < ∞, then Z(·) with values in (the reflexive) Lp (P ) has the (scaler) well-known Radon-Nikod´ ym property. Hence for any bounded measurable f : R × Ω → R (or C) we can define the Lebesgue-Stieltjes type integral by the equation: Z Z dZ(·, ·) (t1 , ·) d(|Z|)(t1 ), (10) f (t, ·) f (t, ·) dZ(t, ·) = d|Z| A A for all A ⊂ R+ , the class of bounded Borel sets. Then we can analyze the vector space of Z(·)-integrable f : R × Ω → R (or C), denoted L1 (dZ). In case the process {Xt , t ≥ 0} is essentially bounded, so that Z(·) defined by the increments of Xt ’s and has a σ-additive extension with finite variation on bounded sets, also with values in Lp (P ), 1 < p < ∞, as above, then using the lifting theorem (recalled in Section 5.4) one can regard Z(·, ω) as a scalar Stieltjes measure and define the integral (10) for each ω ∈ Ω and proceed. R R 1 Now kf kp = [ Ω R |f |p )(t) d|Z|(t)dP ] p is a norm with which Lp (Z) becomes complete. However either of the above methods excludes the Brownian motion which does not have finite variation on any non-degenerate interval of R. This exclusion is unacceptable since the condition also excludes almost all the work of the preceding chapter. So we need to look for alternative methods to which we now turn. Let us recall and generalize the L2,2 -boundedness principle of Definition 4.3.1 to analyze the relation between the integrator (i.e., the random measure) and the integrand (a random function) for which a general (random) integral can be defined to meet our needs. Observe that given an adapted process {Xt , Ft , a < t ≤ b} ⊂ L2 (P ) we associate a function (a ‘pre-measure’ determined by it) Z(·) by the equation on the class of sets Q = {(t1 , t2 ] × A : a ≤ t1 < t2 ≤ b, A ∈ Ft1 } on I × Ω, where I = (a, b], and (Ω, Σ, P ) is the basic probability space. If At is the semi-ring of half-open intervals of the form a < t1 < t2 ≤ t, then Rt = At × Ft is a familiar semi-ring on which one can define Z ˜ 1 , t2 ] × A) = (Xt2 − Xt1 ) dP = E(χA (Xt2 − Xt1 )), A ∈ Ft1 (11) Z((t A

and if t1 = a, then we express it as = E(χA Xa ). This is well-defined and is a (finitely) additive scalar function. [If a = +∞, then we set X∞ = 0.] It is useful ˜ × A) is finitely additive for each A ∈ Fa and Z((t ˜ 1 , t2 ] × ·) to observe that Z(· ˜ × ·) is not a bimeasure in the sense we studied is σ-additive on Ft1 . Thus Z(· earlier. However on the semi-ring R it is finitely additive. This property has to be exploited, and the (restricted) integral developed. But integration with finitely additive set functions, on simple measurable functions is summation in a different notation. Now the Bochner boundedness concept for adapted processes and additive random measures is meaningful. We state this as follows.

226

6 Martingale Type Measures and Their Integrals

Let X = {Xt , t ∈ I} ⊂ Lp (P ), I = (a, b], and a = t0 < t1 < . . . < tn+1 = b. If f :I →P R is a bounded Borel function, define the formal ‘integral’ as the sum, n for f = i=0 ai χ(ti ,ti+1 ] , Z

f (t) dXt =

I

n X i=0

ai (Xti+1 − Xti ) ∈ Lp (P ).

(12)

The left side symbol is uniquely defined for each (finite) partition of I, and X is said to be L2,2 -bounded relative to some µ, assumed to exist, if for a constant C > 0, Z 2 Z E f (t)dXt ≤ C |f (t)|2 dµ(t) (13) I

I

holds, where µ : B(I) → R can be taken a (Borel) measure. Here µ may be the Lebesgue measure in which case this is a specialization of (4) that includes the B.M. process case with C = σ 2 > 0, and there is even equality. In general we assume the existence of µ and constant C > 0, for the X-process and then X is termed L2,2 -bounded when only the inequality is demanded in (13). The result given for simple functions, extends to all f in L2 (µ) into L2 (P ). This extended purely classical integral thus includes the original Wiener definition. We can present the result as follows: +

Proposition 6.2.3 Let Z : B(I) → L2 (P ) be a vector measure. If X = {Xt = Z((a0 , t]), t ∈ I} for an arbitrarily fixed a0 ∈ I, then X is L2,p -bounded for all 1 ≤ p ≤ 2 relative to a measure µ ˜ : B(I) → R+ . In particular, a 2,2 process X of orthogonal increments is L -bounded relative to some µ ˜ and C > 0. Proof. If p = 2 the result was established in detail in Theorem 4.2.3 and Proposition 4.3.2, the former being the basic Grothendieck inequality for bounded bilinear forms. But if Z : B(I) → Lp (P ), p ≥ 1,R then by Theorem 1.3.4 there is a controlling measure µ for Z(·) and hence I f dZ is well-defined for each bounded Borel function f : I → R. By another important extension of the Grothendieck inequality used above [the result depends on a crucial version of this inequality due to Lindenstrauss and Pelezy´ nski (1968) detailed also in Rao (2004), Theorem 7.4.7 pp. 527–529 where we take Y = Lp (µ), then for a finite measure µ : B(I) → R+ which is a concrete realization of Y, the so-called Lp -spaces for 1 ≤ p ≤ 2, and these cover all Lp (µ)-spaces for p in that range], it follows that

Z

Z 12

(14) ≤ kf kL2 (µ) = |f |2 dµ , 1 ≤ p ≤ 2.

f dZ I

Lp (P )

I

This is just L2,p -boundedness for Z(·) relative to µ. [The existence of such a µ (not unique) is the key contribution of Grothendieck’s inequality.]

6.2 Random Measures and Stochastic Integrands

227

The result becomes simpler if p = 2 and Z(·) has orthogonal values, so ˜ that µ ˜(A, B) = E(Z(A)Z(B)) = µ ˜(A ∩ B) since Z(A − B) ⊥ Z(B − A) ⊥ Z(A ∩ B) and µ : B(I) → R+ is σ-additive. This implies, as seen in Theorem 4.3.3, that Z Z Z

Z

2

¯ ¯ =E f (s)f (t) dZ(s) dZ(t) = |f (t)|2 dµ,

f dZ L2 (P )

I

I

I

I

with C = 1 and even equality, is obtained. 2 We want to extend this result if f is a random function, i.e., f : I × Ω → C, and connect it with the measure Z : B(I) × Σ → C. It is now necessary to relate this to algebras, the filtrations Ft ⊂ Σ and consider subalgebras of B(I), with the fundamental procedure used by K.Itˆo for the B.M. which was generalized by P.A.Meyer and his associates showing many possibilities whose work can be further extended with Bochner’s boundedness principle (somewhat generalized) along with Grothendieck’s theorem. The procedures and ideas of the Itˆ o-Meyer school is more probabilistic and the Bochner-Grothendieck analysis is functional analytic in nature. An extension of the latter work is needed to generalize the former analysis. We first restrict to the L2,2 -boundedness case, and consider an interesting method of I. Cuculescu (1970) which shows that the resulting stochastic (or random) integral is a kind of spectral integral in Hilbert space. This may be anticipated from the appearance of bilinear forms in (14) when p = 2. These methods use only the functional analysis ideas and have special interest. They are elaborated and applied here. The important idea in extending Itˆ o’s definition of the general B.M. integral to a random measure Z : B(I) → Lp (P ) and a random integrand f : I × Ω → R (or C) is the matching of their measurabilities in addition to the existence of a controlling measure µ on B(I) ⊗ Σ. The L2,2 -boundedness criterion of X = {Xt , t ∈ I} now takes the form: Z

Z

2

≤ C |f (t, ω)|2 dµ(t, ω), Ω ′ = I × Ω, (15) f (t, ·) dX

t I

2

Ω′

for a measure µ on Ω ′ and an absolute constant C > 0. Let Z : B(I) → L2 (P ) be the induced vector measure by X. Keeping the B.M. to be included here which does not have finite variation but which has a finite quadratic variation, defined as (taking Z((a, t]) = Xt − Xa ): [[Z]](I) = p lim

n→∞

kn X j=1

|Z(Ijn )|2

(16)

n Ijn = I denotes a partition of the interval I with max1≤j≤kn |Ijn | → 0 where ∪kj=1 as n → ∞, |Ijn | denoting the length of the subintervals Ijn ⊂ Ij , so that the partitions are refined. Of course the limit in (16) need not exist in general. But for the B.M. it exists and is a constant = σ 2 (b − a). If the variation is

228

6 Martingale Type Measures and Their Integrals

finite (i.e., |Z|(I) < ∞) then factoring out the maximum term in (16), which tends to zero, it follows that the quadratic variation is automatically zero. Now the point shown by B.M. is that |Z|(I) = +∞, and its quadratic variation [[Z]] < ∞. For simplicity let a = 0, bi > 0 and I = (0, b]. Consider the dyadic which is automatically refined as n inpartition of I, i.e., Ijn = 2jbn , (j+1)b 2n creases. Then Xb2 − X02 = =

kn kn h i2 X X − X jb X jb X X (j+1)b + 2 (j+1)b − X jb n n n n n kn X j=1

Let fn =

Pkn

j=1

2

j=1

|Z(Ijn )|2 + 2

2

kn X j=1

2

j=1

2

2

X jbn Z(Ijn ).

(17)

2

X jbn χIjn , a simple function on Ω ′ → C, and (17) becomes 2

kn X j=1

|Z(Ijn )|2 = Xb2 − X02 − 2

Z

fn dZ

(18)

I

where the integral of the simple function fn is uniquely defined. Thus the left side, giving the quadratic variation of Z, has a limit in L2 (P ) if and only if the right side integral is definable. This holds when Z(·) is L2,2 -bounded. We now detail the relations of the underlying σ-algebras for this (right side) limit to exist uniquely. See Corollary 6 below on the existence of the limit in (18) for martingales in L2 (P ). The simple functions fn defined above have the property that fn (s, ω) = X jbn (ω) if s ∈ Ijn and ω ∈ Ω, so that fn is measurable, for F jb , for s ∈ Ijn , 2 2n and for each ω ∈ Ω it is B 0, (j+1)b -measurable, also jointly for the σ-algebra n 2

B((0, (j+1)b , n ≥ 1. This is generated by the class of sets (u, v] × A 2n ]) ⊗ F 2jb n for a ≤ u < v ≤ b, and A ∈ Fu , for u, v ∈ I = [a, b], A ∈ Fu ⊂ Σ, and Z(Ijn ) of (17) is Z(A) in this notation and is Fu -measurable. This corresponds to Itˆ o’s assumption of the integrand being “non-anticipating”. Recall now the classical statistical terminology: the integrand fn is said “predictable” and Z(·) Skn is “charging” only the corresponding sets A, if it is σ-additive on j=1 F jbn , n ≥ 2 ′ 1. To make this precise, consider the σ-algebra of sets of Ω = I × Ω of the form {(u, v] × A : 0 ≤ u < v ≤ b, A ∈ Fu } of the filtration {Fu ↑, u ∈ I = (0, b]}, and to include u = 0, we add {0} × A, A ∈ F0 the collection of the former forming a semi-ring and this family determines a σ-algebra denoted by P termed the predictable σ-algebra contained in B(I) ⊗ Σ on which Z(·) extends to be a random measure. This is the primary object of our study. [Following the general measure theoretical analysis, this will be the same σalgebra if the open or any semi-closed sub-intervals of I are used to obtain P. If left closed right open intervals are used, we get what is called an optional

6.2 Random Measures and Stochastic Integrands

229

σ-algebra O. But the present analysis is simplified if the left open-right closed ones are used which we follow hereafter although P ( O, and it conforms to the original Itˆ o formulation where the Xt -process is path-wise continuous.] With this the limit exists in (18) as n → ∞. We shall see that it also exists for the continuous parameter, path continuous, square integrable martingales since every such process is a certain transform of B.M. due to an important theorem by P. L´evy. Because of its significance we state it later in a general form, which needs some further concepts. Our main aim is to obtain stochastic integerals as linear operations. To understand the problem more fully, we start with the (generalized) L2,2 boundedness case. Thus if {Xt , Ft , t ∈ I} where 0 < t ≤ b, (I = (0, b]), is L2,2 -bounded, let fn be a simple function defined as fn =

n X i=0

ai χ(ti ,ti+1 ]×Ai , Ai ∈ Fti , 0 ≤ t0 < t1 < . . . < tn+1 = b

(19)

denoted as S(I, P). Consider the mapping τ : S(I, P) → L2 (P ) and [motivated by (17)] defined as: τ (fn ) =

Z

fn dX =

I

n X i=0

ai χAi (Xti+1 − Xti ).

(20)

Since Xt ∈ L2 (P ) by hypothesis, and Ft ⊂ Ft′ for t < t′ so that the conditional expectation operators E Ft , E Ft′ commute giving E Ft E Ft′ = E Ft , we have with L2,2 -bounded hypothesis, on using the identity E(Y ) = E(E G (Y )) for any Gt ⊂ Σ and Y ∈ L1 (P ) the following with G = σ(Xs , s ≤ t): E(|τ (fn )|2 ) =

n X i=0

+2

X

a2i E(χAi E Gti (Xti+1 − Xti )2 )

0≤i<j≤n

ai aj E χAi ∩Aj (Xti+1 − Xti )E Gti (Xtj+1 − Xtj ) . (21)

To simplify (21) we specialize the X-process, since in general we cannot go further. For the B.M. and more generally for a square integrable martingale [we are using probability theory crucially here] E Gti (Xti+1 ) = Xti with probability one so that, the second term on the right of (21) vanishes. By the same hypothesis E Gti (Xti Xti+1 ) = Xt2i , a.e., and the first term of (21) inside E(·) becomes E(χAi (Xt2i+1 − Xt2i )). Hence (21) reduces to (µxt is implicitly defined below): 2

E(|τ (fn )| ) =

n X i=0

a2i E(χAi (Xt2i+1

−

Xt2i ))

=

n X i=0

a2i µxt ((ti , ti+1 ] × Ai ).

But {Xt2 , Gt , t ≥ 0} is an integrable collection and for s < t

(22)

230

6 Martingale Type Measures and Their Integrals

E Gs (Xt2 ) ≥ (E Gs (Xt ))2 = Xs2 by the martingale hypothesis of the Xt -process and the conditional Jensen’s inequality, so that {Xt2 , Gt , t ≥ 0} is a (positive) submartingale. In (22) it is observed that µxb on (0, b] × Ω is an additive (nonnegative) function on the semi-ring of sets of the form (ti , ti+1 ] × Ai , Ai ∈ Gi ⊂ Gi+1 . We now need to show the nontrivial fact that µxb (·) is a measure implying that the square integrable martingale process {Xt , 0 ≤ t ≤ b} is L2,2 -bounded. We establish this under the condition that the Xt -process is left continuous. More generally that a positive submartingale which is left continuous determines a measure, following Dol´eans-Dade (1968). This is a key measure theoretical technicality. Here we use an argument due to Pellaumail (1973) to show how the classical analysis of measures (and no probability results are used) can be employed: Proposition 6.2.4 Let {Xt , Ft , t ∈ I = [0, b]} be a positive submartingale satisfying limt↓s E(Xt − Xs ) = 0. Then the (positive) additive set function µx : (s, t] × A 7→ E(χA (Xt − Xs )) for A ∈ Fs is σ-additive on the ring generated by the sets {(s, t] × A, 0 < s ≤ t ≤ b, A ∈ Fs } and has a unique σ-additive extension to the (predictable) σ-algebra P ⊂ B(I) ⊗ Σ. Proof. First note that µx ((s, t] × ·) : Fs → R is a measure since the Xprocess is a submartingale and µx is additive on the semi-ring generated by the sets of the form (s, t] × A, 0 < s ≤ t, A ∈ Fs . It has a unique additive extension to the ring generated by these sets known from classical measure theory (cf., Rao (2004) Proposition 1.2.6). Thus it suffices to show that, under the (left) continuity hypothesis on the first coordinate variable, µx is also σ-additive in the first component so that µx is essentially (but not quite) a bimeasure. We actually show that µx (· × ·) is a measure on P ⊂ B(I) ⊗ Σ, using the topology of R (its Heine-Borel property) similar to the classical theory (cf., Rao (2004), Theorem 1.2.8). The (left) continuity hypothesis of the Xt -process implies for µx the conditions (i) limt↓s µx ((0, s] × Ω) = µx ((0, s] × Ω) and (ii) for any sequence An ∈ Σ, An ↓ ∅, then limn→∞ {µx (B) : B ⊂ An × (0, b]} = 0 uniformly in B ∈ R the ring generated by the sets {(s, t] × A, A ∈ Fs , 0 < s ≤ t ≤ b}. These conditions imply at once that µx is continuous from above at ∅ which establishes its σadditivity on R and hence on P which it generates. Since the additivity of µx on R implies its σ-super additivity, it suffices to show that it is σ-subadditive on R, which then establishes the desired result from elementary Measure Theory. S∞ Thus let B, Bn ∈PR, B = k=1 Bk be a disjoint union. We have to ∞ x show that µx (B) ≤ µ (B ) k to complete the argument. By definition k=1 B = (s, t] × A while Bn = (sn , tn ] × An for some An ∈ Fsn and A ∈ Fs . By the left continuity of the process we can find s < s′ < t so that µx ((s, s′ ] × A) < 4ǫ and similarly for each n, µx ((tn , t′n ] × An ) < ǫ/2n+1 where 0 < sn < tn < t′n ≤ b. Since the An ’s need not be monotone (or disjoint)

6.2 Random Measures and Stochastic Integrands

231

let a(ω) = the (at most finite) set of n’s such that ω ∈ An , and let Cn ⊂ Ω be chosen to satisfy: ω ∈ Cn if and only if (this idea is due to Pellaumail (1973)) n [ [s′ , t] ⊂ {(sk , t′k ) ⊂ a(ω)}. k=1

Then Cn ∈ Σ (i.e., Cn is measurable by the left continuity again of Xt , t ∈ I) for each ω, since the Bn ’s are disjoint, we get [ [ (sk , t′k ). (sk , tk ] ⊂ [s′ , t] ⊂ (s, t] ⊂ 0
0
But the extreme right is an open cover of the compact interval [s′ , t], so that (the Heine-Borel property) there is a finite subcover, implying the existence of a number n such that ω ∈ Cn . If Dn = Ω − Cn , then the Dn ↓ ∅, since Cn ’s increase to Ω. Hence by condition (ii), there exists an integer n0 so that sup{µx (B) : B ∈ R, B ⊂ (0, b] × Dn0 } < ǫ. Sn0 ˜ n0 ⊂ (0, b] × Dn0 If we set Dn0 = k=1 (sk , t′k ] × Ak , B ′ = (s′ , t) × A, then B ′ − D x ′ ˜ so that µ (B − Dn0 ) < ǫ. Consequently, ∞ ǫ ǫ X x µ ((sk , t′k ] × Ak ) + µx (B ′ ) ≤ + 4 4 k=1 X X ǫ ǫ X x =ǫ+ µx (Bk ). µ (Bk ) + ≤ + k+1 4 2

µx (B) ≤

k≥1

k≥1

k≥1

Since ǫ > 0 is arbitrary, µx (·) is countably subadditive, being additive it is a measure. 2 Remark 6.2 We presented all these details with an eye to introduce a concept that takes care of the various decompositions depending on ω where approximations should be made. This shows how an additional argument is needed, as compared with the classical analysis without ω’s. The desired notation, now depending also on ω as well as the order condition of R, is called the stopping line which is special to probabilistic analysis. It helps one to keep the resulting book-keeping in perspective. First we set down some important applications of this work itself. As a consequence of the preceding discussion and Proposition 4, we introduce the following extension of a random measure and its integral which implies a substantial application of the L2,2 -boundedness property. A process X = {Xt , Gt , t ∈ I} ⊂ L2 (P ) will be called a semi-martingale if it admits a representation as (there are other definitions and generalizations, but they are not needed now): Xt = Mt + Bt , t ∈ I, (23)

232

6 Martingale Type Measures and Their Integrals

where {Mt , Gt , t ∈ I} is a square integrable martingale and {Bt , Gt , t ∈ I} is a process of integrable variation. As usual Gt− = σ(∪s
I

by the same Proposition 4 and the CBS-inequality, Z (|f |2 [d˜ µ + |B|∞ d|Bt |dP ]) =2 Ω′ Z |f |2 dα =2

(24)

Ω′

RR µ(t, ω)+|B∞ |(ω) dt |B|(t, ω)dP (ω)], A ∈ P), is a finite meawhere α(A) = A [ d˜ sure. Taking (C, µ) as (2, α) in the definition of L2,2 -boundedness, we conclude that our semi-martingale X is L2,2 -bounded. 2 Remarks. Some interesting consequences of this theorem follow. 1. The result implies that the controlling measure α is not always of product type. In the martingale case (22) shows that there is equality in (24) so that α(·) is a minimal dominating measure, but in general the above obtained α(·) is not necessarily minimal. However, simple functions belong to all such spaces. 2. The representation (23) says that the variation process Bt is a ‘signal’ and the martingale process Mt is a ‘noise’, as in the B.M. case. Typically the ‘noise’ is not ‘smoother’ than the signal. If I = [0, b], M0 = 0 and Gt = {∅, Ω} all t, then M ≡ 0, and the integration relative to X reduces to the Lebesgue-Stieltjes definition.

6.2 Random Measures and Stochastic Integrands

233

The square integrable class is simpler, and forms a building block of the theory. It can then be compared with the Grothendieck inequality. In both cases the dominating measure(s), α to build up the general theory will be necessary. Grothendieck’s analysis asserts existence and Bochner’s boundedness principle assumes existence of such a measure. We now present an analysis showing the existence of an α if an integral can be defined at all with any reasonable (and useful) properties, which implies that these two conditions are two aspects of the structure that support the work here. We can now establish the existence of the limit for fn ∈ S(I, P) of (18) as an immediate consequence of the preceding result: (‘plim’ is limit in probability). Corollary 6.2.6 Let {Xt , t ∈ I} be a square integrable c` adl` ag martingale. Then its quadratic variation exists in measure and is given by: p lim

n→∞

kn X i=1

|Z(Ijn )|2 = Xb2 − X02 − 2p lim

n→∞

Z

fn dZ, I

P n n 2 and the left side is denoted by [X, X] = p limn→∞ K i=1 |Z(Ij )| ≥ 0, where [X, X] is a uniquely defined positive quadratic form, Z((s, t]) = Xt − Xs determines a random measure and Z is L2,2 -bounded relative to the predictable σ-algebra P and the measure α. (If X is continuous path-wise usually one uses hX, Xi and the general case is denoted as [X, X], as a matter of distinguishing the two.) It should be observed that for the B.M. process X = {Xt , t ∈ I}, which has continuous paths, this variation is hX, Xi(·) = σ 2 dµ(·) where µ is the Lebesgue measure and σ 2 > 0, so it is even a non-stochastic increasing function. For (even) the general case of a square integrable (c`adl` ag) martingale [·, ·] is an increasing random function. The result extends to vector valued processes. It plays an important part in the study of stochastic integration, in general, leading to various types of differential and integral equations with many applications. We include an example in the complements section. An important consequence of Corollary 6 is that the quadratic variation of X, denoted [X, X](·) or [X](·) is a positive definite form, and hence by the polarization identity one can obtain a (non negative or positive) definite bilinear form for a pair of processes X, Y satisfying the above hypothesis as: 1 (25) [X, Y ] = ([X + Y, X + Y ] − [X − Y, X − Y ]) , 4 (also called quadratic covariation sometimes) since X, Y satisfying the same hypothesis implies that of X ± Y and so the corollary applies. [If X and Y are complex processes satisfying the above hypotheses, then X ± iY also obeys the same conditions. So Corollary 6 applies and then [X], [Y ], [X ± iY ] exist and (25) is given by the following equation:

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6 Martingale Type Measures and Their Integrals

[X, Y ] =

1 [X + Y, X + Y ] − [X − Y, X − Y ] + i[X + iY, X + iY ] 4 −i[X − iY, X − iY ]

and thus the bilinear form is obtained in both cases.] The existence of these can also be shown by repeating the proofs of the theorem and corollary with direct but somewhat longer computations. It follows that the quadratic (co) variations have finite (and σ-finite on R) Vitali variations so that they determine dominating measures and then the L2,2 -boundedness principle applies to such martingale classes. To appreciate the generality of Theorem 5, let us introduce the concept of stochastic integrability abstractly as in the following: Definition 6.2.7 Let X = {Xt , Gt , t ∈ I ⊂ R} be an adapted c` adl` ag process and Xt ∈ Lφ (P ) where φ : R → R+ is a monotone function such that φ(x) = φ(−x) and φ(x) = 0 if and only if x = 0, which thus includes φ(x) = |x|p , 0 0 : k Ω [Such a φ is called a generalized Young function and simply a Young function if φ is moreover convex.] Then as in the classical Lp (P ), the (Lφ (P ), k · kφ ) are complete metric spaces when functions differing only on sets of P -measure zero are identified. Let P be a predictable σ-algebra of Ω ′ = I × Ω and let S(Ω ′ , P) be the set of R simple functions, relative to the filtration {Gt , t ∈ I}, and define τ (f ) = I f dX for simple functions f ∈ S(Ω ′ , P). Then X is called a stochastic integrator on S(Ω ′ , P) if the following pair of conditions hold: (i) The set {τ (f ) : f ∈ S(Ω ′ , P), |kf |k = supt {kf (t, ·)k∞ ≤ 1}} ⊂ Lφ (P ) is bounded in the metric of the last space, and (ii) fn ∈ S(Ω ′ , P) : |fn (t, ω)| ↓ 0 a.a.(ω), t ∈ I ⇒ limn τ (fn ) = 0 in P measure. This concept incorporates the conclusion of Theorem 5 above, and the dominated convergence property is included in (ii). Moreover if φ(x) = |x|2 so that Lφ (P ) = L2 (P ), then by the same theorem one concludes that a c` adl` ag square integrable semimartingale is a stochastic integrator. An essential converse to the above statement also holds. Namely a stochastic integrator of Definition 7, is derived from a “local” semi-martingale, and the word “local” can be dropped if the X-process is c` adl` ag and φ(x) = |x|2 . The word “local” signifies that the process X is integrable on bounded ‘time sets’. A precise definition depends on the new idea of “stopping times”, which can be used to extend the subject to a much greater extent. In the next section we shall introduce the concept together with a few properties (of stopping times)

6.2 Random Measures and Stochastic Integrands

235

in presenting another view of stochastic integrals and show again that it is a functional operation although obtained quite differently. In (15) above we have defined a (generalized) L2,2 -boundedness of a process X = {Xt , Gt , t ∈ I ⊂ R} ⊂ L2 (P ) relative to a measure α on P of Ω ′ = I × Ω and a constant k > 0, considered as an extension, L2,p -boundedness in Proposition 3. It is stated in the following more general form for reference and application. Definition 6.2.8 Let φ1 , φ2 be a pair of generalized Young functions on R and X = {Xt , Gt , t ∈ I ⊂ R} be a process such that X : I × Ω = Ω ′ → R is jointly measurable relative to B(I) ⊗ Σ and Xt ∈ Lφ2 (P ), t ∈ I. Let P ⊂ B(I) ⊗ Σ be the predictable σ-algebra. Then X is termed (generalized) Lφ1 ,φ2 ¯ + and bounded relative to φ1 , φ2 , P if there exists a σ-finite measure α : P → R R ′ a constant K(= Kφ1 ,φ2 > 0) such that τ f = I f (t) dXt , f ∈ S(Ω , P)-simple, and Z Z E(φ2 (|τ f |)) = φ2 (τ f |) dP ≤ K φ1 (|f |) dα. (26) Ω

Ω′

Note that if φ1 (x) = φ2 (x) = |x|2 , then this is just L2,2 -boundedness of the earlier definition. Both the preceding concepts are related in the following way: Theorem 6.2.9 Let X = {Xt , Gt , t ∈ I ⊂ R} ⊂ Lφ2 (P ) where {Gt , t ∈ I} is a filtration satisfying the standard hypothesis and Xt is Gt -adapted (hence also to σ(Xs , s ≤ t) ⊂ Gt ). If X is (generalized) Lφ1 ,φ2 -bounded relative to some ¯+ (generalized) Young function φ1 and σ-finite measure α : P(⊂ B(I) ⊗ Σ) → R on S(Ω ′ , P) then X is a stochastic integrator in the sense of Definition 7 and ′ ¯ , P), k · kφ1 } the integral τ : S(Ω ′ , P) → Lφ2 (P ) extends to M φ1 (α) = sp{S(Ω with values in Lφ2 (P ) and for the extended integral, the dominated convergence criterion (classical form) holds. In the converse direction, let (Ω, Σ, P ) be separable, φ2 have a moderate growth in the sense that φ(2x) ≤ Cφ(x), x > 0 for a real C > 0, and X be a stochastic integrator as in Definition 7. Then there exists a convex (Young) ¯ + relafunction φ1 such that φ1x(x) ↑ ∞ as x ↑ ∞, a σ-finite measure α : P → R φ1 ,φ2 -bounded so that (26) holds. The integral tive to which X is (generalized) L τ (f ), f ∈ S(Ω ′ , P), extends to M φ1 (α) for which the dominated convergence criterion (as in the direct part) holds. The proof depends on some properties of the (linear) metric spaces Lφ1 (P ) and Lφ1 (α), and the direct part which is of interest in our treatment follows more or less easily. The converse part uses all the restrictive conditions which imply that Lφ2 (P ) is separable, weakly complete, and τ extended to the space of bounded P-measurable functions, τ : B(Ω ′ ) → Lφ2 (P ); and it can be given an integral representation relative to a weakly σ-additive vector measure Z : P → Lφ2 (P ) by a suitable Riesz type representation, which then coincides with (or induces) the X-process. A further analysis shows that the vector measure Z-has a controlling measure α0 and a finite semi-variation

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6 Martingale Type Measures and Their Integrals

relative to the space Lφ1 (α0 ). Here φ1 is as in the statement and then the measure α(= α0 ⊗ P ) is found to satisfy the given conditions. The details are several using the works of functional analysis (Orlicz spaces) and will not be included. A somewhat detailed sketch is given in my book (Rao (1995), Section 6.2). The important point here is that Bochner’s boundedness principle (slightly extended) and stochastic integrator as introduced in Definition 7 are closely related, and in fact are essentially equivalent. The flexibility of the standard filtration used here is such that {Gt , t ∈ I} satisfies Gt ⊂ Σ, and also that Ft = σ(Xs , s ≤ t) ↑⊂ Gt . The possibility that the inclusion may be proper is useful and can be employed in extending the classical Brownian integrator to some “anticipating” integrands as already observed by Itˆ o (1978). Proposition 4 shows that all c` adl` ag semi-martingales in L2 (P ) are stochastic integrators, and any process of finite variation (and left or right) continuous on compact intervals is an integrator as it defines a Lebesgue-Stieltjes measure. Thus the sum of these two processes which form a semi-martingale (cf. (23)) qualifies to be a stochastic integrator by Theorem 5. What other classes of processes are eligible for this special treatment? In 1980–81, C. Dellacherie and K. Bichteler independently showed that under some conditions that such “semi-martingales” and a few of their direct modifications essentially exhaust all stochastic integrators. We state a form of their result for comparison and reference. 0 Theorem 6.2.10 Let X = {Xt , Gt , t ∈ I} be a c` adl` ag process in R L (P ). It is ′ then a stochastic integrator relative to S(Ω , P) so that τ : f 7→ I f (t) dXt for f ∈ S(Ω ′ , P), is a continuous mapping, as in Definition 7, if and only if X is a “semi-martingale” (to be discussed later after Definition 2 in Section 4 below) hence Xt = X0 + Yt + At , Yt is a “local” martingale and At is of (locally) finite variation on I.

6.3 Random Measures, Stopping Times and Stochastic Integration It was seen in the proof of Proposition 2.4 above that the argument uses both its topological and order structures of the interval I ⊂ R and the abstract measure properties of (Ω, Σ, P ). This point of view distinguishes the classical Lebesgue-Stieltjes approach compared to that of the stochastic one in the subject. To simplify and consider the important use of order in such an analysis we turn to a new element, called stopping times which helps to streamline the predictable σ-algebras, so important in the current work on integration. We also can generalize and conceptually somewhat simplify the work with ‘random integrands’ as well as solve several new problems. For all of this we need to introduce the new concept as follows.

6.3 Random Measures, Stopping Times and

Stochastic Integration

237

Definition 6.3.1 Let {Ft ↑, t ≥ 0} be the standard filtration in the complete probability space (Ω, Σ, P ). The mapping T : Ω → R+ is termed a stopping time of the filtration if {ω : T (ω) ≤ t} ∈ Ft for all t ≥ 0. [For the general definition, only Ft ↑ is necessary and the filtration need not be standard in the sense of its ‘right continuity’.] Now clearly the constant times are stopping times. If {Tt , t ∈ J ⊂ R} is a family of stopping times of the given filtration {Gr , r ∈ I}, then the family is called a stopping time process if for t1 , t2 ∈ I, t1 ≤ t2 ⇒ Tt1 (ω) ≤ Tt2 (ω), ω ∈ Ω, and Yr (ω) = YTr (ω) (ω) defines Yr as a random variable if X(·) (·) is B(I) ⊗ Σmeasurable where Tr : Ω → I, for each r ∈ J. Thus {Yr , r ∈ J} is a new process, also called a superposition of X and T , when their ranges and domains match. Note that although T1 , T2 are stopping times, of {Gt , t ∈ I}, implies that T1 + T2 , min(T1 , T2 ), max(T1 , T2 ), are stopping times of the filtrations but in general their linear combinations are not necessarily so and care is needed in dealing with their calculus. Stopping times appear in applications of the following very common and nontrivial problems. An adapted process X = {Xt , Gt , t ≥ 0} is progressively measurable when X : [0, t] × Ω → R is B([0, t]) ⊗ Gt -measurable for each t ≥ 0. If {Gt , t ≥ 0} is a standard filtration in (Ω, Σ, P ) and {Xt , Gt , t ≥ 0} is a progressively measurable process then for each Borel set B ∈ B(R), the hitting time of B by the process is defined as DB (ω) = inf{t ≥ 0 : Xt (ω) ∈ B}, ω ∈ Ω, (1)

also called “the debut of B”, is a stopping time as in Definition 1. However the proof is nontrivial and it depends on Choquet’s capacity theory! This fact suggests that one may be able to develop the whole theory without using the concept of stopping times but only using Real Analysis (and its extension to the complex case) techniques since these are essentially a form of truncation. In fact L.C. Young (1970) has obtained several results of stochastic integration theory in the context of Riemann-Stieltjes formulation. A consequence of this method is to forego much of the motivation for the subject and, what is worse, the contact and inspiration with new real applications and results will be missing. For this reason we shall try to keep the “random part” of the subject in clear view and remember the stochastic motivation on the anvil all the time. In the middle, there is another method developed by E.J. McShane which we consider later. It also does not depend on the stopping time calculus, and yields somewhat less general results. ¯+ Let {Ft , t ≥ 0} be a standard filtration from (Ω, Σ, P ) and T : Ω → R be a stopping time of this filtration. Then the collection FT ⊂ Σ defined as ¯ +} FT = {A ⊂ Ω : A ∩ [T ≤ t] ∈ Ft , ∀t ∈ R (2)

is seen to be a σ-subalgebra of Σ and the ‘time’ T is measurable rela¯ + is replaced by R+ , so that the tive to FT . It should be noted that if R set has no last element, then FT is still a σ-algebra but may not be contained in Σ. To include all the cases and obviate other problems we can

238

6 Martingale Type Measures and Their Integrals

take A ∈ Σ in (2). If T is replaced by {Tj , j ∈ J} to be a stopping time process, J being a partially ordered index set, {Ftj , j ∈ J} will be a filtration and an adapted process {Xt , Ft , t ≥ 0} is transformed into a new (“superposed”) process Yj = XTj where Yj (ω) = XTj (ω) (ω) = X(Tj (ω), ω), ω ∈ Ω, and {Yj , FTj , j ∈ J} is an adapted (new) process for the (per˜ j = FT . Thus a new technology arises haps a not standard) filtration F j here, and this will be briefly considered, because it is frequently employed below. We observed just prior to (20) in Section 2 above that the square integrable martingale processes generalize the Brownian motion, and were characterized by P. L´evy. The precise result can be stated as follows for a convenient reference. Theorem 6.3.2 (P. L´evy) Let X = {Xt , Ft , t ≥ 0} be a square integrable, continuous (in t), martingale (so for 0 ≤ s < t, E F (Xt ) = Xs a.e.) adapted to a standard filtration {Ft , t ≥ 0}. If for some stopping time sequence Tn ↑, (n) of the filtration such that Xt = X(Tn ∧ t), t ≥ 0, is also a square integrable martingale relative to {Ft , t ≥ 0} for n = 1, 2, . . . , and the quadratic variation of X is given as hXit = hX, Xit = t, t ≥ 0, then the process X is simply a Brownian motion. This result gives some “reasons” why the stochastic integral determined by X generalized the Itˆ o-Wiener integral, and opened up a chapter on integration in this context and the L2,2 -boundedness as well as various extensions that followed. The process of the theorem is usually termed a local martingale. Proof of this result depends on Itˆ o’s version of the integration by parts formula which gets an extra term to the familiar calculus result. A short demonstration using Fourier transforms can be found, for instance, in the author’s book (cf. Rao (1995), p.403), and will be omitted here. The result has been extended independently and simultaneously by Dambis (1965), and Dubins and Schwartz (1965), even for the vector case. [This is also included in the above reference of the author as Theorem 5.3.23 and the interested reader may find the details there.] This and (vector) Dellacherie-Bichteler theorem discussed in the last section show how the martingale theory extends Brownian (ItˆoWiener) integrals. It can even be extended to (semi-) martingales using the L2,2 -boundedness concept. We now establish another approach to this analysis, namely that this integration is also related to the spectral representation of certain continuous linear operators in a Hilbert space, essentially following Cuculescu (1970). The aim is to emphasize its basic functional analytic character. Here the quadratic (co-) variation (cf. (25) of the preceding section and Corollary 2.6), having finite Vitali variation on compact intervals, plays a key role by associating scalar measures in the analysis and to invoke some classical abstract analysis results in illuminating the structure of the stochastic analog of the work which was established earlier.

6.3 Random Measures, Stopping Times and

Stochastic Integration

239

First we recall a classical functional analysis result which will be identified in our treatment. Thus let A be a complex bounded function algebra with identity ‘e’ on a measurable pair (S, S), which is complete relative to the uniform norm k · ku , and let H be a Hilbert space. For each f ∈ A, consider a linear multiplicative mapping τ : A → B(H), the algebra of bounded linear mappings in H, such that τ : f 7→ τf is a homomorphism into B(H), i.e., τf ∈ B(H), f ∈ A, and suppose this satisfies (τf )∗ = τf ∗ , τe =identity, τf g = τf τg and kτf kB(H) ≤ kf ku . If Ff (x, y) = (τf x, y), x, y ∈ H and (·, ·) as its inner product, then F(·) (x, y) : A → C is a continuous linear functional for each x and y for which we can apply a Riesz representation theorem to find a unique bounded additive set function µx,y on the σ-algebra S of S to get Z f (s) dµx,y (s), f ∈ A, (3) Ff (x, y) = (τf x, y) = S

such that |µx,y |(S) = kF(·) (x, y)k ≤ kxkkyk where µx,y (·) is linear in x and conjugate linear in y. In general µx,y (·) is just additive and if τf is a positive operator, so that (τf x, x) ≥ 0, then µx,x ≥ 0 also. The integral in (3) for finitely additive measures is defined in the usual manner (cf., Dunford-Schwartz (1958), Section III.2). On the other hand let τf be defined as in (3) by the integral for an additive set function µx,y with µx,x ≥ 0. Then τf satisfies (f ∗∗ = f) Z (τf ∗ f x, y) = (τf∗ τf x, y) = (τf x, τf y) =

f ∗ f dµx,y

(4)

S

R so that kτf xk2 = S |f |2 dµx,x and f 7→ τf is an isometric isomorphism of A into L2 (S, S, µx,x ), x ∈ H. From (3) and (4) above, we conclude that τf and τf ∗ commute and that τf ∗ = (τf )∗ . For each µx,y , that such a τf exists is established in many books on the subject (cf., e.g., Loomis (1953), p.93). We now show that this “representation” of A on a Hilbert space gives our integrator by a suitable specialization. To utilize stopping times efficiently, we consider the following properties which are easy consequences of the definition although not entirely trivial. If T1 , T2 are stopping times of a filtration {Ft , t ≥ 0} the stochastic interval [T1 , T2 ] is defined on Ω ′ = R+ × Ω, often denoted as, [[T1 , T2 ] = {(t, ω) ∈ Ω ′ : T1 (ω) ≤ t ≤ T2 (ω)}.

(5)

¯ + in such a context. Thus if T2 (ω) = ∞ for an ω, then We always use R+ not R we take only those ω for which T2 (ω) < ∞. The graph of a stopping time T1 is denoted [[T1 ] and is the set {(t, ω) ∈ Ω ′ : T1 (ω) = t}. In general the graph is an element of the optional algebra O. We have the four types of the intervals [[T1 , T2 ], ((T1 , T2 ], [[T1 , T2 ), ((T1 , T2 ) where the end points are sometimes included and sometimes not. A stopping time is called predictable if there is a sequence of times Tn such that 0 ≤ Tn ≤ Tn+1 ≤ T, Tn (ω) < T (ω) on [ω : T (ω) > 0], and Tn (ω) ↑ T (ω) for ω ∈ Ω as n → ∞. The following is of interest to our work.

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6 Martingale Type Measures and Their Integrals

Proposition 6.3.3 If T1 , T2 are stopping times of a filtration {Ft , t ≥ 0} from (Ω, Σ, P ), then the intervals [[0, T1 ], ((T1 , T2 ] are predictable and the same is true of [[T1 , T2 ] if T1 is predictable. If both T1 , T2 are predictable then ((T1 , T2 ) is also predictable. Further the predictable σ-algebra P ⊂ B(R+ ) ⊗ Σ is generated by the set of all stochastic intervals of the form ((T1 , T2 ] or of all the intervals of the form [[T1 , ∞) where T1 is predictable. Sketch of Proof. If R is the semi-ring formed of sets (s, t] × A, A ∈ Fs and since every constant s ∈ R+ is a stopping time so that if T1 = s on A and = t on Ac , T2 (ω) = t, ω ∈ Ω, we see that (s, t] × A is representable as (T1 , T2 ] and if R′ is the semi-ring of the stochastic intervals of the form [[T1 , T2 ], then R ⊂ R′ . The opposite inclusion is obtained by approximation as follows. If α > 0 and S 0 < tj < tj+1 , so nj=1 Aj is a partition of Ω with Aj ∈ Ftj of the filtration, let P T1n = nj=1 χAj tj . Then Tj is a stopping time, and A˜j = (tj , α] × Aj , α > T1n ⇒ ∪nj=1 A˜j = ((T1n , α), for α > Tjn , is a decomposition of this interval, A˜j ∈ R′ . Since ((T1 , T2 ] = ((T2 , α] − ((T1 , α], it follows that ((T1 , T2 ] ∈ R′ for α ≥ T2 and so R′ ⊂ R and the semi-rings coincide so that the σalgebras they generate are identical, i.e., P is generated by both R and R′ . 2 Recall that for B = (s, t] × A ∈ R, if {Xt , Ft , t ≥ 0} is a Brownian motion, then with Ω ′ = R+ × Ω, (and A ∈ Fs ) we have by definition: Z Z χ(s,t] χA dXt = χA (Xt − Xs ). χB dXt = YB = R+

R+

Since Xt − Xs is independent of Fs , one has (with σ 2 = 1 for BM): E(YB ) = E(χA (Xt − Xs )) = P (A)E(Xt − Xs ) = 0,

E(YB2 ) = E(χA (Xt − Xs )2 ) = P (A) · E(Xt − Xs )2 = P (A)(t − s) = (P × µ)((s, t] × A) = µ((s, t])P (A) = µX (B),

(6) (7)

where µ is the Lebesgue measure and µX (·) is a (product) measure on R which has a unique σ-additive extension to P. This fact is to be generalized to processes X = {Xt , Ft , t ≥ 0} that have properties enabling us to get a σ-additive measure µX on P. We R then associate a random measure Z(·) induced by X, and the integral R+ Y (t)dZ(t) is defined for a wide class of processes Y which are (predictable or) P-measurable and Z(·)-integrable, through the scalar measure µX , i.e., µX : P → R induced by X or Z. We now show that this can be done if X is a square integrable martingale which inherits many of the properties of a B.M., and yet generalizes the Itˆ o-extension of the Wiener integral for the predictable integrand processes. It will be executed using stopping time techniques and the ideas of abstract analysis indicated in (3) and (4) above. This shows that the extension is substantial connecting it with other well-known abstract areas while presenting a new perspective of our subject. In fact, if X is also a martingale, then

6.3 Random Measures, Stopping Times and

Stochastic Integration

241

(6) is automatic. If further X is square integrable, then (7) becomes on using the conditioning identity E(V ) = E(E B (V )), for any integrable random variable V and σ-subalgebra B ⊂ Σ, with B = Fs , we get for B above: E(YB2 ) = E(χA (Xt − Xs )2 ) = E(E B (χA (Xt − Xs )2 )) = E(χA E Fs (Xt2 − 2Xt Xs + Xs2 ))

= E(χA (E Fs (Xt2 ) − 2Xs E Fs (Xt ) + Xs2 )) = E(χA E Fs (Xt2 ) − E Fs (Xs2 )) (8) = E(E Fs (χA (Xt2 − Xs2 )) = E(χA (Xt2 − Xs2 )) ≥ 0. Here we used the fact that for Xt , a square integrable martingale E Fs (Xt2 ) ≥ Xs2 (Jensen’s inequality) so the squared martingale is a submartingale. Then (7) is replaced by (8) and if the right side of (8) is set as µX 2 ((s, t] × A), then µX 2 (·) is actually a σ-additive positive measure on P by Proposition 2.4. With this background (and confidence) we now show that it leads to using (4) and its abstraction in obtaining a generalization of the Itˆ o integral for martingales. For this purpose we need to derive two commutation relations of conditional expectations determined by the particular σ-algebra generated by a pair of stopping times T1 , T2 . Proposition 6.3.4 Let T1 , T2 be stopping times of a standard filtration of (Ω, Σ, P ), and let FT1 , FT2 be the σ-subalgebras of Σ determined by T1 , T2 as in (2), also written as F(Ti ) for convenience. Let B1 , B2 be defined as: B1 = {A1 ∪ A2 : Ai ∈ F(Ti ), A1 ⊂ [ω ∈ Ω : T1 (ω) ≤ T2 (ω)],

A2 ⊂ [ω ∈ Ω : T2 (ω) < T1 (ω)]} B2 = {A1 ∪ A2 : Ai ∈ F(Ti ), A1 ⊂ [ω ∈ Ω : T2 (ω) < T1 (ω)], A2 ⊂ [ω ∈ Ω : T1 (ω) ≤ T2 (ω)]}.

Then Bi ⊂ Σ are σ-algebras and the following relations between their conditional expectation operators E F(Ti ) and E Bi , i = 1, 2 hold: E F(T1 ) (E F(T2 ) (f )) = E F(T2 ) (E F(T1 ) (f )) = E B1 (f ), f ∈ L1 (P ), a.e.,

(9)

and further E F(T1 ) (f ) + E F(T2 ) (f ) = E B1 (f ) + E B2 (f ), a.e., f ∈ L1 (P ).

(10)

[Here B1 = F(T1 ∧ T2 ), B2 = F(T1 ∨ T ), for the ‘min’ and ‘max’ of times Ti .] Proof. Recall that if V ∈ L1 (P ), G ⊂ Σ any σ-subalgebra, then by the RadonR Nikod´ ym theorem for the scalar measure ν : A 7→ A V dP , we have Z Z νV (A) = V dP = V ′ dPG , PG = P |G, A

A

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6 Martingale Type Measures and Their Integrals

dνV , PB -uniquely. We write E G : V → V ′ and see that where V ′ = dP B E G : L1 (Ω, Σ, P ) → L1 (Ω, G, PB ) is a positive linear contraction operator, which is the conditional expectation, E G |L1 (Ω, G, PB ) = identity. Moreover, if G1 ⊂ G2 ⊂ Σ are σ-algebras, then E G1 E G2 = E G2 E G1 = E G1 which fails if G1 6⊂ G2 . It follows that E G is a positive contractive projection. Several properties of this operation are in standard books (cf., e.g., Lo´eve (1963), Ch.7) and a more detailed analysis in the author’s book (Rao (2005), 2nd ed.). These results will be used in the proof here as well as later. The fact that B1 , B2 are σ-algebras will be left to the reader which is a standard manipulation. Using these we deduce that B1 and B2 are given by B1 = F(T1 ) ∩ F(T2 ) and B2 = σ(F(T1 ) ∪ F(T2 )). With these relations (9) and (10) are verified as follows. Since A1 ∪ A2 is a disjoint union and is a generator of B1 , we have: Z Z f dP, f ∈ L1 (P ) E B1 (f ) dPB1 = A1 ∪A2 Z ZA1 ∪A2 f dP f dP + = A2

A1

=

Z

E F(T1 ) (f ) dPF(T1 ) +

A1

=

Z

Z

A2

E F(T1 ) (f ) dPF(T1 )

E F(T2 ) (E F(T1 ) (f )) dPB1

A1

=

+ Z

Z

E F(T2 ) (E F(T1 ) (f1 )) dPB1

A2

E F(T2 ) (E F(T1 ) (f )) dPB1

(11)

E F(T1 ) (E F(T2 ) (f )) dPB1 , by symmetry,

(12)

A1 ∪A2

=

Z

A1 ∪A2

and this shows that the integrands in (11) and (12) can be identified with the integrand on the left side, a.e. [PB1 ], establishing (9). Next (10) is obtained using (9) and the facts that Qi (·) = E F(Ti ) (·), i = 1, 2 are commuting (contractive) projection operators on Lp (P ), p ≥ 1. Thus consider Q = Q1 + Q2 − Q1 Q2 . Then Q1 − Q1 Q2 and Q2 are orthogonal projections on L2 (P ), and Q is also a projection which is again orthogonal (hence a contraction there). On L2 (P ), let RQ (RQi , i = 1, 2) be the range of Q (and Qi , i = 1, 2). Then the classical functional analysis implies: ¯ 2 (F(T1 ), P ) ∪ L2 (F(T2 ), P2 )) = L2 (B2 ) RQ = sp(R ¯ Q1 ∪ RQ2 ) = sp(L

(13)

by the characterization of contractive projections with such ranges (cf., e.g., Rao (2005), p.241), implying that Q = E B2 . But by (9) Q1 Q2 = E B1 . Hence

6.3 Random Measures, Stopping Times and

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243

Q1 + Q2 = Q + Q1 Q2 which is (10) on L2 (P ), and then extend to L1 (P ). 2 We now proceed to use the representations outlined in (3) and (4) to derive stochastic integration of a random measure obtained from a square integrable martingale and an integrand that is adapted to a predictable σ-algebra. This uses the preceding analysis with H = L2 (P ). Here we recall the fact that an L2 (P ) bounded class of functions {Xt , t ∈ I} ⊂ L2 (P ) is uniformly integrable since by the CBS-inequality Z Z χA |Xt |dP ≤ P (A)kXt k2 ≤ KP (A), (14) Xt dP ≤ Ω

A

which tends to zero uniformly in t and since kXt k2 ≤ K all t ∈ I by our supposition. This is of course also true if L2 (P ) is replaced by Lp (P ), 1 < p < ∞. If {Xt , Ft , t ∈ I} is moreover a c` adl` ag martingale and it is uniformly integrable, then a basic result in martingale theory states that there is a (unique) random variable X ∈ L2 (P ) such that Xt → X a.e. and in L2 (P ) as t → ∞. Moreover, Xt = E Ft (X), a.e. [This is true for all c` adl` ag uniformly integrable martingales.] The most important consequence is that every S such L2(P )-bounded martingale can be represented by a function 2 X ∈ L (σ t≥0 Ft , P ). The result from functional analysis noted for (3) and (4) above can be established in our special but important case as follows: Theorem 6.3.5 Let Ω ′ = R+ × Ω and P be the predictable σ-algebra on Ω ′ relative to a complete probability space (Ω, Σ, P ) and a standard filtration {Ft , t ≥ 0} with F0 as the trivial field equivalent to {∅, Ω}. If A = B(Ω ′ , P) is the algebra of bounded complex P-measurable functions with uniform norm, and H = L2 (P ), then the mapping I : A → B(H), the space of bounded linear operators on H, defined for step functions g = χA ∈ A, A = [[0, T ) a generator of P (and extend linearly for all A later) as Ig ∈ B(H), Ig X = E B(T ) (X), where X ∈ H, represents a martingale: {Xt = E Ft (X), Ft , t ≥ 0}. Further there is a scalar measure µX,Y : P → C such that Z f (t, ω) dµX,Y (t, ω), f ∈ A(Ω ′ , P), (15) [If X, Y ] = Ω′

for all X, Y ∈ A. In fact µX,Y (dt, dω) = hX, Y i(dt) × P (dω), where [X, Y ](·) is the quadratic covariation of the martingales determined by X and Y relative to the (common) standard filtration {Ft , t ≥ 0}, so that {E Ft (If X) = (If X)t , Ft , t ≥ 0}, ∀f ∈ A is a martingale in H. [See Cor. 2.6 above on [·, ·].] Remark 6.3 Following the representation (15), we denote (If X)t as: (If X)t =

Z

0

t

f (s, ·) dXs , Xs = E Fs (X),

t ≥ 0, X ∈ L2 (P ),

(16)

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6 Martingale Type Measures and Their Integrals

and the proof of the theorem shows that it is the desired stochastic integral of the bounded predictable f relative to the martingale {Xt , Ft , t ≥ 0}. We include a derivation of (16), after establishing the theorem and (15). In abstract analysis, the measure µX,Y is usually termed a spectral measure. Proof. Since {Ft , t ≥ 0} is a standard filtration and X ∈ L2 (P ) as well as F∞ = σ(∪t≥0 Ft )-measurable, (otherwise we take X = E F∞ (X ′ ) for X ′ ∈ adl` ag martingale so L2 (P )), {Xt = E Ft (X), t ≥ 0} is uniformly integrable c` that limt→∞ Xt = X a.e. and in L1 (P ). The work on martingale theory implies E F(T ) (X) = XT a.e. for any stopping time T of the filtrations and for predictable T , we even have T replaced by (T −) since by predictability, on 0 < T < ∞, there is a sequence Tn ↑ T of stopping times Tn of the filtration such that {XTn , F(Tn ), n ≥ 1} is a martingale, XTn = E F(Tn ) (X), and E F(T −) (XT ) = XT − a.e., as well as E F(T ) (XT − XT − ) = 0, and so for such T , we have XT = XT − a.e. The necessary results used are from the classical martingale theory. Here we may and do assume these facts. Taking A = [[0, T ), a generator of P, define µX,X as, µX,X (∅) = 0 and using the inner product of H set: µX,X (A) = (E F(T ) (X), X) = (E F(T ) (X), E F(T ) (X)) Z |XT |2 dP, = (XT , XT ) =

(17)

Ω

where we used the result that E F(T ) (·) is a self adjoint projection on H. Also [[0, T1 ) ∩ [[0, T2 ) = [[0, T1 ∧ T2 ), and similarly [[0, T1 ) ∪ [[0, T2 ) = [[0, T1 ∨ T2 ) for stopping times T1 , T2 of the filtration. A similar relation holds for a sequence {Tn , n ≥ 0} and the collection of all such sets of stochastic intervals forms a σ-lattice, say S0 which thus is closed under countable unions, intersections, having ∅ and Ω ′ . Moreover σ(S0 ) = P. By (17), µX,X : S0 → R+ is defined and µX,X (Ω ′ ) < ∞. By a simple normalization we take, for simplicity, that µX,X (Ω ′ ) = 1. Thus (i) µX,X (A) ≤ µX,X (B) for A ⊂ B, (ii) µX,X (·) of (17) is strongly additive in the sense that µX,X (C ∪ D) − µX,X (C ∩ D) = µX,X (C) + µX,X (D) for C, D in S0 , and (iii) An ↑ A in S0 ⇒ µX,X (An ) ↑ µX,X (A). Such a µX,X (·) is called a “modular” set function (since S0 is a σ-lattice), and by a classical theorem of B.J.Pettis and some of its known refinements, it admits a unique σ-additive extension to P. Since this is not directly given in books (c.f., Rao (2004), Theorem 10.2.8 for a basic extension) we include details in a series of steps here, which is somewhat different (and hopefully simpler than Cuculescu’s original argument that tied closely to probability theory). I. Since {XTi , F(Ti ), i ≥ 1} is a martingale so that {XT2i , F(Ti ), i ≥ 1} is a submartingale it follows that for A = [[0, T1 ), B = [[0, T2 ) where T1 ≤ T2 , µX,X (A) ≤ µX,X (B). Similarly Ti ↑ T implies µX,X (Ai ) ↑ µX,X (A) where Ai = ((0, Ti )

6.3 Random Measures, Stopping Times and

Stochastic Integration

245

and A = ((0, T ). Thus (i) and (iii) hold, and we need to show the nonobvious property of strong additivity (also called modularity) to invoke Pettis’s theorem. For any stopping times T1 , T2 of the filtration {Ft , t ≥ 0} we have seen in Proposition 4 above (cf. relation (10)) that XT1 + XT2 = XT1 ∨T2 + XT1 ∧T2 and then with the martingale property, XTi = (XTi − XT1 ∧T2 ) + XT1 ∧T2 is an orthogonal decomposition for i = 1, 2. Hence E F(T1 ∧T2 ) (XTi ) = XT1 ∧T2 , a.e.,

(18)

and using the properties of martingale increments we have E[(XT2 − XT1 ∧T2 )(XT1 − XT1 ∧T2 )]

= E(E F(T1 ∧T2 ) (XT2 − XT1 ∧T2 )(XT1 − XT1 ∧T2 )) = 0.

(19)

With these relations, we have for Ai = ((0, Ti ], i = 1, 2, and (17), µX,X (A1 ∪ A2 ) − µX,X (A1 ∩ A2 ) Z Z |XT1 ∧T2 |2 dP |XT1 ∨T2 |2 dP − = Ω ZΩ 2 [XT1 ∨T2 − XT1 ∧T2 ] dP, by (18) and (19), = Ω Z 2 [(XT1 − XT1 ∧T2 ) + (XT2 − XT1 ∧T2 )] dP, by (10) = ZΩ = ([XT1 − XT1 ∧T2 ]2 + [XT2 − XT1 ∧T2 ]2 )dP, by (18), Z Z ZΩ XT22 dP, by (10) again, XT21 ∧T2 dP + = XT21 dP − 2 Ω

Ω

Ω

= µX,X (A1 ) + µX,X (A2 ) − 2µX,X (A1 ∩ A2 ).

(20)

This shows that µX,X (·) is modular, and (i)-(iii) are valid. II. The underlying ideas of proof, of σ-additive extension of µX,X , are although similar to those of the extended Pettis’s theorem (cf. Rao (2004), Sec.10.2), our result cannot be deduced easily and will sketch the detail because of its importance. We obtain a σ-additive extension of µX,X defined above on P, and also get a useful representation on the way. First define for each F ⊂ Ω ′ , a function νX,X (·) as: νX,X (F ) = inf{µX,X (A) : A ⊃ F, A ∈ S0 }. (21)

Then νX,X |S0 = µX,X and 0 ≤ νX,X (F ) ≤ µX,X (Ω ′ ) = 1. It is asserted that νX,X (·) is subadditive. In fact for Fi ⊂ Ω ′ and ǫ ≥ 0, there exist Ai ⊃ Fi such that νX,X (Fi ) ≥ µX,X (Ai ) − 2ǫ , so that using the strong additivity of µX,X , one finds that

246

6 Martingale Type Measures and Their Integrals

νX,X (F1 ) + νX,X (F2 ) ≥ µX,X (A1 ) + µX,X (A2 ) − ǫ = µX,X (A1 ∪ A2 ) + µX,X (A1 ∩ A2 ) − ǫ ≥ νX,X (F1 ∪ F2 ) + νX,X (F1 ∩ F2 ) − ǫ, by definition of νX,X .

(22)

Since νX,X (Ω ′ ) = µX,X (Ω ′ ) = 1, νX,X (∅) = 0 and F2 = F1c one has νX,X (F1 ) + νX,X (F1c ) ≥ 1.

(23)

From definition of νX,X (·), it is clearly increasing, and let D = {F ⊂ Ω ′ : νX,X (F ) + νX,X (F c ) = 1}. Then ∅, Ω ′ ∈ D and F ∈ D ⇒ F c ∈ D. Proceeding as in Real Analysis, consider for Fi ∈ D, i = 1, 2, by (22) and (23), with F1 , F2 and adding: 2 ≤ νX,X (F1 ∪ F2 ) + νX,X (F1 ∩ F2 ) + νX,X (F1c ∪ F2c ) + νX,X (F1c ∩ F2c ) ≤ 2, (24) so that there is equality and F1 ∪ F2 ∈ D, F1 ∩ F2 ∈ D implying D is an algebra. Now claim that D is closed under monotone limits (increasing suffices) to conclude it is a σ-algebra. For this we need to show that νX,X (·) is strongly additive (or modular), so from the fact that νX,X |S0 = µX,X one can conclude that S0 ⊂ D. Let Fn ⊂ Ω ′ and Fn ↑ F . Then given ǫ > 0, by (21), there exist An ∈ S0 , An ⊃ Fn , such that νX,X (Fn ) > µX,X (An ) − 2ǫn . But An ’s need not increase. So let Bn = ∪ni=1 Ai (ǫS0 by the lattice property). Then Fn (⊂ Bn ) ↑ B, say, and Fn = Fn ∩ Fn+1 (⊂ Bn ∩ An ∈ S0 ). We claim that 1 (25) νX,X (Fn ) ≥ µX,X (Bn ) − ǫ 1 − n , n ≥ 1. 2 This is clear for n = 1, and, using induction consider µX,X (Bn+1 ) = µX,X (Bn ∪ An+1 ) = µX,X (Bn ) + µX,X (An+1 ) − µX,X (Bn ∩ An+1 ), by (20), 1 ǫ ≤ νX,X (Fn ) + ǫ(1 − n ) + νX,X (Fn+1 ) + n+1 2 2 −µX,X (Bn ∩ AN ), by induction hypothesis, 1 ≤ νX,X (Fn ) + ǫ 1 − n+1 , since 2 Bn ∩ An+1 ⊃ Fn ∩ Fn+1 = Fn and µX,X (Bn ) ≤ νX,X (Fn ), 1 ≤ νX,X (Fn+1 ) + ǫ 1 − n+1 . 2 Letting n → ∞ one finds from (21) and (25) that νX,X (F ) ≤ µX,X (A) ≤ lim νX,X (Fn ) + ǫ. n→∞

6.3 Random Measures, Stopping Times and

Stochastic Integration

247

Hence νX,X is σ-subadditive and since it is additive, one concludes its σadditivity, and that from νX,X (Fn ) + νX,X (Fnc )) = 1 and Fn ↑ F ⇒ νX,X (F ) + νX,X (F c ) = 1 ⇒ F ∈ D so that D is a σ-algebra on which ν is σ-additive. III. Since µX,X = νX,X |S0 , it is σ-additive by the above step and so has a unique σ-additive extension to P = σ(S0 ) by the classical Hahn extension theorem. This extension, µ ˜X,X say, is µ ˜X,X = νX,X |P, which is seen as follows. Suppose if possible µ′X,X is another extension to P. Then by definition of νX,X one has for F ∈ P, µX,X (F ) = inf{µX,X (A) : A ∈ S0 , A ⊃ F }

= inf{µ′X,X (A) : A ∈ S0 , A ⊃ F } ≥ µ′X,X (F ),

(26)

by the monotonicity of µ′X,X . But we also have 1=µ ˜X,X (F ) + µ ˜X,X (F c ) ≥ µ′X,X (F ) + µ′X,X ′ (F c ) = µX,X (Ω ′ ) = 1, by (26). Hence there is equality here and in (26) so that µ ˜X,X = µ′X,X . We denote this extension again by µX,X . If gi = χAi , Ai = [[0, Ti ) and Igi X = E F(Ti ) (X) for X ∈ H, then Igi ∈ B(H). Also Ig1 g2 (X) = Ig1 (Ig2 X) by (9). Thus by linearity of I(·) , one can conclude that I : f 7→ If is defined for all f ∈ A and it is a ∗ -representation. Using polarization, we may define for X, Y ∈ H the measure µX,Y (·) as: µX,Y =

1 [µX+Y,X+Y − µX−Y,X−Y + iµX+iY,X+iY − iµX−iY,X−iY ] 4

which is seen at once to be sesqui-linear in (X, Y ). This implies (15). 2 S Derivation of the integral representation (16). Let F0 = σ t≥0 Ft and define ζt : F∞ → R+ as: ζt (F ) = µX,X ([0, t) × F ),

t ≥ 0, F ∈ F∞ .

Then 0 ≤ ζt ≤ µX,X (Ω ′ ) < ∞, ζ0 = 0, ζt (·) is σ-additive, and P -continuous, since if P (F ) = 0, the set [0, t] × F can be identified with [[0, TF ) where TF = 0 a.e. is a stooping time vanishing on F (and = +∞ on F c ). So dζt by the Radon-Nikod´ ym theorem there is a PFt -unique function At = dP Ft and that At ≥ 0 a.e., since ζt (·) ≤ ζt′ (·) for t ≤ t′ . We get At ≤ At′ a.e. If tn ↓ t ≥ 0 then Atn → At a.e. and in L1 (P ). Hence one can define {At , t ≥ 0} to be an increasing and (by a standard modification technique) P-measurable process. This may be expressed (with Fs = (0, s) × F ) as: Z µX,X (Fs ) = χF dAs (ω) dP (ω), F ∈ P. (27) Ω′

248

6 Martingale Type Measures and Their Integrals

Since supt E(At ) = µX,X (Ω ′ ) < ∞, for each stopping time T of the filtration, A∞ (= limt At a.e. and in L1 (P )) is F(T )-adapted, so that {At , t ≥ 0} is uniquely defined and P-measurable. Taking X = Y in (15), and f = χ[[0,T ) for predictable time T it follows from (15) and (27) (if F = ((0, T ) = (0, t) × B ∈ P), f = χF(·) ), that [If X, X] =

Z

Ω′

χF(·) dµX,X =

Z

χF ds As (ω) dP (ω).

(28)

Ω′

It also results by uniqueness that As = µX,X ((0, s) × ·) = [X, X]s , a.a.(ω). Hence, by polarization, we get Z Z t χF (ω) dXs (ω) (29) [IχF X, Y ] = Ω

0

and considering P-measurable simple functions f , this extends to give: Z t f (s, ·) dXs , t ≥ 0, Xs = E Fs (X), a.e. E Ft (If X) = (If X)t = 0

This is our (16) as defined earlier, and it is a martingale integral. The following technical result is established in Steps II and III of the above proof and is of interest to state it separately for applications. We give it as a proposition for a convenient reference, and it was established by Cuculescu (1970) differently. Proposition 6.3.6 Let S be a lattice of subsets of a set S, containing ∅ and S. If µ : S → R+ is a finite strongly additive (or modular) and increasing function then µ is uniquely extendable to an additive measure on the algebra generated by S. If moreover S is a σ-lattice and for An ∈ S, An ↑ A ∈ S, and µ also satisfies µ(An ) ↑ µ(A), (called µ is right continuous) then µ has a unique σ-additive extension onto the σ-algebra generated by S. The same conclusion holds for “σ-finite” µ, i.e., if there is Sn ∈ S, Sn ↑ S and µ(Sn ) < ∞ for all n ≥ 1. The point of considering the above, Hilbert space valued (H = L2 (P ) here) integral is to show the basic structure of this extension and to contrast it with the familiar abstract integrals. Also note that once this is established, several extensions of it are possible. They include processes with such properties only locally although not square integrable globally. This is similar to the Grothendieck bilinear representations, and some others to be discussed later on. Remark 6.4 In the integral (16) we assumed that the integrand f (s, ·) is Pmeasurable so that it is of predictable class. Choosing f (s, ·) appropriately, it is desirable to be able to recognize (If X)t -process. A great deal of work has been done on this question. It can be seen that P is also generated by,

6.3 Random Measures, Stopping Times and

Stochastic Integration

249

not only stochastic intervals of the type ((T1 , T2 ] where T1 , T2 are stopping times of the (standard) filtration {Ft , t ≥ 0}, but is also generated by processes {Xt , Ft , t ≥ 0} which are continuous. Several properties of predictable processes and σ-algebras have been discussed in the book by Dellacherie ((1972) Chapter IV). It is now natural to ask for some familiar classes of P-measurable processes that are integrands of the integral with martingale integrators, and more particularly for analogs of the Fundamental Theorem of Calculus corresponding to Stieltjes integrals in the present case. We shall just indicate one possibility. Recall that the simple form of the stochastic integral relative to a c` adl` ag square integrable martingale X = {X , F , t ≥ 0} is of the form for f = t t Pn -adapted, assumed bounded (0 < t < , I = [t , t ), and f is F f χ 1 i i i+1 i t i I i i i=1 · · · < tn+1 ≤ a < ∞), so that Z a n X (30) f (ti )(Xti+1 − Xti ). (τf X)a Ya = f (t)dXt = 0

i=1

This defines a martingale, since f is a predictable function. The mapping τ : f → L2 (P ) is a predictable transform, and the general case is an extension both of this and of L2,2 -boundedness case. In this form it is sometimes called a predictable transform of the X-process by f of the above type. Thus {Yt , Ft , t > 0} is a martingale, and the integral may be regarded as a “pathwise extension” of the Stieltjes integral. Observe that the “process” {f (t, ·), t ≥ 0} itself can be a c` adl` ag L2 (P )-martingale. Can we describe the Yt process in a more concrete way? For instance do there exist martingales {Yt , Ft , ≥ 0} such that the following equation holds: Z t Yt − 1 = Ys− dXs , t > 0. (31) 0

We shall present it as an exercise with some details proving that there is a unique martingale {Yt , Ft , t ≥ 0} which satisfies this type of equation, implying that many other L2,2 -bounded processes can similarly be treated. In the same way, we may consider the integral Z a Ya = f (s, ·) dXs , f ∈ L2 ([X, X]), (32) 0

where [X, X](= [X]) is the quadratic variation of {Xs , Fs , s ≥ 0} and ask for an analog of the fundamental theorem of calculus. In the non-stochastic case, we know that the integral will be well-defined only when f and X do not have common discontinuities. Thus in this general case, the X-process will be assumed continuous in s for a.a.(ω), (in particular if it is a B.M.) and obtain an analogous result in the stochastic context. This will also be sketched in the exercises section. These questions are meaningful for all L2,2 - (or even Lφ1 ,φ2 -) bounded processes and are of interest to study further. (See also Sec. 7.1 later.)

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6 Martingale Type Measures and Their Integrals

6.4 Generalizations of Martingale Integrals The preceding work covers integrators which are c` adl` ag square integrable martingales {Xt , Ft , t ≥ 0}. An extension to the closely related submartingales is of interest in applications as well as theory, but it is not yet considered. It depends on an interesting decomposition of such a process into a martingale and a special increasing process, first observed by Doob in the easy discrete case and established by Meyer in the difficult continuous parameter case, now known as the Doob-Meyer decomposition. Thus a c` adl` ag uniformly integrable submartingale {Xt , Ft , t ≥ 0} can be uniquely decomposed, when it satisfies a technical condition called “condition (DL)” (detailed in Rao(1995), Sec. 5.2, pp.338ff), as: Xt = Mt + At , t ≥ 0 (1) where {Mt , Ft , t ≥ 0} is a c` adl` ag martingale and {At , Ft , t ≥ 0} (A0 = 0 a.e.) is an increasing predictable integrable R ∞ process. The last condition is also equivalent to the statement that E( 0 Ys− dAs ) = E(Y∞ A∞ ) for each bounded c` adl` ag martingale {Ys , Fs , s ≥ 0}. Several proofs of this decomposition appeared later, improving (or relaxing) the integrability hypothesis. But the decomposition is a useful result. [A recent interesting proof from ‘first principles’ is given in Bass (1996).] An important consequence is that when the submartingale lies in a ball of L2 (P ), then the martingale integration of the last section and the Stieltjes integral with respect to the At -process give us the stochastic integration with respect to (c`adl` ag) submartingales, and the work of the preceding section admits a complete extension. [For details the reader is again referred to, e.g., Rao(1995), Sec. 5.2 as noted above. This result is given here only for a comparison and will not be needed later on.] The analysis of that section along with the Doob-Meyer decomposition essentially implies the following: Theorem 6.4.1 Let X = {Xt , Ft , t ≥ 0} be a c` adl` ag submartingale in L2 (P ) where the filtration is standard and the process is uniformly integrable on compact subintervals of R+ . Then X defines a random measure such that for each bounded predictable f (t, ·) relative to P (the predictable σ-algebra), (If X)t =

Z

0

t

f (s, ·) dXs ,

t>0

(2)

holds and the process X defines a stochastic integrator. This result shows that we may consider, using the standard extension procedures, stochastic integration relative to submartingales for an independent study. The question is whether we can extend the analysis further. To understand this, let us introduce a more inclusive class of processes defined as follows:

6.4 Generalizations of Martingale Integrals

251

Definition 6.4.2 Let X = {Xt , Ft , t ≥ 0} be a c` adl` ag process with a standard filtration {Ft , t ≥ 0}. Then it is called a quasi-martingale if for 0 ≤ a < b, and partition a ≤ t1 < t2 < · · · < tn+1 ≤ b, we have E

n X i=1

n X E(|Xti − E Fti (Xti+1 )|] ≤ K b < ∞ |E Fti (Xti+1 − Xti )| =

(3)

i=1

where the constant K b is independent of the partitions used. The expression (3) may be thought of as a kind of (locally) finite conditional variation of the process. (Observe that if the X-process is B.M. then this is satisfied with K b = 0 for b > 0, and also true more generally for martingales. Indeed X is a semi-martingale if Xt = Yt + Zt , t ≥ 0, where {Yt , Ft , t ≥ 0} is a martingale and {Zt , Ft , t ≥ 0} is an adapted process whose sample paths t → Zt (ω) are of bounded variation on each compact interval. Then for a.a.(ω), the Z is, under mild conditions, the difference of two predictable increasing processes for a standard filtration {Ft , t ≥ 0}. It can be shown that a quasi-martingale is a semimartingale if and only if it is uniformly integrable on each compact subinterval of R+ . These results are part of the general theory of stochastic process, and will not be detailed here. (For references, and complete details, see M´etivier (1982), the author’s book, Rao (1995), Sec. 5.2, or Dellacherie and Meyer, Part B (1982) among others.) A useful result for our purposes here is that every quasi-martingale as defined above is the difference of two nonnegative supermartingales with the same stochastic base termed the standard filtration. This is a kind of Jordan decomposition for quasi-martingales. Since a super martingale by definition is the negative of a submartingale, this decomposition along with Theorem 1 above implies that each (second order) quasi-martingale of a standard filtration which is uniformly integrable on compact intervals of R+ determines an integrator with respect to which each bounded predictable integrand process has a well-defined stochastic integral(not the Lebesgue-Stieltjes type). From this it can be shown that a similar statement holds for semi-martingales as well. On the other hand Theorem 2.10, of Dellacherie and Bichteler, implies that these are essentially the most used results of the subject. We summarize the above discussion and state it for some information and a convenient reference as follows: Theorem 6.4.3 Let X = {Xt , Ft , t ≥ 0} be a second order c` adl` ag process with a standard filtration {Ft , t ≥ 0}. If the process is uniformly integrable on compact intervals of R+ , and is either a sub (or super)-martingale, or more generally a quasi-martingale or a semi-martingale, then it is a stochastic integrator and all bounded P-measurable functions {f (t, ·), t ≥ 0} are integrable relative to X. Moreover X is L2,2 -bounded relative to a σ-finite measure on the predictable σ-algebra P ⊂ B(R+ ) ⊗ Σ. On the other hand a stochastic integrator X = {Xt , Ft , t ≥ 0} based on a standard filtration and c` adl` ag is necessarily a (local) semi-martingale.

252

6 Martingale Type Measures and Their Integrals

While these integrators are L2,2 -bounded, the latter is more inclusive, i.e., there exist integrators which are (generalized) L2,2 -bounded and which integrate some non-predictable (or “anticipating”) integrands. In fact a classical example that is anticipating is the one called Hitsuda-Skorokhod integral. This is defined in terms of series expansion using “Wiener’s chaos” and multiple Wiener-Itˆ o integrals. It is outlined in the author’s book (cf. Rao (1995), pp.531–532) and that it satisfies the Bochner L2,2 -boundedness condition, although not necessarily the complete converse or the full conclusion of the above theorem. In all the preceding analysis, we are concerned with the structural aspect of the theory, and its fine points are discussed using the stopping time techniques. However, for applications of these results in practical problems it is difficult to verify these conditions and no algorithms are in sight. This important question on the implementation for general classes of problems involving stochastic integration, slightly more general than Brownian motion, has been considered by E.J. McShane and he was able to find usable and implementable conditions on a class of processes (with continuous sample paths) including the B.M. as well as many square integrable martingales and processes of (locally) integrable variations. We shall describe this and present a key result. From the fact that (local) semi-martingales are essentially the most general class of stochastic integrators, it follows that the McShane integrators are also included in the class of semi-martingales. The key concept isolating the desired class, but without using stopping times in the formulation, is introduced as follows: Definition 6.4.4 Let X = {Xt , Ft , 0 ≤ t ≤ a}, a ∈ R+ , be an adapted process with values in L2 (P ), {Ft , 0 ≤ t ≤ a} as (standard) filtration. Then: (a) X satisfies a K.δt condition if there is an absolute constant 0 < K < ∞, such that for each 0 ≤ u ≤ v ≤ a one has (for v − u < δ) F E u (Xv − Xu | ≤ K(v − u), E Fv (Xu − Xv )2 ≤ K(v − u), a.e. (4)

(b) X satisfies nearly a K.δt-condition if for each ǫ > 0, there is an adapted process X ǫ = {Xtǫ, Ft , 0 ≤ t ≤ a} such that it is K.δt for each ǫ > 0, and P {ω : Xtǫ (ω) = Xt (ω), 0 ≤ t ≤ a} > 1 − ǫ. (5) The measurability of the set in (5) is implied by the fact that the process is adapted and the other conditions on the filtration are assumed. An important aspect of this definition is that, as stated below, it is not dependent on any stopping time process of the filtration. That is implied by (4) itself. Another important point, often not appreciated in probability contexts, is that the methods of proof used by McShane are the extended Riemann-type integration, generalizing the Henstock-Kurzweil approach applicable to nonabsolute integration which McShane (1969) himself has extended and used here. This allowed him to largely bypass the stopping time technology and replace

6.4 Generalizations of Martingale Integrals

253

the (fixed) ǫ − δ arguments with varying gauges (δ, δ ∗ ) essential for the nonabsolute integration. [Here δ(·) is a nonnegative (set) function and δ ∗ > 0 is a number.] From (4) we deduce immediately the following K(u − v) ≤ E Fu (Xv ) − Xu ≤ K(v − u), a.e., 0 ≤ t ≤ a.

(6)

Let Yt = Xt + Kt, so that (6) implies that {Yt , Ft , 0 ≤ t ≤ a} is a right continuous submartingale (on taking such a version). Also it is uniformly integrable on [0, a], a compact interval. By considering the process and its negative parts one can conclude that a right continuous process satisfying the K.δt condition is a semi-martingale. This conclusion is seen to hold even if K.v is replaced by a monotone increasing right continuous function F (·) on R so that K(v − u) becomes F (v)−F (u) ≥ 0, bounded on compact sets. Then if {f (t), Ft , 0 ≤ t < a} is a predictable process relative to the standard filtration {Ft , t ≥ 0} of (Ω, Σ, P ) and if the second part of (4) is similarly replaced by F (v)− F (u), then it follows that the process {Xt , 0 ≤ t < a} is adopted to this filtration. By our work in Rt Section 3, the integral 0 f (s−)dXs is well-defined for this K.δ or F (t)-condition and satisfies the L2,2 -boundedness relative to the measure dF ⊗ dP = dα(t, ω), Rt R Rt since we now have E( 0 f (s−)dXs )2 ≤ K Ω 0 |f (s−)|2 dF (s)dP (ω). The above considerations can be extended for processes satisfying nearly K.δt-(or in our generalization for nearly δF (·)-) conditions where these existence results are obtained from our stochastic integration analysis of the preceding section. They are actually established directly by McShane without stopping times. The following is the final result thus obtainable: Theorem 6.4.5 Let X = {Xt , Ft , 0 ≤ t ≤ a} be a right continuous process adapted to the standard filtration {Ft , t ≥ 0} satisfying the nearly K.δt (or ∆F (·) with F strictly increasing finite on compact intervals) condition. Suppose that {f (t, ·), 0 ≤ t < a} is a predictable process in L2 (P ), or f ∈ L2 (P, dF dP ). Then for each 0 < α ≤ t ≤ β < ∞, we can find a version of the process defined by Z t

G(t) =

f (s) dXs ,

(7)

α

such that G(·, ω) is continuous or right (left) continuous for each ω at which X(·, ω) has the same property on [α, β]. Moreover, the X-process is L2,2 bounded relative to dF ⊗ dP and some constant which may be absorbed in the measure. As already noted, this important result has been established directly in McShane (1975), without using the machinery that went into our work, and hence is immediately applicable for stochastic calculus. On the other hand, if Y = {Yt , Ft , t ≥ 0} is a bounded predictable process and X = {Xt , Ft , t ≥ 0} is a continuous L2 (P )-valued martingale

254

6 Martingale Type Measures and Their Integrals

then for each a > 0, and 0 = t0 < t1 < · · · < tn ≤ a a partition, denoted πn , of [0, a] with max πn = max1≤i≤n |ti − ti−1 | → 0 as n → ∞, the work of Section 2 above (cf. Proposition 2.4) implies the following limits n−1 n X X (8) (Yti − Yti−1 )(Xti+1 − Xti ) lim Yti (Xti+1 − Xti ), lim πn →0

i=0

πn →0

Rt

i=1

exist in probability. The first one is 0 Ys dXs which is a continuous martingale Rt and the second one becomes 0 d[Y, X]s , the first being the usual (Itˆo)-integral and the second is a pathwise Stieltjes integral. If the processes X, Y are continuous (pathwise), then recall that [Y, X] is written as hY, Xi. These two statements can be combined to get a “new” type of stochastic integral as follows. Now the limits (8) imply the following: 1 X (Yti + Yti+1 )(Xti+1 − Xti ) lim πn →0 2 ti ∈πn X X 1 Yti+1 (Xti − Xti−1 ) + = lim (Yti − Yti−1 )(Xti − Xti−1 ) lim πn →0 2 πn →0 t ∈π ti ∈πn i n Z t Z 1 t = Ys dXs + d[Y, X]s , 2 0 0 Z t 1 (9) = Ys dXs + [Y, X]t , 2 0

where the first of (9) is the stochastic integral and the second is a Stieltjes Rt integral, for 0 < t ≤ a. This new object is denoted by 0 Ys ◦ dXs , and is sometimes termed the symmetric integral, and it was defined in an equivalent form by R.L. Stratonovich (1968), and it is often called the Stratonovich integral. The {Xs , Fs , s ≥ 0} gives a well-defined (scalar) random measure µ′ on P, the predictable σ-algebra. The following analysis admits on extension using the localization technique involving stopping times. [An adapted process X = {Xt , Ft , t ≥ 0} is termed local if there is a stopping time process {Tn , n ≥ 1} of the given filtration {Ft , t ≥ 0} if Tn ↑ ∞, (Tn < ∞, a.e.) and if Xn (t) = {X(t ∧ Tn ), Ft , t ≥ 0} is uniformly integrable (in t) for each n. It is denoted {Xtloc , Ft , t ≥ 0}. Thus if X is a martingale then X loc is a local martingale, and the like. For simplicity here only the general class is considered without local extensions.] Now denote by A, M and S the classes of processes adapted to a standard filtration {Ft , t ≥ 0} of (Ω, Σ, ν), that are respectively of bounded variation, of martingales, and of semi-martingales. Thus each Z ∈ Sc , continuous, admits a decomposition as Zt = Xt + At , t ≥ 0, where {Zt , Ft , t ≥ 0} is a continuous semi-martingale,{Xt, Ft , t ≥ 0} is a continuous martingale and {At , Ft , t ≥ 0} is a predictable process of bounded variation. Our previous analysis indicates that the decomposition of Z = X + A, as above, is unique. Thus (9) can be expressed in the symmetric integral notation as:

6.4 Generalizations of Martingale Integrals

Z

0

t

Ys ◦ dXs =

Z

0

t

1 Ys dXs + hY, Xit , 2

t > 0.

255

(10)

It was noted that the Itˆ o-integral, generalizing the Lebesgue-Stieltjes classical case is one of a nonabsolute type (as a Riemann-Stieltjes generalized integral) and it is also distinct. The symmetric extension noted above is again different. To clarify these points we recall the classical integration by parts formula in each of these types, to conclude our discussion. This will highlight the reasons why the random measures and their integrals introduce some really new terms and analysis. We thus discuss briefly the HenstockKurzweil-McShane procedure. Let f : R → R be a nondecreasing function and for each interval I = (a, b) ⊂ R, let µf (I) = f (b) − f (a). Denote by µ ˜f , the Carath`eodory generated measure using µf . Then all Borel sets are µ ˜f -measurable. This µ ˜f is usually called the Lebesgue-Stieltjes measure. Suppose on the other hand for an interval [a, b], we consider A1 , . . . , An subsets of [a, b], such that Ai ∩ Aj = ∅, (i 6= j) or just a single point. Now let x1 , . . . , xn ∈ [a, b], called tags, and consider the pairs {(xi , Ai ), i = 1, . . . , n} called a Perron (or P )-partition of [a, b] if xi ∈ Ai , and if the last condition on the xi ’s is not required, it is a Lebesgue (or L)-partition of [a, b]. A function δ : [a, b] → R+ is a gauge if Ai ⊂ (xi − δ(xi ), xi + δ(xi )) in addition and the resulting partition is then called δ-fine. It is a consequence of the HeineBorel theorem, established by P. Cousin in 1895, that every nonempty compact interval I of R admits a δ-fine P -partition of a mapping δ : I → (0, ∞). A function h : I = [a, b] → R is generalized Riemann (or just P -) integrable relative to an increasing function f : [a, b] → R+ if for given (δ, ǫ) where δ is a gauge function as defined above and ǫ > 0, there is a δ-fine P -partition πn = {(xi , An ), i = 1, . . . , n} for a gauge δ(·) and a constant c ∈ R such that for the Riemann sum S(h, πn , f ) =

n X

h(xi )f (Ai ),

i=1

where f (Ai ) is a given Stieltjes measure of Ai , we have |S(h, πn , t) − c| < ǫ, (11) R and then we denote c = (P ) I h(t) df (t), called the Perron or P-integral. This concept of generalized Riemann relative to the pair (δ, ǫ) was the basis of McShane’s stochastic integral, using δ(·) as a function and not a constant necessarily as in the classical R−S or L−S integration. It is clear that there are more L-partitions, and hence fewer functions will satisfy (11) and thus h may be P -integrable but not |h| and we have a non-absolute integral. The distinction shows up clearly in the integration by parts formula, which we state below, and then present its analog for the stochastic integral. That will exemplify the differences. Proposition 6.4.6 Let f, g : I = [a, b] → R be a pair of functions.

256

6 Martingale Type Measures and Their Integrals

(i) If f, g are both nondecreasing and µ ˜f , µ ˜g are the measures determined by the Carath`eodory process, of f and g, then we have Z Z µf (x). (12) f (x) d˜ µg (x) = f (b+)g(b+) − f (a−)g(a−) − g(x−) d˜ I

I

(ii) If g is a bounded continuous increasing function with µ ˜g (·) as its generated measure (cf.(i)), and f is P -integrable (or Perron integrable) relative to µ ˜g , Rx then letting F (x) = a f (t)d˜ µg (t), we have Z Z (P ) f (t)d˜ µg (t) = F (b) − (L) F (x)d˜ µg (x), (13) I

I

where the left side one is the (nonabsolute) P -integral and the right one is the Lebesgue-Stieltjes integral. Thus in (i) we have the standard integration by parts result for the L−S integral and in (ii) the corresponding one for the nonabsolute (Perron or) P -integral. The proof of (ii) is less direct as it uses the (δ, ǫ)-pairs and depends on a somewhat involved argument. [For details, see e.g. the author’s book (Rao(2004), Sec.7.1.] All the results of McShane’s discussed above use a generalization of this type of argument in place of the stopping times considered in the alternative methods. The corresponding result for stochastic integrals is central for the resulting calculus. A key extension for Brownian integrators was obtained by Itˆ o, and the result was generalized to square integrable martingales by Kunita and Watanabe (1967) which was taken over and presented thereafter for semi-martingales by Meyer and his associates. We include a sketch of this development, and then using (10) give an interesting result (analogous to the classical non-stochastic form in its appearance) for the symmetric or Stratonovich integrals. The real difference between the Lebesgue-Stieltjes and the stochastic integration (including B.M. and more generally square integrable martingales) is that the measures in the former have finite variations on compact sets of R, whereas the latter measures have just finite quadratic (co-) variations so that some extra terms appear in this process. The preceding analysis implies that for any bounded continuous f : [a, b] → R, and a square integrable martingale X = {Xt , Ft , t ≥ 0} relative to a standard filtration {Ft , t ≥ 0}, the (stochastic) f (Xt ) is integrable, so Rb that a f (Xt ) dXt is well-defined for [a, b] ⊂ R+ since {f (Xt ), a ≤ t ≤ Rb b} is predictable, i.e., measurable for P, and, of course, a f (Xt )dAt exists as a classical Stieltjes integral where At = [X, X]t so that the expression Z t

(If X)t =

f (Xs ) dXs ,

a

0 < a ≤ t < ∞,

Rt Rt is well-defined. More generally a g1 (X, A)(s) dXs and a g2 (X, A)(s) dAs are Rt meaningful for continuous g1 , g2 . Thus by (10), a g(X, A)(s)(dXs + dAs ) has

6.4 Generalizations of Martingale Integrals

257

a definite meaning for continuous g : [a, b] × R+ → R (or C), and the right side of (10), for continuous square integrable martingales, J=

Z

b

a

g1 (Xs , As ) dXs +

Z

b

g2 (Xs , As ) dAs ,

(14)

a

is a well-defined random variable. This observation motivated K. Itˆ o to obtain the following comprehensive result which was generalized by Kunita-Watanabe, Meyer, McShane and many other researchers, leading to further advances in the subject. The following basic formula is essentially due to K. Itˆ o: Theorem 6.4.7 Let X = {Xt , Ft , t ≥ 0} be a continuous square integrable martingale, and A = {At , Ft , t ≥ 0} be a continuous P-adapted or predictable process of whose sample paths are a.e. of bounded variation, relative to the standard filtration {Ft , t ≥ 0} from (Ω, Σ, P ). Let Z = X + A, be a continuous semi-martingale. If f : R → C is a twice continuously differentiable function, and Z is as given above, then {f (Zt ), Ft , t ≥ 0} is a semi-martingale, integrable on all compact intervals. Moreover one has: f (Zt ) − f (Z0 ) Z t Z Z t 1 t d2 f df df (Zs ) dhXis . (Zs ) dXs + (Zs ) dAs + = 2 0 dx2 0 dx 0 dx

(15)

Remark 6.5 If f (x) = x, then this is simply Z = X + A, the given relation, and if f (x) = x2 , this can be identified with the right side of (10). The general case depends on the fact that X has finite quadratic variation, and so in using the usual Taylor’s expansion of f one has to retain also the second order terms and thus the proof necessarily becomes somewhat more complicated! Proof. Consider the closed interval [0, t] and its partition πn : 0 = t0 < t1 · · · < tn = t. To use the differentiability hypothesis on f , let us start, using the Taylor expansion, as: f (Zt ) − f (Z0 ) = =

n X

i=1 n X i=1

(f (Zti ) − f (Zti−1 )) n

f ′ (Zti−1 )(Zti − Zti−1 ) +

1 X ′′ f (ci )(Zti − Zti−1 )2 2 i=1

(16)

where ci is an intermediate point between Zti and Zti−1 . We now use the fact that Zt = Xt + At and simplify (16) as the partitions πn are refined so that πn = max1≤i≤n (ti − ti−1 ) → 0, as n → ∞, and that the

258

6 Martingale Type Measures and Their Integrals

Zt -process is a (bounded continuous) semi-martingale. We need to simplify the two sums of (16) separately. Now the first term becomes, as πn → 0, n X i=1

f ′ (Zti−1 )(Xti − Xti−1 ) + →

Z

t

n X

f ′ (Zs )dXs +

0

i=1 Z t

f ′ (Zti−1 )(Atk − Atk−1 ) f ′ (Zs )dAs

(17)

0

since the second term is just the Stieltjes sum, and the first one may be exRt pressed as h (Z n s )dXs . Here hn (·) is a predictable simple function given by Pn 0 hn (s) = i=1 χ(ti−1 ,ti ) (s)f ′ (Zti−1 ) which is P-simple and if h(s) = f ′ (Zs ), then khn − hk2,hXi → 0 as n → ∞, by Theorem 3.5 and the remark following its statement as well as the fact that Z0 = 0 by our assumption (in addition to the continuity of t → Zt (ω) for a.a.(ω)). Let us simplify the second term of (16) which is special to this subject (i.e., the use of finite second or the quadratic variation). Hence consider it as (omitting 21 ): 2nd term of (16) =

n X i=1

f ′′ (ci ){(Ati − Ati−1 )2 + (Xti − Xti−1 )2

+ 2(Ati − Ati−1 )(Xti − Xti−1 )}

= I1n + I2n + I3n

(say).

(18)

Since the At -process is of bounded variation on the compact set [0, t], and f ′′ is continuous (hence bounded) the sum converges to zero pointwise, i.e., I1n → 0 a.e., and so also is I3n , the product term, since |I3n | ≤ sup |f ′′ (x)| sup |Xtk − Xtk−1 |kAkt → 0 a.e. x

tk

Here we use the boundedness of f ′′ on compacts and the Xt -process is uniformly continuous, whereas kAt k is bounded a.e. Thus I3n → 0 a.e. and so consider I2n . Corollary 2.6 establishes a special case of this (if f ′′ = 1 there), but we need the general case here, and because of its importance the details will be included.

I2n =

n X i=1

f ′′ (cn )(Xti − Xti−1 )2

n X (f ′′ (cn ) − f (Xti−1 ))(Xti − Xti−1 )2 = i=1

+

n X i=1

f (Xti−1 )(Xti − Xti−1 )2 .

(19)

6.4 Generalizations of Martingale Integrals

259

The first term tends to zero in mean as n → ∞, since {Xt , Ft , t ≥ 0} is a bounded continuous martingale on [0, t] by hypothesis and hence has finite quadratic variation hXit . Thus it is dominated in mean by 1

1

[E(max |f ′′ (ci ) − f ′′ (Xti−1 )|2 )] 2 [E(hXit − hXi0 )2 ] 2 , the CBS-inequality, 1

1

≤ (E(hXt i2 )) 2 (E(max |f ′′ (ci ) − f ′′ (Xti−1 )|2 ) 2 → 0,

as n → ∞ since f ′′ (·) is uniformly continuous and bounded and E(hXt i2 ) is bounded by our initial simplification. So consider the second term of (19) to find its limit which thus is that of I2n . We asset that n X i=1

f ′′ (Xti−1 )(hXiti ) − hXiti−1 ) and

n X i=1

f (Xti−1 )(Xti−1 − Xti )2

(20)

Rt have the same limit as n → ∞ and it is 0 f (Xs )dhXis , which implies (15). To establish the above assertion, note that for the Xt -martingale, E Fti−1 (Xti − Xti−1 )2 = E Fti−1 (Xt2i ) + Xt2i−1 − 2Xti−1 E Fti−1 (Xti ) = E Fti−1 (Xt2i ) − Xt2i−1 ,

( martingale property),

= E Fti−1 (Mti + hXiti ) − (Mti−1 + hXiti−1 ), since {Xt2 , Ft , t ≥ 0} is a submartingale for which the

Doob-Meyer decomposition is applicable with {Mt , Ft , t ≥ 0}as a martingale and {hXit , Ft , t ≥ 0}

as its increasing natural process, = E Fti−1 (hXiti − hXiti−1 ).

(21)

Using the fact that f ′′ is bounded and continuous, and Xt is a bounded martingale, the difference of the two expressions in (21) is bounded, we get: E[max |f ′′ (x)|2 x∈R

n X i=1

{(Xti − Xti−1 )2 − (hXiti − hXiti−1 )}2

n X ≤ 2 max |f ′′ (x)|2 sup E( [(Xti − Xti−1 )4 + (hXiti − hXiti−1 )2 ]), x

πn →0

2

2

i=1 2

since (a ± b) ≤ 2(a + b ), → 0, as n → ∞,

by the continuity of the Xt -process (and |πn | → 0). This establishes (20) and finishes the argument.

(22) 2

260

6 Martingale Type Measures and Their Integrals

Remark 6.6 It is possible to extend the above result for locally bounded and locally integrable martingales using the classical truncation arguments suitably, as pioneered by McShane for the K.δt generalization or using stopping times techniques as in the original Meyer’s work. Also one can extend it to the locally bounded semi-martingales adapted to standard filtrations. This is available in the books by Ikeda-Watanabe (1981), the author (Rao (1995), Chapter 5) or Dellacherie-Meyer (1982) among others. We introduced in (10) a symmetric integral and remarked that the integration by parts takes a familiar simple form compared to (15). This can be established easily under a slightly strengthened differentiability condition. Proposition 6.4.8 Let X = {Xt , Ft , t ≥ 0} be a continuous martingale, and A = {At , Ft , t ≥ 0} be an increasing continuous (adapted) process with Z = X + A, f : R → R be thrice continuously differentiable. Then Z t f (Zt ) − f (Z0 ) = f ′ (Zs ) ◦ dXs . (23) 0

where the ‘circle’ notation denotes the symmetric or Stratonovich integral. Proof. Consider the right side and simplify the integral as: Z t Z t 1 ′ f (Z0 ) ◦ dXs = f ′ (Zs ) dXs + hf ′ (Z), Xit , 2 0 0 ′ since f (Z) satisfies the hypothesis of Theorem 5, Z t 1 f ′ (Zs ) dXs + hf ′ (X0 + A0 ), Xit = 2 0 +hf ′′ (X + Z), (X + Z)(X + Z)it + 0, since f ′ is twice continuously differentiable and Theorem 5 can be applied, Z

t

1 f (Zs ) dXs + {0 + 0}, 2 0 since the bracket for (semi) martingales shown vanishes, and a formal calculus is given below, Z t = f ′ (Zs )dXs = f (Zt ) − f (Z0 ), =

0

which is the usual form of the integration by parts formula. 2 Here is the symbolic calculus for semimartingales set out by Itˆ o and Watanabe (1978), which we state just for information. Let Ac , Mc , Sc be continuous elements of processes respectively of (locally) bounded variation, square integrable martingales and similarly of semimartingales. Also let B be the set of

6.5 Complements and Exercises

261

(locally) bounded predictable processes, all of which are adopted to the same (standard) filtration {Ft , t ≥ 0} in (Ω, Σ, P ). Thus for Y ∈ B and X ∈ M, Rt we set (Y · X)t = 0 Ys dXs , so that Y · X ∈ M and this integral defines a ‘dot’ product. NowPY 7→ (Y · X) is linear and if X, Y ∈ M then Theorem 5 n implies that limπn i=1 (Xti − Xti−1 )(Yti − Yti−1 ) = hX, Y it as n → ∞. We then have: (i) Z1 , Z2 ∈ S ⇒ d(Z1 + Z2 ) = dZ1 + dZ2 , dZ1 · dZ2 = dhX1 , X2 i, if Zi = Xi + Ai ∈ S, i = 1, 2, (ii) Y ∈ B, Z = X + A ∈ S ⇒ Y · Z = Y · X + Y · A, and Zi ∈ S, i = 1, 2 as above ⇒ Z1 ◦ dZ2 = Z1 · Z2 + 12 dZ1 · dZ2 . With this symbolism the following multiplication table is verified: Proposition 6.4.9 Let A, M, S, B be the classes defined above. Then the differential elements, denoted dA, dM and dS, satisfy: (a) dS · dS ⊂ dA, (b) dS · dA = 0 (so dS · dS · dS = 0). These relations can be expressed in more detail if Y1 , Y2 ∈ B, Z1 , Z2 , Z3 ∈ S, so that (i) Y1 · (dZ1 + dZ2 ) = Y1 · dZ1 + Y1 · dZ2 , (Y1 + Y2 ) · dZ1 = Y1 · dZ1 + Y2 · dZ2 , Y1 · (dZ1 · dZ2 ) = (Y1 Y2 ) · dZ2 = Y1 · (Y2 · dZ2 ) (ii) Z1 ◦ (dZ2 + dZ3 ) = Z1 ◦ dZ2 + Z1 ◦ dZ2 , (Z1 + Z2 ) ◦ dZ3 = Z1 ◦ dZ3 + Z2 ◦ dZ3 , Z1 ◦ (dZ1 · dZ2 ) = (Z1 ◦ dZ1 ) · dZ3 = Z1 · (dZ2 · dZ3 ) (Z1 Z2 ) ◦ dZ3 = Z1 ◦ (Z2 ◦ dZ3 ), These relations show that dS is a commutative algebra over B. This algebraic fact may be verified first and then the Itˆ o formula can be derived. The amount of work is essentially the same, but the algebraic background can be of interest. This was pointed out in Itˆ o-Watanabe (1978), the reverse procedure minimizes the probabilistic content, but the final result will be essentially the same. We now include some important complements to the above work.

6.5 Complements and Exercises 1. Let {Ft , t ≥ 0} be a standard filtration of (Ω, Σ, P ) and A = {At , Ft , t ≥ 0} a left continuous adapted process of locally finite variation on R+ . Then there exists uniquely a predictable process {A˜t , Ft , t ≥ 0} such that {At − A˜t , Ft , t ≥ 0} is a martingale and both A and A˜ processes generate the same σ-algebra. (A˜ is often termed a “compensator” of A.] (Hint: Apply first the Jordan decomposition and then the Doob-Meyer decomposition suitably). 2. Suppose that X = {Xt , Ft , t ≥ 0} is a continuous square integrable martingale with X0 = 0 a.e. on (Ω, Σ, P ) and the filtration is standard.

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Verify that X is Brownian motion if and only if its quadratic variation satisfies hXit = t a.e., with the ensuing sketch. [Hints: For the B.M. X, it is easily shown that hXit = t ≥ 0. The important converse is obtained with Itˆ o’s formula as follows. For y ∈ R, let fy (t) = eity , and we have Z Z t 1 t d2 f dfy (Xs ) dhXis . (Xs ) dXs + fy (Xt ) − fy (Xs ) = 2 s dx2 s dx The first integral on the right is a martingale increment so E Fs applied annihilates it, and for each B ∈ Fs we get Z Z Z Z t 1 iyXs iyXt dP = dP − e e (−y 2 · eiyXs ) ds dP, 2 B s B B with the definition and properties of conditional expectations. This implies, since eiyXs is a ‘constant’ for E Fs , and dividing by it: Z Z Z 1 2 t iy(Xt −Xs ) eiy(Xr −Xs ) dP dr. (∗) e dP = P (B) − y 2 s B B If the left side of (∗) is qy (B; t, s), then we get the functional equation 1 qy (B; t, s) = P (B) − y 2 2

Z

t

qy (B; r, s) dr,

s

with boundary conditions qy (B, s, s) = 0, q0 (B; t, s) = P (B), 0 < s < t, and 1 2 that |qy (B; t, s)| ≤ P (B). The solution is qy (B; t, s) = P (B)e− 2 y (t−s) . Taking B = Ω, deduce that Xt − Xs is Gaussian with mean 0, variance t − s, and that E[(Xt − Xs )(Xr − Xs )] = 0 for s < r < t. Now complete the desired result. This is an interesting application of Itˆ o’s formula.] R1 3. In Exercise 5.5.5 we have seen that the Wiener integral 0 f (t)dZ(t), is welldefined for a bounded Borel function f : [0, 1] → R and Z(·) the random measure determined by a process that is L2,2 -bounded relative to Lebesgue’s measure, and in particular if Z(·) is obtained from B.M., so that it can be R1 given by 0 f (t)Y (t)dt where Y (·) is a vector function and the integral is a (vector) Bochner integral, but Y (t) = dZ(t) is not valid in the classical dt sense, except when Z has finite variation which is impossible for the B.M. The problem is that we are demanding the pointwise limit be true here. Motivated by the classical F. Riesz example of a sequence of bounded measurable functions not converging at any point, but converging in measure, we may ask if the Fundamental Theorem of Calculus is valid for this integral if the computations and limits are not in the pointwise sense, but are weakened to ‘in L2 -mean or in measure’. A positive solution to this conclusion can be given not only for the Wiener integral, but also for the

6.5 Complements and Exercises

263

Itˆ o generated stochastic integral and we outline it here which is due to Isaacson (1969). Consider the Itˆ o formula for the B.M. {Xt , Ft , t ≥ 0} with f (x) = x2 so that Y = f (X) is integrable relative to the B.M. On [0, t], t > 0 we have Z t Z 1 t ′′ f ′ (Xs ) dXs + f (Xs ) dhXis Xt2 = 2 0 0

which becomes since hXis = s for the B.M., Xt2 = 2

Z

t

Xs dXs + t, or Yt =

0

Z

t

Xs dXs =

0

t Xt2 − . 2 2 ∆Y

The fundamental theorem of calculus asks for p lim∆Xs →0 ∆X s , and its relation to the integrand where ∆Yt = Yt+δ − Yt . Similarly ∆Xt and ∆Yt → 0 as δ → 0. Since we already know that the pointwise limit does not exist, we ask for the (weaker) limit in probability. Verify that this limit exists not only for the process of f (Xs ) above but the following more general integrands, namely if f is continuous adapted (hence predictable) for the Brownian filtration {Ft , t ≥ 0}, where Ft ≡ σ{Xs , 0 ≤ s ≤ t} and (t) completed, then p lim ∆Y ∆Xt = Rf (t), t ≥ 0 holds. t+h [Sketch: The process {Iht = t (Us − Ut ) dXs , s, t ≥ 0} is a continuous martingale for the process {Us , Fs , s ≥ 0} which is integrable relative to the quadratic variation {hXit , t ≥ 0} of the given (continuous) square integrable martingale X. (Our problem is for the B.M. process {Xt , Ft , t ≥ 0} and Ut = Xt so that hXit = t, but we outline a general argument, explaining the structure better.) In this form Rt ∆Xt → Ut in probability Yt = 0 Us dXs and it is to be shown that ∆Xt ∆Yt 1 t for t ≥ 0. But ∆Xt − Ut = ∆Xt Ih using the above notation where {Iht , Ft , t ≥ 0} is a continuous square integrable martingale. But for any ǫ > 0,   !2 t 1 2 2 1 h ǫ |Ih | > 2 .k 2  > ǫ = (h− 2 |Iat |)2 |∆Xt | |∆Xt | k ⊂

(Iht )2 (∆Xt )2 ǫ2 1 > 2 ∪ < 2 . h k h k

so that t 2 t Ih (∆Xt )2 ǫ2 1 (Ih ) >ǫ ≤P > 2 +P < 2 P ∆Xt h k h k

(∗)

Since {(Iht )2 , Ft , t > 0} is a (positive) submartingale we can get an upper bound using the classical extension of Kolmogorov’s inequality by Doob to

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submartingales to have (because of continuity hypotheses on the processes and there are no real measurability problems) 2 k2 δǫ2 t 2 P sup |Ih | > 2 ≤ 2 E Iǫt k δǫ 0≤h≤δ Z Z k2 1 t+δ = 2 (Us − Ut )2 ds hXidP. (+) ǫ Ωδ t But µt = hXit ⊗ P is a (positive) product measure on (t, t + δ] × Ω, and hence by the classical (Lebesgue-type) differentiation, this tends to zero as δ → 0 for a.a.(t). Thus for each ǫ > 0 and k > 0 given, we can choose δ > 0 independently, so that (+) can be made small. For the 2 t) B.M. (∆X , as a normalized chi-square variable, the probability that it h is < k12 can be made arbitrarily small by choosing k > 0, large enough. Hence the right side of (*) can be made arbitrarily small (< ǫ) if δ > 0 is small enough and k > 0 is large enough as desired. This establishes the result for continuous (predictable) integrands Ut (= Xt is possible). Our assertion uses only the facts that {(Iht )2 , Ft , t ≥ 0} is a submartingale with c` adl` ag paths and the Xt -process is such that the increments ∆Xt have distributions which do not charge the origin, and need not be Gaussian. This is the generalization for which the (weak) differentiation applies.] 4. In this exercise we consider solutions of (stochastic) equations of the form Rt (∗)Yt = 1+ 0 Ys− dXs where {Xt , Ft , t ≥ 0} is a continuous martingale. The resulting process, (the existence and uniqueness to be established here) is termed an “exponential martingale”. Show that there is a unique solution of the above integral equation (*) when the X-process is a square intes| ≤a → 0 grable continuous martingale satisfying the condition P |∆X h as a → 0 for h > 0, (this is automatic for the X-process if it is B.M.). The solution is an exponential semi-martingale {E(X)t , Ft , t ≥ 0}, and is due to Dol`eans-Dade (1970), in analogy with the ODE, dy = ydx or y = y0 ex with y0 as the initial value. If the X-process is continuous (i.e., has no jump), then we get the expression for Yt = E(X)t as: 1 (∗+) Yt = Y0 exp{Xt − X0 − hXi}. 2 [Sketch: The fact that (∗+) satisfies (∗) is verified by using Itˆ o’s formula for f (·) as the exponential where Y0 = 1 when X0 = 0. For uniqueness, let Y˜ = {Y˜t , Ft , t ≥ 0} be another process satisfying (*) so that Z t Ut = Yt − Y˜t = Us− dXs 0

Rt is also an L (P )-martingale. Set ϕ(t) = E(Ut2 ) = E( 0 Us2 dhXis ) so Rt that ϕ(t) = 0 ϕ(s)ds in the case of B.M. Since ϕ(·) is nondecreas2

6.5 Complements and Exercises

265

ing as the mean of a submartingale and ϕ(0) = 0, we get on iteration Z tn Z t2 Z t1 ϕ(s) ds. ··· dt2 ϕ(t1 ) = 0

0

0

C where C = max0≤s≤t ϕ(s) < ∞ so that ϕ(t) = 0, Hence 0 < ϕ(t) ≤ n! ˜ and Yt = Yt a.e. (or indistinguishable).] [Actually the result holds for all second order semi-martingales. We need to use the Itˆ o formula for multidimensions and anticipate jump discontinuities, even if X, Y are complex (i.e., X = X 1 + iX 2 and Y = Y 2 + iY 2 where X 1 , X 2 , Y 1 , Y 2 are semimartingales for the standard filtration {Ft , t ≥ 0} of (Ω, Σ, P )). We state the solution (∗+) in this case for a comparison, but its proof is nontrivial, and is omitted. The solution of (*) in this generality is an exponential semi-martingale given by (with X = X 1 + X 2 ):

1 Yt = E(X)t = Y0 {exp(Xt − X0 − [X˙ 1c , X˙ 2c ]t + [X˙ 2c , X˙ 2c ]t 2 −F˙ [X˙ 1c , X˙ 2c ]t )}Π0<s≤t (1 + ∆Xs )e−∆Xs ), the infinite product converges absolutely a.e., and [·, ·]t is the quadratic (co-) variation symbol introduced earlier, ∆Xt = Xt − Xt− is the jump and if Yt = Y c +Y d is the continuous and discrete parts of the decomposition, then X˙ c = Y c . The result was established in this generality by C. Dol´eans-Dade (1970).] 5. The multivariate Itˆ o formula, needed in the general statement above is a relatively easy extension of a one-dimensional case. This is given as follows: Let f : Rn × Rn → R be a twice continuously differentiable function relative to all n + n coordinates, and Xt = (Xt1 , . . . , Xtn ), At = (A1t , . . . , Ant ) be n and n vector c` adl` ag square integrable martingale and locally bounded variation component processes. Then the Itˆ o formula for the n + n variables is given as: f (X, A)t −f (X0 , Z0 ) =

Z t i ∂f ∂f (Xs , As ) dXsi + (Xs− , As− ) dAis 0 ∂yi 0 ∂xi i=1 Z n n t ∂2f 1 XX (Xs− , As− )d[X i , X j ]s . + 2 i=1 j=1 0 ∂xi ∂yj

n Z X

t

The proof is a standard extension of the case m = n = 1 done in the text, although the necessary work needs some care and is thus not entirely trivial. The formula can be extended for quite general (local) semi-martingales where the At -processes can be added with several Stieltjes type integrals without disturbing the structure. We included it here only for information. 6. Let H1 be a Hilbert space and H2 = L2 (P ). A mapping V : H1 → H2 with domain all of H1 and range all of H2 , such that (V h1 , V h2 )H2 = (h1 , h2 )H1

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6 Martingale Type Measures and Their Integrals

for all h1 , h2 ∈ H1 , is called isometric and in case H1 = H2 it is unitary. Observe that every isometric (hence unitary) operator is linear, as is known. In fact, if h1 , h2 ∈ H and h = ah1 + bh2 , (a, b scalars) then ¯ ∈ H1 we have, using the sesqui-linearity of the inner prodfor any h uct, ¯ H ¯ H + b(h2 , h) ¯ H = a(h1 , h) ¯ H = (h, h) (V h, V h) 1 1 1 2 ¯ H , ¯ H = (aV h1 + bV h2 , V h) ¯ H + b(V h2 , V h) = a(V h1 , V h) 2 1 2 ¯ ∈ H2 being arbitrary, this imand since the range of V is H2 , with V h plies V h = aV h1 + bV h2 , so that V is linear. If now we use the fact that H2 is L2 (P ) where we assume that f ∈ L2 (P ) is centered, i.e., E(f ) = 0, and P is Gaussian so that f is N (0, σ 2 ), then {V h, h ∈ H1 } is centered Gaussian family which is linear. The mapping V is then called an isometric Gaussian operator on H1 onto H2 (⊂ L20 (P )) is possible. Verify that Z : B(R) → H2 defined by Z(A) = V χA , A ∈ B(R), is a random measure such that, if H2 is real Hilbert space and H1 = L2 (R, B(R), λ) where λR is Lebesgue measure, then Z(·) is the Wiener measure. Thus V h = R hdZ, h ∈ H1 is a vector (H2 -valued) integral, and it is the Wiener integral. It cannot be defined pathwise as Stieltjes integral, since Z(·) does not have finite variation. This idea was generalized by Itˆ o for multiple Wiener integral which will be considered in the next chapter. 7. Using the notation of the preceding problem, let Σ0 = σ(eV h , h ∈ H1 ). We then assert that the set {eV h , h ∈ H1 } is fundamental in H2 . Moreover, H2 admits the following direct sum decomposition obtained from the fact that the classical Hermite polynomials in Gaussian random variables furnish the desired splitting of H2 as follows. Since this will be useful for the next chapter, we sketch the desired points and urge the reader to fill in the standard detail which will be refined in the next chapter. Recall that the generating function of Hermite Hn (·)i hpolynomials 2 P dx − (x−t) tx tx−x2 /2 x2 /2 2 |t=0 is defined from G(t, x) = e = e n≥0 n! dtn e as: (−1)n x2 dn − x2 e 2 , n ≥ 1, H0 (x) = 1. e2 Hn (x) = n! dxn These polynomials Hn (·) satisfy the relations for n ≥ 1: (i) Hn (−x) = (−1)n Hn (x), n (x) (ii) Hn′ (x) = Hn−1 (x), and the recurrence: [as usual Hn′ (x) = dHdx ] (iii)(n + 1)Hn+1 (x) = xHn (x) − Hn−1 (x); and if X, Y are standard normal N (0, 1), random variables, then using the moment generating function 2 2 1 for X, Y , verify that E(etX+sY ) = e 2 (s +t +2stE(XY )) and with (m + n times differentiating at s = t = 0, verify that E(Hn (X)Hm (Y )) = δmn

1 [E(XY )]n , n!

Bibliographical Notes

267

where δmn is the Dirac delta. If Ln = sp{H ¯ n (V h) : h ∈ H1 }} ⊂ L2 (P ), then Ln ⊥ Lm for n 6= m and we have the decomposition ∞ M H2 = L2 (Σ0 , P ) = Ln , n=0

with L0 consisting of constant functions. The interest in this decomposition is in understanding the subspaces Ln . It will be seen that they are characterized as the spaces of Wiener integrable scalar functions on Rn spaces, relative to products of Wiener measures Z(A1 )Z(A2 ) . . . Z(An ) integrating f (x1 , . . . , xn ) leading to multiple Wiener integrals. [The computations omitted above may be obtained from Hida-Hitsuda (1982) or Nualart (1995), where some of the above (omitted) detail is discussed. The use of Hermite polynomials in this work was discussed by Kakutani (1961) who derived and then generalized the original Itˆ o analysis. The affinity of the above generating function with that of the Gaussian random variable is crucial. We consider some of this in the following chapter.

Bibliographical Notes The readers would have noticed by now that this chapter continues, expands and deepens the work of the last chapter especially as it relates to square integrable (and then local) martingales. The need for extended analysis of the stochastic integrals was prompted by the need to consider the convolution of random measures discussed in Chapter 5. Here the general framework for such integrals initially based on (second order)martingales is treated in detail. The first section concentrates on an extended study of stochastic integrals with deterministic integrands, and then the (long) second section considers the case of the stochastic counterpart of the integrals, still of non-anticipative type. A comparison of Bochner’s boundedness principle and Grothendieck’s representation is included and contrasted as it was not done in the literature before. To explain the functional analytic aspect of these integrals, we have detailed Cuculescu’s (1970) method to show that the second order stochastic integral is closely related to the spectral theory in Hilbert space, and this aspect illuminates the basic structure of this (stochastic) integration. The other approach to the study of random measures is that of Dol´eansDade’s (1970) formulation and the conditions for its σ-additivity are detailed. The importance of stopping time tools, as some types of truncation devices, is explained. Earlier both E. J. McShane (1969) and L. C. Young (1970) with a more detailed and advanced analysis have studied, and the former carried it out in his monograph of (1974) as well as an extension of some aspects of that work in (1975), without using stopping times tools. The interesting point of

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6 Martingale Type Measures and Their Integrals

McShane’s approach is to use the Denjoy-Perron non-absolute generalization of the Riemann-Stieltjes integration as detailed in modern terms by Henstock and Kurzweil. This is clarified and juxtaposed in our analysis, since so far it is ignored or overlooked in most probabilistic treatments. Hopefully this is made explicit here to a significant extent. All the known stochastic integrals are seen to satisfy the Bochner boundedness principles. Here the problem is ultimately reduced to finding a dominating or controlling positive measure for such integration. This is the crucial part of stochastic integration which is shown by the above principles. Hilbert space analysis naturally plays a critical role in this study. Now the work going beyond the martingale integration is to study the sub (or super, semi, or quasi-) martingales. The point here is to decompose these into martingale parts (which are not necessarily of finite variation) and those of (locally) finite variation. The key aspect played by Meyer’s analysis on the basic decomposition, generalizing Doob’s idea, exhibited in the simple discrete case comes out clearly and emphatically. This point was illuminated afresh in Bass (1996). The ultimate result of this integrator as a (local) semi-martingale under reasonable conditions, automatically obeys the Bochner boundedness principle. This is explained and the algebraic properties of the related stochastic differentials, introduced by Itˆ o and S. Watanabe (1976) are clarified. Many of these conditions are in the nature of existence theorems and for explicit applications, a reasonably general result currently available is McShane’s (nearly) K.δt-condition whose members are semi-martingales but which (so far) appears to be the only one of constructive type known. New ideas and methods are needed for left continuous with right limits, instead of right continuous with left limits integrands. This has been detailed in Brennan (1978), and it will be studied in the next chapter. Most of the exact references are given in the text proper and need not be repeated here. There are some novel applications included in the complements section with a few of the details to be completed by the reader using the extended hints and references. These are supplements to the text and also are important. For instance, the subject of stochastic integrals is related to Bochner’s vector integral in some aspects, but cannot be reduced to an absolutely continuous case unless the random measure has finite variation on compact intervals, (analogous to σ-finiteness). Similarly a weak form of the fundamental theorem of calculus holds (not pointwise). The decomposition of L2 (P ) into an infinite (direct) sum of spaces through Hermite polynomials leads to an analysis of multiple WeinerItˆ o integrals. The useful part played by Hermite polynomials in this analysis, observed by Kakutani (1961) and generalized by MeKean (1973), is discussed here as it motivates a study of multiple stochastic integrals in the following chapters.

7 Multiple Random Measures and Integrals

This chapter is devoted to an extension of the development of random measures and integration with classes of random integrands which may also depend on stochastic elements. Here we present some multidimensional time as well as tensor products of several integrators which are random measures. A number of new problems arise, and we study them both for applications as well as comparing with product and multi- (or poly-) measures to be detailed here and in the following chapters. These have applications for ‘stochastic partial differential equations’ which are being considered in the literature by several types of researches. Thus the next section is devoted to a direct approach to introducing the standard stochastic integral as an extension of the Riemann-Stieltjes method. Somewhat technical, it applies also to ‘quasi-martingales’ that contain the (semi-) martingale integrals discussed at length in the preceding chapter with c` adl` ag integrands. It now includes, with a slight modification, the c` agl` ad (= continuous on the left with right limits – acronym for the French, continue ` a gauche et limite ` a droite integrands) integrals, following the procedure developed by Brennan (1978 and 1980). Then in Sections 2 and 3 the integration problem is extended for multi-parameter processes (or fields) where several new questions have to be answered. The work has to consider ‘products’ of random measures and those on multiple (or cartesian product) fields. Later this is given an extensive treatment for the corresponding quasi-martingales, based on the works of Cairoli-Walsh (1975) together with those of Brennan (1979) and Green (1997). Some extensions and prospects for unified treatment in higher-dimensional time indexes using combinatorial methods as well as various types of applications of these results are included in Section 6. As usual a few complements to all these considerations are given often with sketches of proofs for the less simple problems. Considerable amount of the material in this chapter appears, in a monograph treatment in such detail, perhaps for the first time.

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7 Multiple Random Measures and Integrals

7.1 Basic Quasimartingale Spaces and Integrals Let {Xt , Ft , t ≥ 0} ⊂ L1 (Ω, Σ, P ) (= L1 (P ) for short) be a process. It is called a quasimartingale (cf. Section 6.4) if for each πn : 0 ≤ t1 < t2 < . . . < tn , ) ( n X F t = K < ∞. (1) E E i (Xti+1 − Xti ) sup i=0

The constant K is termed the variation bound of the process. If the Xt process is a martingale so that E Fti (Xti+1 ) = Xti , then condition (1) is automatic with K = 0, while for submartingales ‘=’ is here replaced by ‘≥’ and it will be finite again when the process is L1 (P )-bounded, i.e., supt E(|Xt |) < ∞. Since a super martingale is the negative of a submartingale, it follows that all these three classes are included in the quasi-martingale family. From now on we assume that the filtration {Ft , t ≥ 0} is complete for P and is also standard, i.e., Ft = ∩s>t Fs , for completed Ft , t ≥ 0. Moreover, this filtration is fixed for all the work unless explicitly stated otherwise. In fact we have already discussed (in the preceding chapter) quasi-martingales and semi-martingales along with their integrals for c` adl` ag integrands as well as random measures induced by the above general classes of martingales. We concentrate here the c` agl` ad classes (left continuous with right limits) which necessarily have new problems since this is somewhat different (and going in the other direction) from the standard filtrations. This matter has been investigated by Brennan (1978) and it complements also McShane’s (1975) analysis. (See Section 6.4 for the details of this.) To consider stochastic integrals of processes that are either left or right continuous with limits from the right or left sides on the line (motivated also by the work on random convolutions considered in Chapter 5), we start with a transform of such processes applied to the integrators each of which will be taken as a quasi-martingale or a semi-martingale. We begin with the discrete parameter case. Definition 7.1.1 If X = {Xn , Fn , n ≥ 0} is a quasi-martingale sequence in L1 (P ) and V = {Vn , Fn−1 , n ≥ 1} ⊂ L∞ (P ) is an Fn−1 - adapted set, then V is called a transform of X, defined as: (V ∗ X)n = V1 X0 +

n X

k=1

Vk (Xk − Xk−1 ),

n ≥ 1,

(2)

where {Fn , n ≥ 0} is a (completed) filtration from (Ω, Σ, P ). The motivation here is to replace the sum in (2) by a suitable limit process and get an integral of the V -process relative to the X-process for which some inequalities will be derived. Here one uses the fact that any L1 (P )valued adapted sequence relative to a filtration is decomposable into a martingale and a “predictable sequence” of random variables of “locally finite”

7.1 Basic Quasimartingale Spaces and Integrals

271

Vitali variation uniquely, as observed by Doob (1953). We set down this property to use it effectively later on to produce integrals of the V -processes relative to X, quasi- or semi-martingales, on R+ and the V -processes to be correspondingly generalized. Thus if X = {Xn , Fn , n ≥ 0} ⊂ L1 (P ) and Pn−1 Fn ↑⊂ Σ is P -complete, let A0 = 0, An = k=0 E [(Xk+1 − Xk )|Fk ], and set Yn = Xn − An . It is immediate that {Yn , Fn , n ≥ 1} ⊂ L1 (P ) is a martingale and {An , Fn−1 , n ≥ 1} is a process of theP ‘V’ type described above. ∞ Moreover X is a quasi-martingale whenever E ( n=0 |An+1 − An |) < ∞. The representation (+) Xn = Yn + An , n ≥ 1 is also unique. This follows from the fact that any other representation of Xn = Yn′ + A′n satisfies Yn − Yn′ = A′n − An with A0 = A′0 = 0. Then the martingale property of ′ the Y ’s implies Ym − Yn′ = Yn−1 − Yn−1 = Y0 − Y0′ = A0 − A′0 = 0 so that ′ ′ Yn = Yn and then An = An , n ≥ 1, the An sequences being Fn−1 -adapted, is termed predictable. The continuous parameter version of this decomposition is of course, the celebrated Doob–Meyer decomposition and some extension of it. We have: Proposition 7.1.2 If X = {Xn , Fn , n ≥ 0} is a quasi-martingale satisfying (1) above and V = {Vn , Fn−1 , n ≥ 1} ⊂ L∞ (P ) the predictable transform, then {(V ∗ X)n , Fn , n ≥ 1} is again a quasi-martingale whose variation bound corresponding to (1) is at most αK where α = supn |Vn |. In fact β (3) P sup |(V ∗ X)n | > λ ≤ (kXn k1 + αK) , λ > 0, λ n for some absolute constant β > 0. Proof. First observe that {(V ∗ X)n , Fn , n ≥ 1} is a quasi-martingale with bound K, since (taking α = 1 for simplicity) m X

n=1

E (|E((V ∗ X)n −(V ∗ X)n−1 |Fn−1 )|)) = ≤

m X

n=1 m X

n=1

E |E Fn−1 (Vn (Xn −Xn−1 ))| E |E Fn−1 (Xn −Xn−1 )| ,

since |Vn | ≤ 1 a.e., ≤ K < ∞,

(4)

because m > 1 is arbitrary. Next using the elementary Doob decomposition of the quasimartingale {Xn = Yn + Zn , Fn , n ≥ 0} into a martingale {Yn , Fn , n ≥ 0} and a predictable process {Zn , Fn , n ≥ 1} of bounded variation recalled above, we have, since the V -transform is linear, the following:

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7 Multiple Random Measures and Integrals

λ λ + P sup |(V ∗ Z)k | ≥ P sup |(V ∗ X)k | > λ ≤ P sup |(V ∗ Y )k | ≥ 2 2 k≤n k≤n k≤n n hX i λ ≤ P [1st term] + P . (5) |Vk ||Zk − Zk−1 | > 2

k=1

To simplify the right side of (5), its second term is: " n # n X 2X λ 2 P ≤ |Vk ||Zk − Zk−1 | > E(|Zk − Zk−1 |) ≤ K, 2 λ λ k=1

(6)

k=1

since |Vk | ≤ 1 and the Zk -sequence has a finite variation bounded by K. The first term is a transform of the martingale {Yk , Fk , k ≥ 0} and is again a martingale which converges. So a bound can be given with the classical Doob inequality. However this involves the combination of the V - and Y -processes. To separate it, some additional argument is needed and it is detailed by Burkholder ((1966), Thm. 6). We shall sketch some ideas in a problem later. Thus we have λ λP sup |(V ∗ Y )k | ≥ ≤ β1 sup E(|Yn |), (7) 2 n k≤n for some absolute constant β1 > 0. Consequently from (5), with (6) and (7) substituted in (4), we get (3) with β = β1 . 2 The point of this proposition is that for the integrators {Xt , Ft , t ≥ 0}, we can consider the ‘transform functions’ {Vn , Fn , n ≥ 0} as stochastic integrands to get a (possibly) more general integral. In fact the preceding chapter, dealing with stochastic integration (on the line), started with a stochastic integrator which determines a random measure, and those stochastic integrands for which a dominated convergence property should hold. Although Bochner’s boundedness principles are more general in formulation, they depend on finding controlling measures which are not simple to get. [In some forms, Grothendieck’s theorem of metric geometry asserts the existence, but this is not general enough for certain other important applications!] The above proposition indicates that both the integrator and integrand processes are really determined by the filtration which is fixed in a given integration procedure. While this is not sufficient for all types of (stochastic) integration processes (for instance, the convolution procedure considered earlier in Chapter 5, still needs further analysis and extensions) we want to find a fairly general class of integration procedures in the following analysis. Let {Ft , t ≥ 0} be a standard filtration from (Ω, Σ, P ) and consider the class of locally bounded (real) processes {Xt , Ft , t ≥ 0} so that sup{|Xs |, s ≤ t} < ∞ a.e., for all t > 0. The class is denoted M(F). This is a vector space when equivalent processes, i.e. P [Xt = Yt , t > 0] = 1 are identified (called indistinguishable), so that the space may be considered as a subset of L0 (P ). To begin

7.1 Basic Quasimartingale Spaces and Integrals

273

with, the following topologies will be introduced for it, and they will be used later on. We use the familiar Fr´echet metric in M(F) as: X ∈ M(F) then d : X → R+ defined by ∞ X sup0≤t≤x |Xt | −n , (8) d(X) = 2 E 1 + sup0≤t≤n |Xt | n=1

where we take the fraction to be unity if supt≤n |Xt | = +∞ on a set of positive P -measure and ‘sup’ is measurable by the separability of the process both here and below. A sequence {Xtn , Ft , t ≥ 0} is said to converge to {Xt , Ft , t ≥ 0} on compact sets in probability if for ǫ > 0, P sup |Xtn − Xt | ≥ ǫ → 0 as n → ∞, (9) 0
for each T > 0. This is termed the T-topology which also makes M(F) a topological space, and in fact both concepts are equivalent: Proposition 7.1.3 With the above notation {M(F), d(·)} is a Fr´echet space whose metric topology is equivalent to T. Moreover the class C of processes with a.e. continuous paths, the class Cd of c` adl` ag (= right continuous with left limits) processes and class Cg of c` agl` ad (= left continuous with right limits) process are closed subspaces of M(F) in its I (or equivalently the d(·)-) topology. Proof. The fact that d(·) is a metric is clear since the mapping Q : t → 1 t 1+t = 1 − 1+t is concave increasing so that each term in (8) satisfies the triangle inequality and so does d(·). Now d(f, g) = d(f − g, 0)(= d(f − g) in (8)) implies d(·) is translation invariant. Also the T- and d(·)-topologies are equivalent, since if d(X n ) → 0 as n → ∞, and N > 0 is fixed, then for sup

|X n |

0≤t≤N t → 0 as n → ∞, because d(X n ) → 0 by n ≥ N, 2N d(X n ) ≥ E 1+sup n 0≤t≤N |Xt supposition. This implies for each ǫ > 0, Z supt≤N |Xtn | 1+ǫ n P sup |Xt | > ǫ ≤ n dP ǫ 0≤t≤N [supt≤N Xtn ] 1 + supt≤N |Xt | 1+ǫ ≤ E (integrand) → 0, ǫ

as n → ∞. Thus X n → 0 in T-topology. On the other hand, if X n → 0 in the T-topology, then sup0≤t≤N |Xtn | → 0 in probability for each N , which implies that each term in (8) tends to zero in probability. But each is bounded by one, and the series converges absolutely. Hence d(X n ) → 0. Thus d(·)-convergence and T-convergence are equivalent. We now observe that the asserted completeness also holds. To see this, let {X n , n ≥ 1} be a Cauchy sequence in d(·), so that for any given N > 0 and ǫ > 0, there are n > N and m > N such that

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7 Multiple Random Measures and Integrals

P sup0≤t≤N |Xtn − Xtm | > ǫ < ǫ. Then there exists a subsequence (by the classical F. Riesz’s theorem) {Xtni , Ft , t ≥ 0} such that sup0≤t≤N |Xtni − XtN | → 0 a.e. as ni → ∞, whence |Xtn − XtN | → 0 as n → ∞, (in the T-topology). If ′ N ′ > N then the same argument shows that |XtN − XtN | → 0 in the T-topology implying completeness of M(F) in either of the equivalent topologies determined by T or d(·). Finally regarding the completeness of the (clearly linear) subsets C, Cd and Cg , it suffices to consider one of these, say Cg , the others follow with a similar argument. Thus let {Xtn , Ft , t ≥ 0} be a T-convergent sequence (as n → ∞) of Cg to a process {Xt , Ft , t ≥ 0}. For any arbitrarily fixed N , the set Ain = {sup0≤t≤N |Xtn − Xt | > 21i } is measurable and for some nk < nk+1 , P (Aink ) < 21k , which is possible by the convergence of the sequence in probability. Then we have by the classical Borel–Cantelli lemma P (Aknk , infinitely often) = 0. Thus Xtnk → Xt a.e. uniformly in 0 ≤ t ≤ N . Since N is arbitrary, this implies that {Xt , Ft , t ≥ 0} also belongs to Cg . An identical argument applies to C and Cd . 2 It is known that a quasi- (or a semi-) martingale relative to a standard filtration admits right and left limits at each t (cf. Orey (1967), p.303). We thus consider integrals of elements of M(F) relative to a c` adl` ag semi(or quasi-) martingale which is an integrator and determines a random measure. We can define an integral for finitely valued or simple (or elementary meaning countably valued) functions and extend it. We sketch the details for convenience: Definition 7.1.4 Let S0 (F) ⊂ M(F) be the set of elementary processes, relative to a fixed standard filtration F = {Ft , t ≥ 0} so that: (i) f ∈ S0 (F), supt |f (t, ·)| ∈ L∞ (P ), (ii)

f (t, ·) = f˜0 χ[0] (t) +

∞ X

fi χ(ti ,ti+1 ) (t)

(10)

i=0

for some 0 = t0 < t1 < t2 < . . . , fi is Fti adapted (f˜0 is F0 -measurable), for such a sequence (or set) of f ’s, and X ∈ M(F), We define the (integral) symbol for f relative to X as: Z

0

t

f (s, ·)dXs = f˜0 X0 +

∞ X i=0

fi Xt∧ti+1 − Xt∧ti .

(11)

It is shown below that the integral in (11) is well-defined, i.e., does not depend on the representation of f by (10), and that it is linear in f for each X also linear in X for each f , i.e., is bilinear. We state and prove the result in the following form for reference and later use: Theorem 7.1.5 Let f ∈ S0 (F)(⊂ M(F)) and X ∈ M(F). Then the integral Rt f 7→ 0 f (s) dXs of the above concept is well-defined and is bilinear in (f, X).

7.1 Basic Quasimartingale Spaces and Integrals

275

Moreover, the linear functional (·, X) : S0 → M(F) is continuous in the Ttopology for each X ∈ M(F), and its variation is bounded by Kk supt |f (t, ·)k∞ Rt where K is the variation of X so that if Ytf = 0 f (s) dXs , t ≥ 0, then Y f ∈ M(F). Proof. First observe that (10) is independent of the representation of the elementary function f . In fact let f have two representations as f (t) = f˜0 χ[0] (t) +

∞ X

fi χ(ti ,ti+1 ) (t) = g˜0 χ[0] (t) +

t=0

∞ X

gi χ[si ,

si+1 ] (t).

i=0

Then (11) becomes for a given X ∈ M(F): Z

t

f dX = f˜0 X0 +

0

∞ X i=0

= f˜0 X0 +

∞ X i=0

= g˜0 X0 +

fi Xt∧ti+1 − Xt∧ti

  ∞ X Xt∧ti+1 ∧sj+1 ∨(sj ∨ti ) − Xt∧(ti ∨si )  fi  j=0

∞ X ∞ X i=0 j=0

= g˜0 X0 +

∞ X j=0

gj Xt∧ti+1 ∧sj+1 ∨(sj ∨ti ) − Xt∧(ti ∨sj )

gj Xt∧sj+1 − Xt∧si =

Z

t

g dX.

0

Here we use the fact that fi = gj , if (ti , ti+1 ] ∩ (sj , sj+1 ] 6= ∅ and that P [supt |ft − gt | = 0] = 1. This shows that the integral (11) is unambiguously defined. The second half depends on property (3) of Proposition 2 above. In fact given ǫ > 0, and a sequence {fk , k ≥ 0} ⊂ S0 (F) such that fk → 0 in the T-topology, and X ∈ M(F), then we assert that for any finite subset J ⊂ [0, N ], N > 0 arbitrarily fixed and for n0 sufficiently large, we have by hypothesis: Z t P sup fn (s) dXs > ǫ < ǫ, n ≥ n0 . (12) t∈J 0

Now by (3) for any N > 0, 0 < ǫ < 1, as above as well as a β > 0 (determined by (3)) such that for some δ > 0 we have the estimate 2 + δβ(kXn k + K) < ǫ2 where K of (3) is the variation of X. Let {ti }∞ 1 ⊂ R be a sequence that includes the finite set J as its initial segment and consider fn as a simple (or elementary) function of S0 (F) represented by the expression ∞ X fjn χ[ti ,ti+1 ] . (13) fn = f˜0 χ[0] + j=0

276

7 Multiple Random Measures and Integrals

Pk δ Let gnk = f˜0 χ[|f˜0 |≤δ] + i=1 fin χ[|fin |≤δ] , if J has k-points, so it is the δtruncations of fn . Then we have the estimate as: Z t P sup f n (s) dXs > ǫ 0≤t≤N 0

"

Z ≤ P sup 0≤j≤k

# ǫ f n (s) dXs > ǫ, sup |f n (t)| ≤ δ + 2 0≤t≤N

tj

0

# " k X ǫ s gni Xti+1 − Xti > ǫ + . ≤ P sup g˜0 X0 2 1≤j≤k i=0

2

The first term on the right is at most δβ(kXN k1 + K) ≤ ǫ2 , by the choice of δ above so that (12) holds for any n ≥ N ≥ n0 . Hence the integral is continuous in f , for each element of S0 (F) and each X ∈ M(F). To see that the integral defines a quasi-martingale consider for any f ∈ S0 (F) represented as: ∞ X ˜ fi χ(s ,s ) , 0 < s1 < s2 < . . . (14) f = f0 χ[0] + i

i+1

i=1

and for any 0 < t1 < . . . < tn where {ti , 0 ≤ i ≤ n} ⊂ {si , 0 ≤ i ≤ ∞} with tn = sk , for some k. Now we estimate the integral: ! Z ti Z ti+1 n X fs dXs |Fti )| fs dXs − |E( E i=0

≤E

i=0

k X

=E

i=0

si+1

0

fs dXs −

Z

0

si

! fs dXs |Fti !

|E(fi Xsi+1 − Xsi |Fsi )|

≤ k sup |f | k∞ E t

0

0

k Z X E

k X i=0

|E((Xsi+1 − Xsi )|Fsi )|

!

≤ Kk sup |f | k∞ . t

R RSince f ∈ Cd , and f dX was shown to be in Cd also, it follows that f dX ∈ M(F). Moreover since the latter is complete, the closure of S0 (F) is in M(F), as desired. 2 Remark 7.1 A similar statement holds for C and Cg . But the actual integrals on Cd and Cg are not the same, as shown below. The work is simpler on C. It should be noted that we need to consider suitable truncations and estimates in the detailed work. While this can be done, as was asserted and implemented by L.C. Young (1974), the work becomes more in tune with the probabilistic analysis by using ‘stopping time’ calculus, specially devised for

7.1 Basic Quasimartingale Spaces and Integrals

277

this purpose. Although this is avoidable, the book-keeping without it becomes more elaborate, as already reflected in the above proof, and so we shall once again consider the stopping times technique as introduced with details in Section 6.3, and use them systematically in the following analysis. As will be seen later, this is helpful for integration on R (or R+ ). However it is less useful on Rn , n ≥ 2, i.e. for random fields due to the lack of a linear ordering in it. Recall from Section 6.3 that for an adapted process X = {Xt , Ft , t ≥ 0} relative to a standard filtration {Ft , t ≥ 0} of a (complete) probability space (Ω, Σ, P ), and a stopping time T : Ω → R+ of the filtration, the new process X T = {XT ∧t , F(T ∧ t), t ≥ 0} which is adapted, is usually termed a stopped process at T where for any stopping time T of {Ft , t ≥ 0}, F(T ) = {A : A ∩ [T ≤ t] ∈ Ft , t ≥ 0}, the (obviously) σ-algebra of events prior to T . Then X T is termed the process X stopped at T . Our aim here is to define a stochastic integral for random functions f that are in larger classes so that S0 (F) ⊂ M(F) relative to semi-martingale integrators X. This also holds for f ∈ Cd , the left continuous processes with right limits. It needs special care since our standard filtration is structured to satisfy Ft+ = Ft suitable for c` adl` ag processes and it is therefore necessary to impose additional conditions. This aspect has not been well-studied, except perhaps for the work of Brennan (1978). We thus include a brief account to motivate and aid further research. Definition 7.1.6 A process X ∈ M(F) is said to admit an ε-chain of stopping times Tn if (i) 0 = T0 < T1 < . . . , Tn → ∞ a.e., as n → ∞ and for an ε>0 h i P ω: sup |Xt (ω) − Xs (ω)| < ε = 1, n ≥ 0. (15) Tn (ω)<s, t
Note that if X has right and left limits, i.e. t 7→ Xt (ω) has these limits for a.a.(ω), then for each ε > 0, there exist ε-chains. Indeed, let T0 = 0 and define inductively if Tn is found, n εo Tn+1 (ω) = inf t > Tn (ω) : Xt (ω) − XTn (ω) (ω) ≥ 2

with inf{∅} = +∞, as usual. If X ∈ M(F) is a semi-martingale then it is termed reduced (or reducible) if there is a chain of stopping times 0 = T0 < T1 < . . . and a sequence of quasi-martingales X n such that Xχ[[0,Tn) = X n , n ≥ 1 where [[0, Tn ) is the stochastic interval {(t, ω) : 0 ≤ t < Tn (ω)} ⊂ R+ × Ω. Then the pair (Tn , X n ), n ≥ 1 is said to reduce the process X, so that we can thereafter obtain the X-integral from those of the quasi-martingales X n , n ≥ 1. Definition 7.1.7 An elementary left continuous process f adapted to a standard filtration {Ft , t ≥ 0} ⊂ F of (Ω, Σ, P ) is an essentially bounded process representable as:

278

7 Multiple Random Measures and Integrals

f (t, ω) = f˜0 (ω)χ[0] (t) +

∞ X

fn (ω)χ((Tn ,Tn+1 ) (t, ω)

(16)

n=0

relative to some chain {Tn , n ≥ 0} of stopping times of {Ft , t ≥ 0} where f0 is F0 - and fn is Bn (Tn )-adapted random variables. It is seen that for any refinement {Tn′ , n ≥ 1} of stopping times of the chain {Tn , n ≥ 1} for the filtration {Ft , t ≥ 0}, if one sets g = f˜0 χ[0] +

∞ X

n=0

′ fn (Tn′ +, ·)χ((Tn′ ,Tn+1 ]

then it can be verified that g and f are indistinguishable, so ft = gt a.e. Note that the class of left continuous elementary processes, denoted LS0 (F) ⊂ Cd and the latter contains the former densely in the T-topology. If X is a semimartingale and f ∈ LS0 (Ft ) with representation (16) we set Z t ∞ X fn XTn+1 ∧t − XTn ∧t (ω), t ≥ 0, (17) f (s)dXs (ω) = (f˜0 X0 )(ω)+ 0

n=0

which is an extension of the previous definition of stochastic integral, using the same symbol. One can verify (with computation as in Theorem 7.1.5 above) that the integral is well-defined in (17). An interesting point of it is that the formulation is an extension of the Riemann–Stieltjes method. We now introduce a general concept, motivated by a property of the BM and martingale processes that is useful for stochastic integration, as follows. Definition 7.1.8 Let {Xt , Ft , t ≥ 0} be a c` adl` ag process adapted to a standard filtration and Φ : R → R+ be a continuous increasing symmetric function vanishing only at 0, i.e., Φ(0) = 0. It is called locally Φ-bounded if for each t > 0 and a partition 0 ≤ t1 < t2 . . . < tn ≤ t we have n i h X χ(ti ,ti+1 ] (Xti − E Fti (Xti+1 ) ≤ Kx < ∞, E Φ

(18)

i=1

where Kx is an absolute constant. If Φ(x) = |x|p , p > 0, it is called locally pbounded, and is in a generalized sense if {X0 Tn , n ≥ 1} is Φ-bounded for some stopping time chain {Tn , n ≥ 1} of the filtration. Here “locally” refers to the fact that the condition holds on compact t-sets. Note that (18) is always satisfied for the BM, and more generally for (sub)martingales with Φ(x) = |x|, of which it is an abstraction. Proposition 7.1.9 Let f ∈ S0 (F) and X be a locally bounded semi-martingale. Rt Then Yt : f → 0 f (s) dXs is a well-defined integral and is a locally bounded ¯0 (F), if f is semi-martingale. Moreover the result extends to its T-completion, S ∞ ∞ p ¯ also locally in L (P ), i.e., f ∈ S0 (Ft ) ∩ L (P ), and X ∈ L (P ), 1 < p < ∞.

7.1 Basic Quasimartingale Spaces and Integrals

279

Proof. Let I0 ⊂ R+ be a finite set 0 ≤ t1 < t2 < . . . < tn = T , a partition of [0, t], and if the fi are Fti -adapted let f be defined as ft = f˜0 χ[0] (t) +

n−1 X

0 ≤ t ≤ T.

fi χ(ti ,ti+1 ] (t),

i=0

As noted after Definition 7.1.1, the sequence {Xti , Fti , 0 ≤ i ≤ n} can be decomposed (uniquely) into the sum Xti = Yti + Ati , where {Yti , Fti , i ≥ 0} is a martingale and {Ati , Fti−1 , i ≥ 1} is a sequence of bounded variation. Next Pk−1 letting B0 = 0, Bk = i=0 |Ati+1 − Ati |, it is a bounded (increasing) sequence since 0 ≤ Bk+1 − Bk = |Atk+1 − Atk | = |E Ftk (Xtk+1 − Xtk )| ≤ Kx (t),

(19)

by (18) due to local boundedness of X. Using the fact that for 0 ≤ u ≤ v, and p ≥ 1 we have 0 ≤ (v p − up ) ≤ pv p−1 (v − u), and hence ! n−1 X p p p E(Bn ) = E (Bk+1 − Bk ) ≤ pE

k=0 n−1 X

p−1 Bk+1 (Bk+1

k=0



n−1 k XX

= pE  

k=0 j=0

j=0 k=j



≤ Kx pE 





p−1 Bj+1 − Bjp−1 (Bk+1 − Bk )

 p−1 Bj+1 − Bjp−1  ≤ Kx pkBn kp−1 . p

n−1 X j=0

− Bk )

p−1 Bj+1 − Bjp−1 (Bk+1 − Bk )

n−1 X n−1 X

= pE 

!

Then we have, if kBn kp < ∞, (here Kx ≥ 0 is an absolute bound) kBn kp ≤ pKx ,

and if kBn kp = +∞, replace Bk by Bk ∧ N , to derive the same result and let N → ∞. It then follows that Rs k sup 0 f dXkp ≤ kf k∞ Kx (kXt kp ), (20) t∈I

and since the finite R t set I0 is arbitrary, this shows that by the right continuity of the integral 0 f dX, (20) implies the first part of the proposition. Indeed observe that

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7 Multiple Random Measures and Integrals

k sup t∈I

Rt 0

k X

Rt fi (Yi+1 − Yi ) p f dXk ≤ sup 0 k f dX ≤ sup f˜0 Y0 + k≤n

+

k≤n−1

sup

k X

1≤k≤n−1 i=0 n−1 X

≤ p f˜0 Y0 +

i=0

i=0

fi (Ai+1 − Ai ) p

fi (Yi+1 − Yi ) p + kf k∞ kBn kp ,

and the first term is the martingale transform to which we apply the maximal inequality as in Proposition 7.1.2 above, and the last term is bounded by the preceding analysis. Thus letting the interval [0, t] increase to R+ , we get the desired assertion (the first part) for bounded f ∈ S0 (F). ¯0 (F), the completion in the T-topology, is now easy and The extension to S ¯0 (F) ∩ L∞ (P ), then there is a sequence fn ∈ S0 (F) such is as follows. If f ∈ S that fn → f in probability (the T-topology), and if 0 ≤ t0 < . . . < tk < ∞ then Z ti Z ti+1 k−1 X fn dX ≤ Kx kf k∞ , (21) fn dX − E E Fti i=0

0

0

o is Lp (P )-bounded f dX : n ≥ 0, 0 ≤ t ≤ t n k 0 Rt and since 1 < p < ∞, this implies uniform integrability of the set 0 fn dX which Rt converges to 0 f dX in probability. Hence (by Vitali’s theorem) it is in L1 (P ). Rt Thus (21) implies that 0 f dX is a semi-martingale and is bounded in norm by αKx kf k∞ for an α > 0 fixed. 2 The preceding results can be used employing the stopping time techniques, thereby extending the concept of stochastic integration and obtain a form of the chain rule. This is to be compared with the observations of Exercise 5.5.5 regarding a non-validity of the Radon-Nikod´ ym calculus for random measures and integrals (in the Lebesgue sense). In Definitions 7.1.6 and 7.1.7 above, we introduced the concepts of a chain of stopping times thereby also reducing a given process. Let us call a semimartingale X ∈ M(Ft ), a weak semi-martingale if there is a chain of stopping times {Tn , n ≥ 0} of the filtration {Ft , t ≥ 0} that reduces X so that χ[[0,Tn ) X is a semi-martingale, for each n ≥ 1. by the preceding analysis. Then

nR t

Proposition 7.1.10 A weak semi-martingale X admits a bounded sequence of semi-martingales that reduce it and for each f ∈ Cg (left continuous with right Rt limits) the process Yt = 0 f dX, t ≥ 0, is also a weak semi-martingale for the same filtration. Proof. By definition of weak semi-martingales there exists a sequence of stopping times Tn , n ≥ 1 of F = {Ft , t ≥ 0} and semi-martingales Xn ∈ M(Ft ) such that Xχ[[0,Tn ) = Xn χ[[0,Tn ) , n ≥ 1, the Xn being bounded, i.e., supt |Xn | ≤ n.

7.1 Basic Quasimartingale Spaces and Integrals

281

Consider Xn . Our conditions imply that we may treat each Xn as a quasimartingale, and then by a known generalized Jordan type decomposition of a quasimartingale [cf., e.g., the author’s book (Rao (1995)), Theorem V.2.22 on p. 370, and a different proof is also available by K.M. Rao noted there] Xn = U n − V n where both {U n (t), Ft , t ≥ 0} and {V n (t), Ft , t ≥ 0} are positive supermartingales. Taking σn = inf{t : Utn > n2 or Vtn > n2 } ∧ n Tn ∧ n, and on the set {(t, ω) : 0 ≤ t < σ n (ω)},n wen then have U and n n n V variations bounded. Then Xn = U ∧ 2 − V ∧ 2 is a process such that {(σn , Xn ), n ≥ 1} reduces the process X and for f ∈ S0 (Ft ), left continuous with right limits. Also X ∈ M(Ft ) which has the weak semiRt martingale property and 0 f dX is well-defined. It extends to all elements ¯0 (Sv ) in the S-topology) so that as fn → 0 in T-topology, and of C (= S R (·) d fn dX → 0. 0 Finally let {(σk , Xk ), k ≥ 1} be a reducing sequence of X implying that we get R R h i h i t t P sup 0 fn dX > ε ≤ P sup 0 fn dXk > ε + P [σk ≤ N ] . (22) 0≤t≤N

0≤t≤N

ε 2,

and let n → ∞. Then the right side of R (·) (22) can be made < ε as in Theorem 7.1.5. Thus 0 f dX is well defined for all f ∈ Cd and all weak semi-martingales X. Of course we can only assert that this integrated process just defines a weak semi-martingale for f ∈ Cd . We thus conclude this outline of proof on the (weak) semi-martingale integrals. 2 The point of this detour is to obtain the following useful change of (stochastic) measure result, an analog of the ‘chain rule’ for real measures. It is of interest to contrast this with the result of Exercise 5.5.5.

Fix k such that P [σk ≤ N ] <

Proposition 7.1.11 If f, g ∈ Cd ⊂ M(Ft ), and X ∈ M(Ft ) is a weak semiRt martingale, let Yt = 0 f dX, t > 0, then g is integrable relative to (the weak Rt Rt semi-martingale) Y and we have 0 g dY = 0 f g dX, t ≥ 0 (so that the [weak] chain rule obtains). Proof. Since f, g and f g are in Cd , the integrals given above are well-defined. To establish their equivalence, we observe that (i) for each stopping time σ Rt of {Ft , t ≥ 0}, we see that Ytσ = 0 gχ[[0,σ) dX is defined, and (ii) is measurRt able relative to F(σ), the σ-algebra of the time σ, so that f 0 χ[[σ,∞) dX = Rt 0 f χ[[σ,∞) dX. In fact, it suffices to verify these for step (or elementary) processes f, g, and then the general case follows by taking T-limits of the sequences. ThusP consider one of these (the other being similar), say (i). Let ∞ g = g˜0 χ[0] + n=0 gn χ[[σn ,σn+1 ] , for a chain of stopping times {σn }∞ 1 . Set g˜n = gn χ[σn ≤σ] so that it is F(σn ∧ σ)-measurable, and gχ[[0,σ] = g˜0 X0 +

∞ X

x=0

g˜n χ[[σn ∧σ,σn+1 ∧σ] .

282

7 Multiple Random Measures and Integrals

Then Ytσ for the elementary function g is given by Ytσ = g˜0 X0 + = g˜0 X0 + =

Z

∞ X

n=0 ∞ X

n=0 t

σn +1 σn gn Xσ∧t − Xσ∧t

σ ∧σ g˜n Xt n+1 − Xtσn ∧σ

gχ[[0,σ) dX.

0

We extend it for all left continuous elements gn of S0 (F) which converge to g in Rt Rt the T-topology. Then the corresponding terms Ytn = 0 gn dX → 0 g dX, t ≥ 0, in the T-topology. Now (Y n )σ → Y σ in the T-topology by the previous work. Hence Rt σ Rt 0 gn χ[[0,σ) dX → 0 gχ[[0,σ) dX, in the T-topology, as asserted in (i), and (ii) is similarly verified. Using thePpoints (i) and (ii) we establish the substitution formula. Let ∞ f = f˜0 χ[0] + n=0 fn χ[[σn ,σn+1 ] and g be as above. Then, Z

t

f dY = f˜0 Y0 +

0

= f0 Y0 + = f0 Y0 + = f0 Y0 +

∞ X

fn

n=0

0

Z

t

gχ[[σn ,σn+1 ] dX, by the preceding work,

0 n=0 Z ∞ X t

Z

Z

t

0

fn gχ[[σn ,σn+1 ] dX

∞ X

fn χ[[σn ,σn+1 ]

n=0 t

f χ[[0,∞)g dX = 0

Z

!

g dX

t

f g dX.

(23)

0

Rt Rt But t 7→ 0 f dY and t 7→ 0 f g dX are both a.e. right continuous and agree for each t ≥ 0 so that they are indistinguishable. The interchange of the (infinite) sum and integral is legitimate, the T-limit of the partial sums determine f X(0,∞) for each left continuous f . This establishes the desired substitution statement of the proposition. 2 Remark 7.2 It is of interest to observe that the result does not imply a “differentiation of vector (or random) measures”, and an analog of a Radon-Nikod´ ym type “differentiation” is not available. The actual analog for random (and vector) measures would perhaps involve some additional conditional restrictions, as well as non-uniqueness on the lines of the results in, for instance, Rao ((1973), Section 2). The point of this result is to highlight the basic new elements arising in the present extension. (See Chapter 9 on this problem and a related solution.)

7.1 Basic Quasimartingale Spaces and Integrals

283

So far we considered integrating processes that are left continuous having limits which match with the (order) continuity of the filtration used. If instead we study the right continuous processes with left limits, some significant adjustment by adding extra terms from the left integrals considered above is required. This is similar to the classical Riemann-Stieltjes case. We include some considerations of this difference for completeness, comparison and also for a later application. To define the desired integral, we need the quadratic (co-)-variation of a process X = {Xt− , t ≥ 0} that is right continuous with left limits. Let us denote the left limits process as X − = {Xt−, t ≥ 0} where Xt− = lims↑t Xs , t ≥ 0, with X0− = 0. If X, Y are weak semi-martingales and {σn , n ≥ 1} is an ε-chain (ε > 0) of stopping times of the standard filtration {Ft , t ≥ 0} to which X, Y are adapted, then the corresponding covariation is given by: Vt (X, Y, {σn }, ε) = X0σ0 Y0σ0 + =− P∞

∞ X

∞ X

σ

σn+1

(Xt n+1 − Xtσn )(Yt

n=0

σn+1

Xtσn Yt

∞ X σ Ytσn (Xt n+1 − Xtσn ) + Xt Yt . − Ytσn − n=0

n=0

σn n=0 X(·) χ[[σn ,σn+1 ]

− Ytσn )

(24)

− X(·)

as ε ↓ 0, for the ε-chains, the sums R (·) R (·) − here converge in the T-topology to 0 X dY and 0 Y − dX so that V(·) (X, Y, {σn }, ε) → a limit as ε ↓ 0 independently of σn -chains. Now we denote it by V (X, Y ), called the covariation of X and Y , and variation if X = Y . Thus the covariation process is given by Z t Z t − Vt (X, Y ) = Xt Yt − X dY − Y − dX, (25) But since

→

0

0

which corresponds to the integration by parts in the classical theory. Clearly V (X, Y ) is a bilinear functional and by the parallelogram identity V (X + Y ) = 1 4 [V (X + Y ) − V (X − Y )], and (25) then implies that XY is a weak semimartingale. We now want to define an integral for functions (= process) from Cd , the right continuous process with left limits relative to a weak semi-martingale by postulating some conditions that control the sizes of jumps in compact (sub) intervals. This will show the differences in the c` adl` ag and c` agl` ad cases of integrands for integrator processes. The right continuous elementary (or step) processes f ∈ M(Ft ) are those which are essentially bounded and representable as X f (t, ω) = fn (ω)χ[[σn ,σn+1 ) (t, ω) (26) n≥0

for some chain of stopping times of {Ft , t ≥ 0} and fn is F(σn )-adapted. The class of step process of the type (26) and their T-sequential limits will constitute

284

7 Multiple Random Measures and Integrals

the right continuous integrands with left limits, and will be noted as an alternative form of Cd . The corresponding integral for the weak R t semi-martingales X as integrators of f given by (26) is again denoted as 0 f dX and thus is representable as: Z t ∞ X σ f dX = fn (X − )t n+1 − (X − )σt n + ft ∆Xt , t ≥ 0. (27) 0

n=0

P∞ Note that the representation (26) of f implies f − = n=0 fn χ[[σn ,σn+1 ] , where f − ∈ Cg , and then we have Z t Z t X f dX = f − dX + ∆fs · ∆Xs , (28) 0

0

0≤s≤t

which is well-defined since ∆fs 6= 0 only for a finite number of s points. Here ∆fs = fs − fs− and similarly ∆Xs where fs− = limu↑s fu (likewise for X). This definition is extended for all f ∈ Cg , by using ε-chains and f ε of the type (27) by taking the T-limits. Thus f ∈ Cg is integrable relative to a weak semi-martingale X, if there is an ε-chain {σn , n ≥ 1} and Rt Rt P∞ f ε = n=0 (f ◦ σn )χ[[σn ,σn+1 ) , such that 0 f dX = limε↓0 0 f ε dX whenever this exists in the T-topology. The limit demanded above may not exist for all weak semi-martingales. We isolate the following sufficiently large class of integrators for which the limit does exist. Let us introduce the condition as: Definition 7.1.12 A semi-martingale X = {Xt , Ft , t ≥ 0} is said to satisfy the condition (S) if for any chain {σn , n ≥ 1} of stopping times of the filtration the transformed set {X ◦ σn , F(σn −), n ≥ 1} has a finite variation, in the sense that for some fixed K0 > 0, "∞ # X E |E (X ◦ σn+1 − X ◦ σn | F(σn −))| ≤ K0 < ∞ n=0

where {σn , n ≥ 1} is a chain of finite stopping times and F(σn −) is the sigma algebra of the events {A ∩ [σn > t], t ≥ 0, A ∈ Σ}(⊂ F(σn )). A weak semimartingale Y = {Yt , Ft , t ≥ 0} is said to satisfy condition (S) if there is a reducing sequence {(Tn , X n ), n ≥ 1}, the Tn being stopping times, each X n has to satisfy condition (S). Since F(σn −) can be strictly smaller than F(σn ), we are restricting the class of (weak) semi-martingales as possible integrators for integrating some elements of Cg . The following auxiliary result will be useful for the next key theorem. Proposition 7.1.13 (i) Let X = {Xt , Ft , t ≥ 0} be a weak semi-martingale. Then it has at most a countable set of (left) discontinuities on compact sets, in P 2 the sense that 0≤s≤t |∆Xs | < ∞, a.e.

7.1 Basic Quasimartingale Spaces and Integrals

285

(ii) Let {Zk , Fk , k ≥ 0} ⊂ L2 (P ) be an adapted sequence, and {Ak , k ≥ 0; Vk , k ≥ 0} be predictable sequences of {Fk , k ≥ 0} where 0 ≤ Ak ↑ and |Vk | ≤ 1, a.e., for all k. Then for λ > 0, the following holds: ∞ ∞ ∞ X X i 16 X 2 E(Zk ) + 2E |E(Zk | Fk−1 )| , Vk χ[Ak ≤t] Zk > λ ≤ λP sup λ t≥0 k=0 k=0 k=0 (29) with F1 = F0 .

h

Proof. (i) For the proof we may assume that X is a semi-martingale, and then for each ε > 0, |∆Xs |(ω) = |Xs − Xs− |(ω) ≥ ε for only a finite number of points 0 < s ≤ t, and almost all ω. Let this be N (ω). If 0 < δ < ε, let {σn , n ≥ 1} be a δ-chain for X and if |∆Xs | (ω) ≥ ε, then let r = σn+1 (ω) for some n ≥ 0 and set r = 0 otherwise. Then for r > 0, we have |∆Xr | (ω) ≤ |X ◦ σn+1 − X ◦ σn | (ω) + X ◦ σn − Xr− (ω) σ ≤ Xt n+1 − Xtσn (ω) + δ. Using the trivial inequality (a + b)2 ≤ 2(a2 + b2 ) we get σ

|∆Xr |2 (ω) ≤ 2 Xt n+1 − Xtσn so that X

0≤s
2

(ω) + 2δ 2 ,

∞ X 2 σ Xt n+1 − Xtσn (ω) + 2δ 2 N (ω). |∆Xs |2 χ[|∆Xs |≥ε] (ω) ≤ X02 (ω) + 2 n=0

Since 0 < δ < ε is arbitrary one has on letting δ → 0, X |∆Xs |2 χ[|∆Xs |≥ε] (ω) ≤ 2Vt (X)(ω) − X02 (ω) < ∞. 0≤s≤t

Since ε > 0 is arbitrary, this gives (i) on letting ε → 0. (ii) Put Y0 = Z0 and Yk = Zk − E Fk−1 (Zk ), k ≥ 1 so that E(Yk ) = 0 and E(Yt2 ) ≤ E(Zt2 ). Using the fact that |Vk | ≤ 1, Fk−1 -adapted, we have for λ > 0, the estimate, for given predictable Vt , At sequences, ∞ X h i Vk χ[Ak ≤t] Zk > λ λP sup t≥0

k=0

∞ ∞ X λi i hX Fk−1 ≤ λP sup Vk χ[Ak ≤t] Yk > + 2E (Zk ) . E 2 t≥0

h

k=0

(30)

k=1

P Now {Mn = nk=0 Vk Yk , Fn , n ≥ 0} is a martingale, and define Tt as the first time that Ak+1 > t, i.e., Tt = inf{k : Ak+1 > t}. Thus {Tn , n ≥ 1} is a

286

7 Multiple Random Measures and Integrals

chain of {Ft , t ≥ 0}, each Tn , being a stopping time. Then the stopped process {Mnσt , F(σt ∧ n), t ≥ 0} is a martingale for each n, by the standard results of martingale theory (cf, e.g., the author, Rao (1995), Proposition IV.2.3), and hence by its maximal inequality, for a λ > 0, n λi X h Vk Yk χ[Ak ≤t] > P sup 2 t≥0 k=0 4 16 ≤ 2 E sup |Mnσt |2 ≤ 2 E(Mn2 ) λ λ t≥0 n X 16 Zt2 , since |Vk | ≤ 1. ≤ 2E λ k=0

Substituting this in (30) and letting n → ∞, one has (29). 2 Let us now present the existence of a stochastic integral for the elements (integrands) of Cg relative to weak semi-martingales with property (S). Theorem 7.1.14 Let X = {Xt , Ft , t ≥ 0} be a weak semi-martingale having P property (S). If f ∈ Cg (left continuous with right limit process) such that 0≤s≤t |∆fs ∆Xs | < ∞ a.e., t > 0, then f is integrable relative to X and we have the evaluation as: Z t Z t X f dX = f − dX + ∆fs ∆Xs , t > 0. (31) 0

0

0≤s≤t

Proof. In view of Proposition 7.1.10 above, it suffices to establish the result if X is a semi-martingale which is locally finite satisfying condition (S). P 2 2 < ∞ since X can |∆X | ≤ E 2V (X) − X This implies that E s ∞ 0 s be taken to be a quasi-martingale and using the generalized Jordan decomposition, P it is integrable. Now let {σn , n ≥ 1} be an ε-chain for f and set fε = ∞ n=0 (f ◦ σn )χ[[σn ,σn+1 ) , which is integrable. Then Z

t

ε

f dX =

0

Z

t

(f ε )− dX +

0

=

Z

X

∆fsε ∆Xs

0≤s≤t t

(f ε )− dX +[f0 X0 +

0

∞ X

(f ◦ σn+1 −f ◦ σn )χ[σn−1 ≤t] ∆X ◦ σn+1 ].

n=0

Rt

(32)

But the T-limit of 0 (f ε )− dX exists as ε ↓ 0 since the integrand is in Cd and our previous theory applies to it. Thus the integral on the left of (32) exists as ε → 0, in the T-topology, if the second term [ ] in (32) has a limit. It is dominated by the expression

7.1 Basic Quasimartingale Spaces and Integrals ∞ X (f ◦ σn+1 − f ◦ σn )χ[σn+1≤t ] ∆X ◦ σn+1 + n=0

287

X ∆fs · ∆Xs

0≤s≤t {s6=σn }

∞ X X |∆fs ∆Xs |χ[|∆fs |≤ε]. ≤ (f − ◦ σn+1 − f ◦ σn )χ[σn+1≤t ] ∆X ◦ σn+1 + n=0

0≤s≤t

(33)

Now for any N > 0 the last term above is simplified as: X X |∆fs ∆Xs |χ[|∆fs |≤ε] = |∆fs ∆Xs |χ[|∆fs |≤ε] → 0, sup 0≤t≤N

0≤s≤t

0≤s≤N

P

a.e. as ε ↓ 0 for each N since 0≤s≤N |∆fs ∆Xs | < ∞ a.e. Thus the T-limit of the left term of (33) as ε ↓ 0 vanishes. The first term is simplified by (29) of Proposition 7.1.13 as follows. Let Fk = F(σk+2 −), in that proposition. Set Zk = ∆X ◦ σk+1 and Vk = (f − ◦ σk+1 − f ◦ σk )/ε, Ak = σk+1 . Thus |Vk | ≤ 1, Ak ↑, Fk−1 - adapted, and the inequality (29) gives the estimate for δ > 0, ∞ X i h (f − ◦ σn+1 − f ◦ σn )χ[σn+1 ≤t] ∆X ◦ σn+1 > δ P sup t≥0

≤ 16

≤ 16

n=0

ε 2 δ

ε 2 δ

E E

∞ X

ε |∆X ◦ σn |2 + 2 E δ n=0 ! X 2ε 2 |∆Xs | + K0 δ s

∞ X

n=0

|E(∆X ◦ σn | F(σn −)|

!!

where K0 > 0 is the variation bound coming from condition (S). Hence as ε ↓ 0 the right side quantity tends to zero. Consequently the bound of (33) goes to zero, and (32) implies that (31) and hence the integral of f (∈ Cg ) relative to X satisfying condition (S) is well-defined. The result follows. 2 The point of this theorem is that for integrands of Cg -elements, we decompose the right continuous parts of the integrand, and apply the integration properties of the elements of Cd , plus the jumps as in the classical RiemannStieltjes case. However it naturally has more details to be filled in. Some related points are given as remarks: Remark 7.3P Suppose X is a semi-martingale satisfying condition (S) and f ∈ Cd such that 0≤s≤t |∆fs ∆Xs | < ∞, i.e., the jumps are controlled. Then R (·) R (·) P (1) 0 f dX is a semi-martingale, since 0 f − dX and 0≤s≤t ∆fs ∆Xs both have that property (the last term having paths of bounded variation). R (·) (2) If X is a martingale, 0 f dX need not be a martingale now. If however, R (·) R (·) f ∈ C or Cg , then 0 f dX = 0 f − dX, and the integral reduces to the previously defined case.

288

7 Multiple Random Measures and Integrals

(3) If the integrand f is also a semi-martingale, then we have X

0≤s≤t



|∆fs · ∆Xs | ≤ 

X

0≤s≤t

 12 

|∆fs |2  

X

0≤s≤t

 12

|∆Xs |2  < ∞,

a.e.

(cf. Proposition 7.1.13(i)) and so if X satisfies condition (S), then the semimartingales are integrable in this extended sense. (4) The definition of integration of Cd -elements here is broader than the earlier one given by P.A. Meyer (1976), at least they are different for some elements of Cd (shown in Exercise 2). There are two distinct approaches of extending the above integrals to multidimensions, namely for product (random) measures with general integrands and to multi-parameter integrators (random fields) for the same type of integrands. Both are of interest and they will be considered in the next two sections highlighting the distinctions. If the random measure is nonnegative, then further specializations are possible as in Jacod (1975, 1977). Finally we briefly discuss a random Schwartz distributional approach for this problem, presenting a specialization to BM and white noise processes. (See also Chapter 8 further on this topic.)

7.2 Multiple Random Measures, Part I: Cartesian Products In this section we consider product random measures and their integrals motivated by the classical analysis culminating in the Fubini-Stone-Tonelli theorem and its multidimensional stochastic extension. In fact, in the classical case if (Ωi , Σi , µi ), i = 1, 2, and if S is the collection of measurable rectangles {A × B, A ∈ Σ1 , B ∈ Σ2 } which is a semi-ring or algebra according as Σ1 , Σ2 are σ-rings or algebras, then µ(A × B) = µ1 (A)µ2 (B) which is additive on S and if the µi are (sigma) finite, then µ has a unique extension to be a measure on σ(S). But this is not always true for random measures, which already indicates a need to find different methods. A procedure was indicated, and developed as a key result by Wiener, then a decisive extension by Itˆ o (both authors for B.M.) which we now present for random measures induced by the BM on the line R. Thereafter we intend to discuss integral representations of (continuous) martingales and other stochastic processes. Recall from Definition 1.3.3 that a random measure Z : A → Lp (P ) is a σ-additive function on disjoint sets, in the metric topology for p ≥ 0 which for p = 0 is convergence in probability. When p = 2, and Z has orthogonal values, then using inner product notation, we have (Z(A), Z(B)) = µ(A ∩ B) and µ is a measure dominating Z. We then have for all f ∈ L2 (S, A, µ) = L2 (µ),

7.2 Multiple Random Measures, Part I:

Cartesian Products

289

R

the integral Yf = S f dZ to be well-defined. Also the mapping f → Yf 2 satisfies (f, R g)µ = (Yf , Yg )L22 (P ) for all2 f, g ∈ L (µ), defining an isometry τ : f 7→ s f dZ between L (µ) and L (P ). Suppose that {Z(A), A ∈ A} is a system of Gaussian random variables with E(Z(A)) = 0 and Var Z(A) = E(|Z(A)|2 ) = µ(A). If S = R+ , A is its Borel σ-algebra, and A = (a, b] so that Z((a, b]) = Xb − Xa where {Xt , t ∈ R} is BM with scale parameter unity (i.e. E(Xb − Xa )2 = b − a > R0), then µ((a, b]) = b − a, µ being the Lebesgue measure, and we have A f dZ, as a Wiener integral. If Z1 , Z2 are a pair of random measures (Z1 = Z2 is allowed), it is possible to define Z b1Z b2 f (t1 , t2 ) dZ1 (t1 ) dZ2 (t2 ) = I(f ), f ∈ L2 (R+ × R+ , µ ⊗ µ). a1

a2

Rb If Yf (·,t2 ) = a22 f (t1 , t2 ) dZ1 (t1 ) as in Section 7.1 for simple f , then we have Yf (·,t2 ) to be a random process which need not be adapted to the filtraRb tion of Z2 , and so a11 Yf (·,t2 ) dZ2 is not defined by the one dimensional integrator of the preceding section and so a serious problem arises. To overcome this difficulty we need to consider, fixing the lower limits, and the upper ones to satisfy b2 > b1 ≥ 0, and then extend. Hence the deterministic integrand should be so adjusted that this filtration condition must satisfy with variable upper limits as in Volterra kernels in classical integration. This important step has been successfully implemented by K.Itˆo (1951) who defined a multiple Wiener integral which we now present and then obtain useful extensions of it in the next section for martingales and semi-martingales good enough for PDE and other applications. With the preliminary properties of Gaussian random measure indicated in Exercises 6.5.6 and 6.5.7, we introduce the multiple Wiener integral employing Itˆ o’s method, simplified by Hida and Hitsuda (1993) as follows. Consider a simple function f defined as: r Y X χ[sij ,sij+1 ) (ti ) (1) f (t1 , . . . , tn ) = ai1 ,...,in r>i1 >...>in ≥0

j=1

where the ij are integers from 0 to r and 0 = s0 < s1 < . . . < sr < ∞. Let Z(·) be the standard BM on R+ and for each simple function f of the form (1), consider an n-dimensional object termed, multiple Itˆ o–Wiener integral, introduced as: Z Z f (t1 , . . . , tn ) dZ(t1 ) dZ(t2 ) . . . dZ(tn ) Inr (f ) = . . . (R+ )n

=

X

r>i1 >···>in ≥0

ai1 ,...,in

r Y

j=1

[Z(sij+1 − Z(sij )].

(2)

The n-fold integral Inr (·) given by (2) does not depend on the particular representation of the simple function f of (1) which can be seen from the following

290

7 Multiple Random Measures and Integrals

observation. Using (2) and the mutual independence and stationarity of the increments of BM which are also centered, it follows that E(Inr (f )) = 0. For the same reason one has if n 6= m, since we can assume that Inr = In (fr ) and similarly Im (gr ) to have common division points (by taking some coefficients ai1 ,...,ij = 0), r′ E(Inr Im )

=

Z

∞

0

Z

t1

...

Z

tm−1

fn (t1 , . . . , tm )gn (t1 , . . . , tm ) dtm . . . dt1

0

0

= (f, g)µ δm,n ,

(µ = Lebesgue measure)

(3)

where δm,n is the Dirac delta function, and (·, ·)µ is the inner product of (·) L2 (Rn , µ) = L2 (µ). This shows that the mapping f → In = I(f ), f ∈ L2 (µ), is an isometry of L2 (µ) into L2 (P ). The right side integral in (3) being in the Lebesgue sense, it is unambiguously defined. The isometry implies that the multiple integral of (2) is also well-defined and does not depend on the representation (1) of f . Moreover, this isometry also implies that the integral is uniquely defined for all f which are norm limits of functions of the form (1). If we consider such f which are L2 (µ) limits of all Volterra kernels on n points, i.e., relative to f ∈ L2 (Rn , µ), let Hn be the corresponding space determined by {In (f ), f ∈ L2 (Rn , µ)}(= Hn ). It is of importance to note that if Ft = σ(B(s), s ≤ t), is the σ-algebra generated by the BM {B(s), s ≥ 0} on [0, t], completed for P , then {Ft , t ≥ 0} is a standard filtration. Also the functional I s (f ), for 0 ≤ s ≤ t, is Ft -adapted, since an f t (·) depends only on time points on [0, t]. The preceding discussion may be summarized in the following: Theorem 7.2.1 Let {B(t), t ≥ 0} be the standard BM with {Ft , t ≥ 0} as its generated (and completed, so standard) filtration. If I(·) is a functional defined by (2) for all Volterra kernels, then it is a linear mapping of L2 (R+ , µ) → L2 (P ) and is an isometry into, given by (3). Moreover if 0 ≤ t < t′ , and I t (f ) is de′ ′ fined by (2), and similarly I t (f ), then Int (f ) = E Ft (Int (f˜)) where f = f˜|[0,t] , the restriction and E Ft (·) is the conditional expectation which in L2 (P ) coincides with the orthogonal projection of L2 (P ) onto Hn of the definition given above. The Int (f ) can be represented as: Int (f )

=

Z tZ 0

0

t1

...

Z

tn

f (t1 , . . . , tn ) dB(tn ) . . . dB(t2 ) dB(t1 ).

(4)

0

The above (uniquely) defined n-fold integral has a number of important consequences which we now sketch and use later. It may also be noted that the only properties of Lebesgue measure µ used above are its monotonicity and σ-finiteness. Actually Itˆ o in his original result used this fact simply as a set of finite measure to be decomposable as a finite union of measurable sets of arbitrarily given ε-sizes. For simplicity we use the Lebesgue measure here and below. Also the existence of a BM X = {Xt , t ≥ 0} is obtained by the basic

7.2 Multiple Random Measures, Part I:

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291

Kolmogorov theorem which reduces this to the compatibility of all finite dimensional (Gaussian) distributions on R+ . Consequently the space (Ω, Σ, P ) that + carries the X process, can be taken to be Ω = RR , Σ = σ(Xt , t ∈ R+ ), with P as the Gaussian probability satisfying EP (Xt ) = 0, EP (Xs Xt ) = min(s, t). Here one can actually replace Ω by the space of all continuous real functions on R so that Σ is the cylinder σ-algebra and the finite dimensional distributions are centered Gaussian with covariance r(s, t) = min(s, t). This result will be taken as known (cf., e.g. Chapter 1 of Rao (1995) for all details and extensions). The resulting L2 (Ω, Σ, P ) or L2 (P ) is the basic object under study. Our aim is to decompose this space as a countable direct sum of subspaces each of which is determined by a multiple Wiener-Itˆo integral, called the (Wiener) chaos expansion. This profound result is facilitated by means of Hermite polynomials as an orthogonal basis of L2 (µ) and then showing the desired decomposition of L2 (P ) by means of the isomorphism given by (3) restated in terms of Hermite polynomials instead of Volterra kernel functions introduced as follows. [Now the fact that µ is Lebesgue is decisive.] In Exercise 6.5.7 we recalled Hermite polynomials and their generating functions as well as the recurrence formula. We also announced there about the direct sum decomposition of L2 (P ) where (Ω, Σ, P ) is determined by a standard BM, X = {Xt , t ≥ 0}. Here we amplify afresh this decomposition using the multiple Wiener integrals given above which characterize the component spaces intrinsically. The key Hermite polynomial use goes back to Kakutani. As seen in Exercise 6.5.7, the nth order Hermite polynomial and its generating function are given in a slightly more general form as: 2 n 2 x d x (−t)n exp . n exp − , t > 0, x ∈ R, (5) Hn (t, x) = n! 2t dx 2t where n = 0, 1, 2, . . . , with parameter t (t = 1 was considered there), and G(t, λx) =

h λ2 t i λn Hn (t, x) = exp λx − . 2 n=0 ∞ X

(6)

Its recurrence relation is well-known to be: Hn (t, x) −

x t Hn−1 (t, x) + Hn−2 (t, x) = 0, n n

n ≥ 2.

(7)

R x2 , t > 0, A is Borel in R, then the {Hn (t, ·), n ≥ 0} form If ν : A 7→ A e− 2t √dx 2πt a complete orthonormal system in L2 (R, ν) as a standard computation shows. A k-dimensional version of thisQ statement is also true with a similar argument k by considering all k-products { i=1 Hni (ti , xi ), ni = 0, 1, 2, . . . , t1 , . . . , tk > 0} relative to νk (·) given by (let x = (x1 , · · · , x) ∈ Rk and x′ its transpose below):

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7 Multiple Random Measures and Integrals

νk (A1 × · · · × Ak ) =

Z

1

−1

e− 2 (xRk

A1 ×···×Ak

x′

k

1

(2π)− 2 (det Rk )− 2 dx1 · · · dxk

where Rk = (tj δij , 1 ≤ i, j ≤ k), the νk is product of ν’s above. Thus these products form a complete orthonormal system in L2 (Rk , νk ). We also have the following important relations: Theorem 7.2.2 (a) For 0 ≤ s ≤ t and n ≥ 1 the Hermite polynomials Hn and the n-dimensional Wiener integrals are connected as: Z tn−1 Z t Z t1 kn (t1 , . . . , tn )dB(tn ) . . . dB(t1 ), ... Hn (t − s, B(t) − B(s)) = s

s

s

(8) where k(·, . . . , ·) is the Volterra kernel given by (1) for s ≤ tn < . . . < t1 ≤ t. (b) For a countable dense set {t1 , t2 , . . .} of R+ , the increments {B(ti ) − B(tj ), i, j = 1, 2, . . .} generate the same σ-algebra (completed), Σ, as determined by the filtration {Ft , t ≥ 0} of the BM {B(t), t ≥ 0}. (c) The space L2 (P ) is expressible as the direct sum of multiple Wiener integrals given as: ∞ M Hn , H0 = R (9) L2 (Ω, Σ, P ) = n=0

R t1

R tn−1

where Hn = sp{ 0 . . . 0 kn (t1 , . . . tn )dB(tn ) . . . dB(t1 ) : tn ≥ . . . ≥ t1 ≥ 0}. [However, there exist f ∈ L2 (P ), defining a process {f (t, ·), t ≥ 0} which are not adapted to the (standard) Brownian filtration {Ft , t ≥ 0}.] Proof. We sketch the general ideas, as it involves some fine combinatorial identities which are somewhat involved. (a) Consider the rectangular parallelepiped in Rn namely Qn = [0, T ]n , where T (> 0) is the right end point, and set S = {(t1 , . . . , tn ) : 0 ≤ t1 < t2 < . . . < tn ≤ T }. We have defined (cf.(2)) the integral as: Z Z (10) Y (S) = . . . dB(t1 ) . . . dB(tn ), S

and observe as before that Y (·) is symmetric since B(·) is such. We follow the sketch closely from Engel (1982), the omitted nontrivial details can be found in his memoir. We divide Qn into disjoint subsets indexed by partitions π of the integers {1, 2, . . . , n} and denote by Pn all those partitions. Thus π ∈ Pn if π = (π1 , . . . , πk ), πi ⊂ {1, . . . , n} disjoint and ∪ki=1 πi ⊂ {1, . . . , n} = Pn (say), such a π has rank k(= r(π)). Now let |π| stand for the vector (card π1 , card π2 , . . . , card πk ). The set Pn will have an equivalence relation on (1, 2, . . . , n) by declaring i ∼ j (mod (π)) if i, j ∈ πl , for some l = 1, 2, . . . , k, and π ∈ Pn . Now let ∆(π) = {(t1 , . . . , tn ) ∈ Qn : ti = tj if i ∼ j mod (π)}, so that Qn = ∪π∈Pn ∆(π), as a disjoint union. Let Qn = {(t1 , . . . , tn ) ∈ Qn : ti 6= tj if i 6= j}. We need to calculate (or

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293

count) the disjoint ti -elements, i.e. χQn in terms of χ∆(π) . If π s is the partition of {1, . . . , n} that has just singletons, then χQn = χ∆(πs ) and the general counting is not simple and in fact one finds (as shown in Engel (1982), Theorem 5.1) that there exists an integer valued function c : Pn → N, such that X c(π)χ∆(π) , c(T s ) = 1. (11) χQn = π∈Pn

From (10), the fact that Y (I1 × . . . × In ) = Y (Ii1 × . . . × Iin ) by the symmetry where (i1 , . . . , in ) is a permutation of (1, . . . , n), one gets Z Z Z Z 1 (12) dB1 (t1 ) . . . dB(tn ) = . . . dB(t1 ) . . . dB(tn ), ... n! S Qn where S and Qn are given above. A nontrivial combinatorial argument (detailed in Engel (1982)) shows that c(π) = c(n1 , . . . , nk ) = c(n1 )c(n2 ) . . . c(nk ), Qk and c(n) = (−1)n−1 (n − 1)! so c(π) = (−1)n−k i=1 (ni − 1)! and (11) becomes X

χQn (t1 , . . . , tn ) =

(−1)n−k

k Y

(ni − 1)!χ∆(π) (t1 , . . . , tn ),

i=1

π∈Pn {π=(π1 ,...,πk )=(n1 ,...,nk )}

(13) which uses the nontrivial combinatorial counting. From (12) and (13) one deduces Z Z . . . dB(t1 ) . . . dB(tn ) 0≤t1 <...
1 = n! n

X

n−k

H(n1 , . . . , nk )(−1)

1 +···+nk =n

k Y

i=1

(ni − 1)!

Z k Y

i=1

T

!ni

dB(s)

0

(14)

where H(n1 , . . . , nk ) is the number of distinct partitions of (1, 2, . . . , x) into k-subsets of sizes (n1 , . . . , nk ). In the BM case we use the fact that Z

0

t

dB(s)2 = L2 (P ) − lim

qk X

(k)

B(Ii )2 = t,

i=1

and

Z

0

t

(dB(s))k = 0 for k ≥ 3.

In this case the right side of (14) becomes [ x2 ] n! 1 X (−1)q T q B(T )n−2q = Hn (T, B(T )), = n! q=0 (n − 2q)!2q q! which is the nth Hermite polynomial (cf. (5)) above that gives (8) and establishes (a). The reason we included the general argument is to state below that the method is valid for all processes with independent stationary increments

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7 Multiple Random Measures and Integrals

having two moments. In particular the expression (14) is valid for Poisson processes with independent stationary increments. In that case a corresponding simplification of the right side gives Poisson-Charlier polynomials which we state precisely later to bring out various properties of multiple random measures. (b) This property is easy. Indeed, the measure space (Ω, Σ, P ) is that given by the Kolmogorov existence theorem starting from the compatible collection of finite dimensional distributions of the BM, so that Σ is the (completed) σ-algebra determined by the cylinder sets. But every coordinate function ω ∈ + RR that is continuous, determines Bt (ω) = ω(t), so that for any open set O and t ∈ R+ , let tn → t where tn ∈ R+ . Then we have {ω : ω(t) ∈ O} = ∪k≥1 {∩n>k (ω(tn ) ∈ O)} ∈ σ{ω(t), t ∈ J ⊂ R} where J is a countable dense set of points tn ∈ R+ since such tn ∈ J can be chosen. Hence Σ is determined by σ{ω(t) : t ∈ R+ } as well. Thus both σ-algebras are the same. Note that + so far we only used the fact that the ω ∈ RR are right continuous for this work. (c) Consider a fixed dense denumerable subset D ⊂ R+ , and for any n ≥ 1, let Dn = (t1 , . . . , tn ) ∈ D and let Fn = σ{(B(ti+1 ) − B(ti )), ti ∈ Dn }. Then σ(∪Fn , n ≥ 1) = Σ when completed, as seen above. If f ∈ L2 (P ), then {fn = E Fn (f ), n ≥ 1} is a martingale and E(|fn |2 ) ≤ E(|E Fn (f )|2 ) ≤ E(|f |2 ). Hence fn → f˜ a.e. and in mean. Since Σ is generated by the Fn , it results from the standard martingale analysis that f = f˜ a.e. But the independent increments property of the BM implies that Fn being generated by B(tl ) − B(tl−1 ), 1 ≤ l ≤ n, fn is a function of these variables, and by Theorem 1 and (b) above, it is a function of the Hermite polynomials of B(tl ) − B(tl−1 ). In other words, we have obtained (cf. (7) and below) X aI1 ,...,in−1 Hi1 (t2 − t2 , B(t2 ) − B(t1 )) . . . fn = i1 ,...,in−1

. . . Hin−1 (tn − tn−1 , B(tn ) − B(tn−1 )).

(15)

Hence fn belongs to the subspace generatedL by these n-fold products of Hermitian polynomials, say Hn and hence f ∈ n Hn since fn → f a.e. and in L2 (P ). But we have noted that these generate the whole space and so equals it. Since f ∈ L2 (P ) is arbitrary, we conclude that (9) must hold. Finally we note that not every function f ∈ L2 (P ) which defines a process {f (t1 ·), t ≥ 0} is adapted to the filtration {Ft , t ≥ 0} although it is measurable relative to the completed σ-algebra σ(∪t≥0 Ft ) = Σ. Consequently there are Rt o integral is not defined functions f = ⊕n≥1 fn , for which 0 f (s)dB(s) as an Itˆ since f (t) is “anticipative” i.e., is adopted to Ft+ε , ε > 0 and not if ε = 0. So we can only formally set Z ∞ Z X f= . . . Ki (t1 , . . . , ti )dB(ti ) . . . dB(t1 ), (16) i=0

7.2 Multiple Random Measures, Part I:

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295

where the Ki (·) are Volterra kernels, but the limit f itself is not measurable for the desired σ-algebra and thus not Itˆ o integrable. 2 The last part of the preceding result raises a question of an extension of the Itˆ o integral to all functions which are representable as a series (14). Since every element of L2 (P )(= L2 (R+ , B(R+ ), P )) where P is determined by the BM can be so representable, we obtain an extended integral which is also of interest in this analysis. The next result is a technical step in such an extension procedure, given for n-different independent differential (i.e., also independent increment) processes, which include both the BM and the Poisson classes. It forms a useful step in applying the L2,2 -boundedness principle to obtain the corresponding multiple integral quite generally. The following is the desired result: Theorem 7.2.3 Let {Xj (t), t ≥ 0}, j = 1, . . . , n be real valued stochastic systems satisfying the following conditions on a given probability space (Ω, Σ, P ): 1. E(Xj (t)) = mj (t), where each mj is locally bounded and of finite Vitali variation for j = 1, 2, . . . , n (here locally means, on each compact interval [0, T ] ⊂ R+ ). 2. If I1 , . . . , Ik is any set of disjoint subintervals of [0, T ], and i1 , . . . ik is a permutation of integers of {1, 2, . . . , n}, then {Xi1 (I1 ), . . . , Xik (Ik )} forms an independent system where Xj (I) = Xj (t) − Xj (s) for I = (s, t] ⊂ [0, T ]. [This is an abstraction of the BM, and thus each Xj (·) process must be a ‘differential’ one.] ˜ j (t) = Xj (t) − mj (t), then the variance of Xj is a continuously increas3. If X ing function on [0, T ]n , T > 0. Let for each elementary set A ⊂ [0, T ] of the form [ A= ci1 ,i2 ,...in Ii1 × Ii2 × . . . × Iin

(17)

1≤i1 <...
where I1 , . . . , Iq is a partition of [0, T ] into disjoint intervals defining A and Ij ≺ Ij+1 (the points of Ij are to the left of those of Ij+1 , i.e. “less” using the ordering of R+ ) and where ci1 ,...,in = 1 or 0 according as the corresponding rectangle Iii × . . . × Iin is included or not in the union defining A, let X (18) Y (A) = ci1 ,i2 ,...in X1 (Ii1 ) . . . Xn (Iin ), 1≤i1 <...
so that Y (·) is additive on the class ES of such elementary sets of the form S = {(t1 , . . . , tn ) : 0 ≤ t1 < t2 < . . . < tn ≤ T } ⊂ [0, T ]n .

Then there exists a measure λ : ES → R+ controlling Y (·) in the sense that Z 2 ! 2 kY (A)k2 = E dX1 (t1 ) . . . dXn (tn ) ≤ λ(A), A ∈ ES . (19) A

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7 Multiple Random Measures and Integrals

We now make a key application of this result, whose proof will be given thereafter, in obtaining the σ-additive extension of Y and the integral of f ∈ L2 (P ). Thus let f : S → R be an elementary function defined on sets of the form (17). They are the elementary sets forming a ring ES . The mapping Y : ES → L2 (P ) is additive and hence the integral Z Z Z X f (s)dY (s) = dXn . . . dX1 (20) ... ci1 ,i2 ,...in S

I1

1
In

which, being a finite sum, is well-defined, and does not depend on the representation of f . By Theorem 7.2.3 (especially (19)) there exists a measure λ : ES → R+ , such that for some constant C > 0, Z 2 ! Z E f (s)dY (s) ≤C |f (s)|2 dλ(s). (21) S

S

This holds for f = χA and by linearity for simple functions. Thus the mapping T : L2 (S, ES , λ) → L2 (P ) is continuous and linear on λ-simple f . Hence it has a unique norm preserving extension to the closure of such functions. But it will be seen (in the proof of Theorem (7.2.3)) that ES determines the Borel σ-algebra of R+ , and hence T is defined uniquely on L2 (λ) into L2 (P ), and is L2,2 -bounded. This also implies by the same property that Y (·) is σadditive and integrates all functions of L2 (λ). Although λ is not necessarily unique, the extension of T agrees on the common domains and is unambiguous. Here a nontrivial argument is needed in showing that the σ-algebra determined by the algebra ES is the Borel σ-algebra of Qn . A direct proof of σ-additively of Y (·) on σ(ES ) is quite involved, and it was given by Engel (1982) using combinatorial arguments and constructions. The L2,2 -boundedness appears to be direct and shorter. This application is summarized in the following: Theorem 7.2.4 Let {Xi (t), Ft , t ≥ 0}, i = 1, . . . , n, be a system of adapted processes satisfying Conditions 1 to 3 of the preceding theorem, and define Y : ES → L2 (P ) by (18) which is additive. Then the (additive) integral given by (20) uniquely extends to all bounded Borel or σ(ES )-measurable functions, and can be considered as defined in the Dunford-Schwartz sense. Its domain contains functions that are subject to the dominated convergence criteria. The last statement implies that L2 (R+ , B(R+ ), λ) is contained in the domain of definition of integrals relative to Y . The key implication is the σ-additivity of Y . We therefore present the details of: Proof of Theorem 7.2.3. Let A be an elementary set of the form (17), and define the function Y (·) by the equation (18). Let mi (t) = E(Xi (t)) and ˜ i = Xi − mi : [0, T ] → L2 (P ), so that X

7.2 Multiple Random Measures, Part I:

h kY (A)k22 = E

X

ci1 ,...in

1≤i1
h =E

X

ci1 ,...,in

1≤i1 <...
P′

Cartesian Products

n 2 i Y ˜ Xj (Iij ) + mj (Ij )

297

j=1 n

X′

˜ j (Iij ) ˜ j1 (Iij ) . . . X X k 1 k

n−k Y l=1

k=0

2 i mkl (Iikl )

where denotes the sum of all possible (j1 , . . . , jk , k1 , . . . , kn−k ) for all the choices 1 ≤ j1 < . . . < jk ≤ n and 1 ≤ k1 < . . . < kn−k ≤ n, where {j1 , . . . , jk , k1 , . . . , kn−k } range over the set {1, 2, . . . , n}. Here the summing ˜ j (Iij )’s. Now using Conin k has n + 1 terms each having exactly k of the X ˜ j ′ (Ii ′ ) whose means are zero ˜ j (Iij ) and X dition 2, the independence of X l j l l l Pm P m 2 allows us to conclude that (using | i=1 fi | ≤ Cm i=1 |fi |2 , Cm > 0, a constant) kY (A)k22 ≤ Cn

n X X ′ k=0

 2  n−k Y X   ˜ ˜ mkl (Iikl )  . ci1 ...in Xj1 (Iijk ) · · · Xjk (Iijk ) E  1≤i1 <...
(22)

˜ j (Iij ) are orthogonal, being independent with zero We now use the facts that X k k means, and that the functions mkl (·) have finite (Vitali) variations on [0, T ], so that there is an absolute bound M depending only on n but not on k, q or A, satisfying (23) sup |mki |([0, T ]) ≤ MT < ∞. ki

˜ j (Ik )|2 ), a finite measure, We get from (22) and (23), on writing µj (Ik ) = E(|X with the following domination property: kY (A)k2 ≤ M Cn

n X X ′ k=0

+ |m1 |(Ii )] Qn

X

ci1 ...in [µ1 (Ii )1

1≤i1 <...
Y [µ(Ii ) + |mn |(Iin )] ×

[µn (Iin ) + |mn |(Iin )] = M · Cn · ν(A).

Here ν(A) = i=1 (µi + |mi |)(A), which is the product of the measures (µi + |mi |)(·). If we set λ(·) = M Cn · ν(·), then by the classical Fubini theorem ν (and hence) λ is a (locally finite) measure implying (19). 2 Remark 7.4 If E(Xi (t)) = 0 for each i, as in the case of BM, the work above can be shortened. However, since we would like to include the Poisson case which will have mean measures (6= 0), the additional work done is needed. We outline another procedure in an exercise. The fact that each element of L2 (P ) is expandable as a series of orthogonal terms each of which is a multiple Wiener integral in the case that

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7 Multiple Random Measures and Integrals

X1 = X2 = . . . = Xn = B, the BM, has more information and one then should have special properties of the Y -integral of these elements. This however depends on the particular processes Xk (·) that we use in the above theorems. This involves additional analysis and the fact that the process f (t, ·) is adapted to the given standard filtration {Ft , t ≥ 0}. The cases that it is nonanticipating and anticipating have to be considered separately and nontrivially. Indeed Theorem 7.2.2 shows that there are many anticipating processes for which some extension of Itˆ o’s analysis is needed and is possible. An interesting consequence of the preceding work is the following integral representation of a.e continuous square integrable martingales adapted to the Brownian filtration as an Itˆ o integral: Theorem 7.2.5 Let {Ft , t ≥ 0} be the standard filtration determined by the BM so that Ft = σ(Bs , s ≤ t) and completed for P in (Ω, Σ, P ). If Y = {Yt , Ft , t ≥ 0} is a square integrable (continuous) martingale in the sense that t 7→ Yt is path-wise a.e. continuous, then there exists a (measurable) process f = {f (t, ·), Ft , t ≥ 0} such that Y can be represented as: Z t Yt = f (s, ·)dB(s), t ≥ 0, (24) 0

where this is an Itˆ o-integral relative to the BM {B(s), F(s), s ≥ 0}. The same result holds if L2 (P ) is replaced by L1 (P ) for (24) above.

Proof. Since Yt is Ft -adapted and is in L2 (Ft ) for each t > 0, we consider its (unique) representation as a Wiener-Itˆ o series and then construct the process f (s, ·) from it so that the procedure is somewhat different from the familiar methods. Thus for a fixed t0 ≥ 0, Yt0 in L2 (F) is representable using a family of Volterra kernels Kn , as: Yt0 =

∞ Z X

n=1

t0

0

Z

t1

...

Z

tn−1

Kn (t0 , t1 , . . . , tn )dB(tn ) . . . dB(t1 ).

0

0

Since {Yt , Ft , t ≥ 0} is a martingale, we have for 0 ≤ t ≤ t0 , Yt = E Ft (Yt0 ), by definition, and Yt ∈ L2 (Ft ). Now E Ft is also a contractive, hence an orthogonal projection, and so has the (unique) representation as the above series with t0 replaced by t, it follows that the kernel Kn (t0 ; t1 , . . . tn ) = Kn (t; t1 , . . . , tn ) is independent of t0 or t. Thus the functions f (t, ·), t > 0, defined by f (s, ·) = k1 (s) +

∞ Z X

n=2

0

s

Z

t2

... 0

Z

tn−1

kn (s, t2 , . . . , tn )dB(tn ) . . . dB(t2 )

0

Rt is Fs -adapted for s ≥ 0, f (s, ·) ∈ L2 (Fs ) and satisfies Yt = 0 f (s, ·)dB(s), as an Itˆ o-integral. Note that an L2 (P ) Wiener or BM martingale is a.e. continuous.

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299

[Also the L1 (P ) case detailed below uses truncation which applies to L2 (P ) and is of interest.] Let us extend the result if square integrability is replaced by just integrability of the martingale relative to the Brownian filtration. First observe that such a martingale is necessarily (path wise) continuous. Indeed let t0 ≥ 0, N ≥ 1 be arbitrary and consider the truncation as: = Yt0 χ[|Yt0 |≤N ] + N χ[Yt >N ] − N χ[Yt0 <−N ] , YtN 0 and set Y N = {YtN = E Ft (YtN ), 0 ≤ t ≤ t0 } which is bounded, hence square 0 integrable. But then by the first part we have Z t N f N (s, ·)dB(s), 0 ≤ t ≤ t0 . Yt = 0

Now {YtN , Ft , t ≥ 0} is a continuous L2 (P )-martingale. The continuity for simple processes is clear from the continuity of the BM, and the general case follows from the classical maximal inequality for (sub-) martingales, as follows. Since {(YtN − Yt ), t ≥ 0} is a martingale so that {|YtN − Yt |, Ft , t ≥ 0} is a submartingale, the maximal inequality for it implies P [ sup |YtN − Yt | > λ] ≤ 0≤t≤t0

1 E(|YtN − Yt |) → 0, λ

as N → ∞. So the uniform continuity for (t0 > 0) implies the continuity of the Yt -process. Here the continuity of the BM-process is crucial. Next we let Tk = inf{t : |Yt | > k}. Then {Tk , k ≥ 1} is an increasing stopping time sequence of the filtration and the classical optional stopping transformation of (sub-) martingales implies that {Yt∧Tn , Ft , t ≥ 0} is a martingale, and so Rt by the first part there is a process ftn such that Yt∧Tn = 0 f n (s)dB(s), and R t∧T E 0 n (f n − f n ′ )2 (s)dB(s) = E(Yt∧Tn − Yt∧Tn′ )2 → 0 for large enough ′

n < n′ and f n (s) = f n (s) for 0 ≤ t ≤ t0 (still n < n′ ). Hence the process Rt Rt f (t, ·) = f n (t, ·), n ≥ 1, defines f (t, ·), and 0 f (s, ·)dB(s) = 0 f n (s, ·)dB(s) = Yt∧Tn (= Yt , t ≤ t0 ). Since t0 > 0 is arbitrary, the representation (24) holds for all t > 0 and the Brownian represented martingale is also continuous. 2 It should be observed that a process {f (t, ·), Ft , t ≥ 0}, obtained as an orthogonal sum of multiple Wiener integrals, has nice properties but the sum function need not be adapted to the Brownian filtration. These constitute a sufficiently rich collection that demand an extension of the Itˆ o integral. The problem naturally is related to the concepts of “anticipation” and “nonanticipation” of a process in L2 (P ) relative to the Brownian filtration. Thus if Σ is the σ-algebra generated by {B(t), t ≥ 0}, the BM and then completed relative to P , consider a pair of times 0 ≤ a ≤ b < ∞ and let Fab = σ{(B(t) − B(a)) : a ≤ t ≤ b < ∞} ⊂ Σ = σ{B(t), t ≥ 0}, also

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7 Multiple Random Measures and Integrals

completed, then a process {fab , 0 ≤ a ≤ b < ∞} ⊂ L2 (P ) is called L2+ [L2− ] adapted for the BM if it is Fab -measurable and (i) {ftb , 0 ≤ t ≤ b} is Fab -adapted for all b, [fat , 0 ≤ a ≤ t < ∞, all a ≥ 0] Rb Rb (ii) a E(|ftb |2 )dt < ∞, b ≥ 0, (respectively a E(|fat |2 )dt < ∞, b > 0).

These will be referred to as L2+ (and L2− )-classes. In the early analysis, based on the Itˆ o theory, we have studied the class of L2 -processes with the same BM filtration. For the L2+ -classes the following nontrivial modifications, detailed by Berger and Mizel (1978), are needed. Theorem 7.2.6 Consider the set An (a, b) = {(t1 , . . . , tn ) : a ≤ t1 ≤ . . . ≤ tn ≤ b} ⊂ [a, b]n . Then for an L2+ -adapted process f there exists uniquely a sequence ϕn (a, b; ·) ∈ L2 (An , λ), 0 ≤ a ≤ b, n = 1, 2, . . . , λ = Leb. meas., such that fab ∈ L2 (P ) has the orthogonal expansion (0 ≤ a ≤ b < ∞): ∞ Z X b b fa = E(fa ) + ϕn (a, b; t1 , . . . , tn )dB(t1 ) . . . dB(tn ) = E(fab ) +

n=1 Z b

(25)

An (a,b)

ψ(a, t, b)dB(t)

a

where ψ(a, t, b) is Fat -measurable a.e., for a ≤ t ≤ b, and ψ defined by the above infinite sum satisfies ! Z t Z b 2 2 E |ψ(t, τ, b)| dt < ∞; E |ψ(a, t, b)| dt < ∞, (26) 0

a

for almost all t (Leb.), 0 ≤ t ≤ b. Proof. If we define a shifted BM as β(t) = B(a + t) − B(a), for 0 ≤ t ≤ b − a, then β(·) is again a BM and by substitution of this into series (25) we have the representation of the homogeneous chaos case of Theorem 3 above with m(t) = 0 and Xi = B for all i = 1, . . . , n. So some rewriting and identifications with that result will give this one. We omit the details. 2 To analyze the elements of L2+ , some further restrictions are needed. For example, the following additional conditions are considered by Berger and Mizel (1979, 1980) for the analysis of L2+ class which we state here to indicate what needs to be done for anticipative functions. Using the same notation of the theorem, it is assumed that: ∂E(f b )

(i) the scalar ∂a a and the vector or strong L2 -derivative (both partial), ∂ϕn b ∂a (a, b, ·) should exist for fa of (25);

7.2 Multiple Random Measures, Part I:

Cartesian Products

301

(ii) ϕ(a, b; a) is continuous for 0 ≤ a ≤ b, and for n ≥ 2, ϕn (a, b; a, . . .) ∈ L2 (An−1 (a, b], λ), 0 ≤ a ≤ b. (iii) Using (i), the resulting series XZ ∂ ∂ E(fab ) + ϕn (a, b; t1 , . . . , tn )dB(t1 ) . . . dB(tn ) ∂t1 ∂a An (a,b) n≥1

should exist in L2 (P ), and converge to an L2+ adapted f˜ab . (iv) The series in (25), modified as XZ ϕ1 (a, b; a) + ϕn (a, b, a, t1 , . . . , tn−1 )dB(t1 ) . . . dB(tn−1 ) n≥z

An−1 (a,b)

should converge in L2 (P ) for any 0 ≤ a ≤ b to an element of L2+ . These conditions are listed to show that, for a serious analysis, one has to consider new restrictions for the anticipative case. The latter analysis is needed for stochastic differential or integral equations arising in applications. We omit this specialization here. In the more interesting, but somewhat simpler, nonanticipative case our Theorems 7.2.3 and 7.2.4 are applicable not only for BM, but for discrete processes such as Poisson. It is desirable in this analysis to describe a corresponding orthogonal series representation which involves in place of Hermite polynomials, the Poisson-Charlier polynomials. This opens up the possibility of several other orthogonal polynomials in our analysis and result in many chaos representations and their particular analyses. We discuss the key Poisson case. The Poisson-Charlier polynomials Kn (·, ·) are given by the formula: n 1 X u n q q Kn (u, t) = (−1) t q! , t > 0, u = 0, 1, 2, . . . n! q=0 q q

(27)

and the generating function G(u, t; w) is obtained after a computation for sufficiently small |w|, having the uniqueness property, as: X G(u, t; w) = wn Kn (u, t) = e−tw (1 + w)u . (28) n≥0

A further reasonably straightforward but tedious calculation shows G(u, t1 , λ)G(v, t2 , λ) = G(u + v, t1 + t2 ; λ) which on comparing the coefficients of λn on both sides gives the recurrence relation for these polynomials (analogous to the Hermitian case) as:

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7 Multiple Random Measures and Integrals

Kn (u + v, t1 + t2 ) =

n X

Kq (u, t1 )Kn−q (v, t2 ).

(29)

q=0

There is a relationship between these polynomials and the Hermite and other polynomials such as Jacobi and Laguerre (cf. e.g., Szeg¨ o (1958)). For instance, we may state the following from the latter source for information as well as possible extensions with the respective chaos expansions for interested readers. Using (27) for Kn (u, t) and (5) for Hn (t, x), t > 0, x ∈ R, u = 0, 1, 2, . . ., we have the following relation if u − 12 > n, n = 0, 1, 2, . . ., √ Z 1 √ 1 n!(−1)n Γ (u + 1) (1 − v 2 )u−n− 2 H2n (v u)dv. Kn (u, t) = √ n 1 πt Γ (u − n + 2 )Γ (2n + 1) −1

These polynomials are orthogonal, satisfy (29) and also ∞ X

Km (q, t)Kn (q, t)

q=0

tq −t tn e = δm,n , t > 0. q! n!

(30)

We use these polynomials in obtaining the discrete homogeneous chaos for Poisson processes in place of the continuous case with BM. Recall that the Poisson random measure Z is an integer valued nonnegative additive set function on Σ with independent values on disjoint sets such that P [Z(A) = n] =

(µ(A))n −µ(A) e , n = 0, 1, 2, . . . n!

(31)

¯ + is a σ-finite measure, called a rate function. Now taking where µ : Σ → R Σ as the Borel σ-algebra of R and µ = λ, the Lebesgue measure, one has n −(t−s) Xt = Z((0, t]), and P [Xt − Xs = n] = (t−s) , t > s, with X0 = 0, n! e a.e.. Then {Xt , t ≥ 0} is called a differential process. Hence one can obtain that (in fact this follows from the above conditions), P [Xt+0 − Xt−0 ≥ 1] = 0 and for b > 0, P [sup0≤t≤b (Xt+0 − Xt−0 ) ≥ 2] = 0, E(Xt ) = µ((0, t]) = t. Note that since E(Xt ) > 0 (= m(t) in general), the abstract case presented in Theorem 7.2.3 above will be needed here. We state the corresponding representation for the general discrete homogeneous chaos for a comparison. Let Pn be the set of f ∈ L2 (P ) which are expressible as: f=

X

0≤n1 +···+nq ≤n

ai1 ,...,nq

n Y

i=1

(Z(Ai ) − µ(Ai ))ni

(32)

where µ(Ai ) = E(Z(Ai )), A1 , . . . , Aq are disjoint elements of µ-finite measure from Σ and ai , . . . , aiq , real. Let P0 be the set of constants so that Pi is a strictly increasing sequence and if Qn = Pn − Pn−1 , n ≥ 0, P− = {0} then R = ∪n≥0 Pn is dense in L2 (P ) with Σ generated by the Ztn −filtration. Since

7.3 Multiple Random Measures, Part II:

Noncartesian Products

303

Pn = ⊕ni=0 is an orthogonal sum one can show that f ∈ Pn is precisely of the form n Y X (33) Knl (Z(Al ), µ(Al )), an1 ,...nq ∈ R, f= an1 ,...nq l=1

and A1 , . . . , Aq are disjoint sets from Σ of finite µ-measure, n1 + · · · nq = n. Here the Kn (·, ·) are the Poisson-Charlier polynomials. For comparison, let µ be Lebesgue measure and, D(t) = Xt − t so that E(D(t)) = 0 and set D(t) have independent stationary increments, just like the BM. Then we can state for comparison of both results in terms of Hermite and Poisson-Charlier polynomials, the following: Theorem 7.2.7 (a) Let {Xt , Ft , t ≥ 0} be the BM process. Then Z Z . . . dX(t1 ) . . . dX(tn ) = Hn (X(T ), T ),

(34)

0≤t1 <...
where Hn (·, t) is the nth Hermite polynomial with parameter t > 0. ˜ ˜ (b) Let {X(t), F(t), t ≥ 0} be the differential process for a Poisson class with Lebesgue measure as its rate-function. Then Z Z ˜ 1 ) . . . dX(t ˜ n ) = Kn (X(T ), T ), . . . dX(t (35) 0≤t1 <...
where Kn (·, t) is the nth Poisson-Charlier polynomial with parameter t > 0. It should be noted that the complete details of (33) involve many nontrivial combinatorial arguments corresponding to those needed for the BM case. (For details, see Engel (1982), p.22 ff.) But it is interesting that the orthogonal polynomials with special functions are essential in these chaos decompositions, both of which were created by N. Wiener. Perhaps other orthogonal functions may likewise be considered. We next turn to non cartesian product random measures, to contrast with the above case.

7.3 Multiple Random Measures, Part II: Noncartesian Products So far we considered multiple products dX1 , . . . dXn in developing the integration theory where Xi (t) is a BM or a (semi-) martingale depending on a filtration {Ft , t ≥ 0}. However, if {X(t), t ∈ R2+ } then the basic ordering of R is not available any more and there exist several partial orders in R2+ and serious problems start at this stage itself. We consider some methods of interest to extend the previous theory.

304

7 Multiple Random Measures and Integrals

Since n-parameter random processes {xt , t ∈ Rn } can be studied if initially the two-parameter case is well-understood, here we concentrate on the latter. The analysis for the discrete parameter case i.e., Rn is replaced by Zn , was first considered by Cairoli (1970) for the martingale type processes (more precisely fields) and then an important extension of it was presented by Cairoli and Walsh (1975) for Brownian random fields. This study was continued by Brennan (1978, 1979) for (quasi-) martingales which was further extended by Green(1995, 1997). Some aspects of these works will be detailed to expose the evolving subject and its importance here. For simplicity let the index be the unit square of R2 , denoted S = [0, 1]2 . Thus consider X = {Xt , t ∈ S} ⊂ L20 (P ) as a semi-martingale on a two parameter index set. It is desirable to associate a (signed) measure µX (·) for the random field X and find conditions for its σ-additivity which dominates the stochastic measure associated with X, to define an integral. This will lead to the corresponding line integrals, Green-Gauss-Stokes theorems and various differential forms opening up a deep random analysis study on manifolds. The basic extension when X is a two parameter BM was considered by Cairoli and Walsh (1975) motivated by a desire to establish the holomorphic correspondent for this case. In the classical study one uses the Lebesgue (planar) measure and the related (monotonic but not shift invariant) measure. In the non-finite dimensional case this may be provided by the BM, and it was one of the reasons for the pioneering and penetrating study by these authors. For the corresponding differential forms and analysis and more general fields than BM, a study was initiated by Brennan (1979) and advanced by Green (1997), extending the BM case, which will be detailed here. Since there is no unique linear ordering of the index set S = [0, 1]2 ⊂ R2+ , we consider a partial ordering s = (s1 , s2 ) ≺ t = (t1 , t2 ) to mean s1 ≤ t1 , s2 ≤ t2 for s, t ∈ S and the strong ordering with strict inequalities, denoted ‘≺≺’. If 0 = si0 < si1 < . . . < sin = 1, i = 1, 2, then the set g = {(s1i , s2j ) : 0 ≤ i ≤ n, 0 ≤ j ≤ n} is termed a grid for S and let GS be the collection of grids in S. If sij = (s1i , s2j ) ∈ g and f : S → R is any mapping, then the increment (∆f )(sij ) = f (s1j , s2j ) − f (s1j , s2j ) − f (s1i , s2j ) + f (s1i , s2i ) denotes the value of f on the rectangle sij = (s1i , s2i+1 ] × (s1j , s2j+1 ]. The unit square is partitioned into a finite disjoint collection of such half-open rectangles so that they form a semi-ring under the union, intersection, and the difference being expressible as a finite disjoint collection of the same type of elements, as in classical measure theory. Thus for a (complete) probability space (Ω, Σ, P ), a collection of σ-algebras {Ft , t ∈ S}, where S is the partially ordered index set as above, is a standard filtration if each Ft is P -complete, and for s ≺ t one has Fs ⊂ Ft as well as Fs = ∩s≺≺t Ft . Since there are several types of variations of functions of two (and more) variables, as discussed already by Clarkson and Adams (1933), we find the following two concepts from their list to be of use in our work, and are based on Vitali, Arzela and Hardy (abbreviated here as

7.3 Multiple Random Measures, Part II:

Noncartesian Products

305

V,A and H). Also denote the right and upper boundaries of S by Γ1 and Γ2 where Γ1 = {(s, 1) : 0 ≤ s ≤ 1} and Γ2 = {(1, t) : 0 ≤ t ≤ 1}. Finally the total ‘upper’ and ‘lower’ boundaries are Γu = Γ1 ∪ Γ2 and Γℓ = {(s, t) : s = 0, 0 ≤ t ≤ 1 or t = 0, 0 ≤ s ≤ 1}. With this necessarily involved notation we now introduce: Definition 7.3.1 Let {Xt , Ft , t ∈ S} ⊂ L1 (P ) be an adapted random field (or process with two dimensional indexing). Then it is called respectively: (i) a V -process (or field) if KV = sup{E

X

|E (∆g Xt | Ft ) | : g ∈ GS } < ∞,

(1)

and a V -martingale if KV = 0, (also called a weak martingale by some); (ii) an A-process (or field) if KA < ∞ where ! ) ( n−1 X E Xti+1 −Xt Fti : 0 ≺ t0 ≺ . . . ≺ tn ≺ (t, 1) , KA = sup E i

i=0

(2) and a (planar) martingale if KA = 0, (stronger than the V -martingale concept); (iii) an H-process if it is a V -process and the linear indexed boundary processes {Xt , Ft , t ∈ Γ1 } and {Xt , Ft , t ∈ Γ2 } are just A-process as in (ii) above, i.e., ordinary semimartingales. It is a normal H-process if Xt = 0 a.e. for t ∈ Γ1 ∪ Γ2 . There is a simple relation between the V,H,A-processes, and it will be useful to associate a set function with an adapted X = {Xt , Ft , t ∈ S} by considering it as a function X : S × Ω → R with the class S = B([0, 1]) × Σ = {(s1 , s2 ] × A : (s1 , s2 ) ∈ S, A ∈ Σ}, which is a semi-ring (actually a semi-algebra). Here we define µX (·) for each adapted X as: Z (3) µX ((s1 , s2 ] × A) = (∆X)((s1 , s2 ])dP A

where ∆X is the increment (having four terms) defined above, and seen to be additive on the semi-ring. Hence its variation, denoted |µx |(·) is an additive nonnegative set function on S. Clearly the mapping A 7→ µX ((s1 , s2 ] × A) is σ-additive but µX itself need not be σ-additive. Now, V,H,A processes X with the associated µX are connected as follows: Proposition 7.3.2 Let X = {Xt , Ft , t ∈ S} be an adapted random field in L1 (P ). Then (a) it is an H-process if and only if it is both a V- and an A- process, in the sense defined above, and (b) if it is a V -martingale, then it will be a planar martingale if and only if the restriction to the right and upper boundaries Γ1 and Γ2 are (one parameter) martingales.

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7 Multiple Random Measures and Integrals

The proof follows by juggling the definitions and is left to the reader. It is also detailed in Brennan (1979), and the result is given here to motivate the structural analysis of the class. In our analysis the V-processes are important and sufficiently general. The next result on its structure is thus of interest in our study. Part (b) here may be considered as special kind of the Doob-Meyer and Jordan type decompositions: Theorem 7.3.3 Let X = {Xt , Ft , t ∈ S} be an adapted L1 (P )-process. Then the following structures are true for these classes: (a) X is a V-process if and only if its associated set function µX by (3) satisfies |µX | ((S − Γℓ ) × Ω) < ∞ where Γℓ is the lower boundary (or the boundaries of the coordinate area), and then this value equals = KV of (1). (b) If X in (a) is also right continuous (in the ordering) in L1 (P )-mean, then it admits a decomposition as X = M + Y − Z where M is an L1 -right continuous V-martingale, Y and Z are right continuous in mean nonnegative H-processes which vanish on the right and upper boundaries Γu , and are V-submartingales and A-supermartingales at once. [Here X is naturally called a V-submartingale if for s ≺ t in S, E((∆X)(s, t]) ≥ 0, a.e. (V-supermartingale if the inequality is reversed), and X is an A-submartingale if E Fs (Xt ) ≥ Xs a.e. and similarly Asupermartingale is obtained by reversing the inequality.] Proof. (a) It is useful to note that the key part played by the V-processes is clarified by the following facts. They have the martingale property in the partial order ‘≺’ if the result is true on the upper boundaries. Indeed, let {Xti , Ft , t ∈ S}, i = 1, 2 be a pair of V-martingales and suppose that Xt1 = Xt2 , a.e. for t ∈ Γu . Then Xt1 = Xt2 a.e. for t ∈ S. In fact let t = (t1 , t2 ) ∈ S then by the martingale property 2 2 2 2 1 1 1 Ft (X(1,t a.e. Xt1 = E Ft (X(1,t 2 ) + X(t2 ,1) − X(1,1) ) = Xt 2 ) + X(t1 ,1) − X(1,1) ) = E

As a consequence or even directly, it is seen that a V-martingale is a planar martingale if it is a one parameter martingale when the first and second coordinates are fixed on the upper and right boundaries since the bounds KV1 = KV2 = 0 in (1). For the variation of X and its additive function µX (·), let g ∈ GS be a grid of S, and X be a V-process. Consider for t ∈ g, the sets At = {ω : E Ft (∆g Xt )(ω) ≥ 0} and its complement Act . Then on G0 = S − Γℓ we have

7.3 Multiple Random Measures, Part II:

E

X E Ft (∆g Xt ) t∈g

!

≤E =

X t∈g

X t∈g

Noncartesian Products

307

!

E Ft (|∆g Xt |)

E(∆g Xt (χAt − χAct ))

! Z ∆g Xt dP + ∆g Xt dP c A A t t t∈g Z X Ai ∆X(s1 ,s2 ) dP : ∪Ai × (s1i , s2i ] ∈ S} ≤ sup{ i i

X Z ≤

A

ti ∈g

= |µX |(S × Ω),

so that one has the variation KV ≤ |µX |(S × Ω). For the opposite inequality, let A = ∪ni=1 (s1i , s2i ] × Ai , Ai ∈ Σ. We may assume S0 = ∪ni=1 (s1i , s2i ] with some Ai = ∅ if necessary. Let us take 0 = v0i < i , i = 1, 2 where vj1 contains all s1i and vj2 all s2i . So the grid v1i < . . . < vm i 1 2 gjk = (vj , vk ] is of the form (s1i , s2i ] ∩ gjk ∩ g(j+1)(k+1) , or= ∅. Let Bjk = ∪{Ai : the above intersection is nonempty}. Then set A = ∪ni=1 (s1i , s2i ] × Ai = ∪i,k gik × Bjk so that the right side representation refines the first one. Hence n X X µX (s1 , s2 ) × Ai ≤ |µX (gjk × Bjk )| i i i=1

j,k

≤

XZ

Bjk

j,k



Fgjk × Bjk dP E

 X F ≤ E E gjk (∆Xgjk  ≤ KV . j,k

Taking the supremum on all the grids on the left we get |µX |(S × Ω) ≤ KV and hence the variations agree as asserted, establishing (a). (b) The above part allows us to simplify the discussion. For the given Vprocess X = {Xt , Ft , t ∈ S}, consider the mapping Qt : Ft → R given by Z (4) Qt (A) = µX ((t1 , (1, 1)] × A) = (∆X)(t, (1, 0)]dP, t ∈ S, A ∈ Ft . A

¯t = Then Qt ≪ P and let X and

dQt dP .

¯ t = 0 for t ∈ Γu , Now note that X

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7 Multiple Random Measures and Integrals

µX ((s1 , s2 ] × A) = µX ((s1 , (1, 1)] × A) − µX s11 , s12 ), (1, 1) × A −µX s12 , s22 ), (1, 1) × A + µX (s2 , (1, 1)] × A Z ¯ 1 , s2 ]dP = µX¯ ((s1 , s2 ) × A) ∆X(s =

(5)

A

¯ ¯ is a V-process. Also X ¯ is so that µX = µX¯ , X(t) = 0 for t ∈ Γu , and X ¯ an H-process. In fact by Definition 7.3.1, X is a normal process. Since X is ¯ Indeed for right continuous, we can take a right continuous modification of X. t ∈ S − Γu and A ∈ Ft , since X is right continuous, we have Z Z ¯ t dP = ∆X((t, (1, 1])dP X A A Z = ′lim ∆X((t′ , (1, 1)])dP, by right continuity of X t ≻≻t A Z Z ¯ t dP = ¯ t+ dP = ′lim X X (6) t ≻≻t

A

A

¯t = X ¯ t+ . Hence also M = X − X ¯ by the right L -convergence of X, so that X 1 is a right continuous (in L -metric) process, a V -martingale. We must find a ¯ to fulfill the assertion of (b). decomposition of X For this purpose we associate a measure with X using its increments, for each s ∈ S, as Qns : Fs → R by the expression:  n 2X −1 Z j i j i  , (7) dP : Aij ⊂ A, s ≺ , , Qns (A) = sup  ∆X n 2n n 2n Aij 2 2 i,j=0 1

and if no Aii ⊂ A, we set Aij = ∅. Then we observe that Qns ≪ P and is a measure satisfying Qns (A) ≤ Qn+1 (A) ≤ |µX |(s, (1, 1)] × A) < ∞, for s n ≥ 1, A ∈ Fs . Let Qs (A) = limn Qns (A), A ∈ Fs . Then Qs (·) exists as an additive function. It is actually σ-additive by the Vitali-Hahn-Saks theorem and is bounded. We now assert that |µX |(s, (1, 1)] × A) = Qs (A). Indeed, for s ∈ S, A ∈ Fs and ε > 0, we can find a grid {rij } ≻ s such that Aij ⊂ A, Aij ∈ Frij (take Aij = ∅ if no such Aij ⊂ A exists), satisfying X Z (8) ∆Xrij dP + ε. |µX |((s, (1, 1)] × A) ≤ Aij i,j k l k l k−1 l−1 n Set rij = 2kn , 2ln if rij ∈ , 2n , 2n . By the right con2n , 2n , 2n , 2n tinuity of X, there exists an n0 (= n0 (ε)) such that n ≥ n0 ⇒ X X Z n n dP + ε ≤ Q (A) + ε. ∆Xrij ∆Xrij dP ≤ s Aij i,j

i,j

The preceding two inequalities imply, since ε > 0 is arbitrary, the result that − |µX |((s, (1, 1)] × A) = Qs (A), A ∈ Fs and |µ0 | = µ+ X + µX by the classical

7.3 Multiple Random Measures, Part II:

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309

Jordan decomposition as well as that both µ± X satisfy the hypothesis of (a). Thus Z Z ¯ s dP = µX¯ ((s, (1, 1)] × A) = (Ys − Zs )dP, A ∈ Fs , X (9) A

A

dµ+ X dP Fs ,

¯ 1 X Zs = dµ where Ys = dP Fs , and the Ys , Zs - processes are L -right continuous since t 7→ |µX |((t, (1, 1) × A) has that property. Now by (a), Y, Z can be taken normal H-processes so that in this form they are unique. 2 The decomposition given in the above theorem helps to find conditions on the X-process in order that the associated additive set function µX may be σ-additive. However the two parameter V-martingale M need not easily give a σ-additive function, contrary to the one-parameter case. To explain the difficulty we now introduce the ‘marginal’ processes related to X and discuss some consequences. 1 Definition 7.3.4 Let 2X = {Xt , Ft , t ∈ S} be an adapted process, Ft = σ ∪t2 >0 F(t1 ,t2 ) and Ft = σ ∪t1 ≥0 F(t1 ,t2 ) be the ‘sectional’ σ-algebra families on t1 and t2 . If X is in L1 (P ), it is called a weak martingale provided i i E Fs1 (X((s1 , s2 ]) = 0 a.e., it is an i-martingale if E σ(Fs1 ) (X(s1 , s2 ]) = 0, i = 1, 2 1 2 and a strong martingale if E σ(Fs ∪Fs ) (X((s, t])) = 0, a.e.

The weak martingale concept is a natural extension of the one-parameter version - namely the increments are conditionally (and hence generally also) centered. In the linear index case we saw that a square integrable martingale satisfies Bochner’s L2,2 -boundedness principle and so qualifies to be a stochastic integrator as seen before. But an example due to Bakry (1981) shows that such a result in the two parameter case is no longer true. We shall introduce a suitable condition that overcomes this difficulty, and allows a development of (stochastic) line integrals leading to Green-Gauss-Stokes type results for the Wiener-BM, and certain other developments towards many analogs of differential forms. We hasten to add here that in multiple indexed processes, there is another BM, called the L´evy BM, which is of interest in many applications. However, it is not a martingale, as opposed to the Wiener BM which is a martingale. We discuss the L´evy-BM later for comparison and point out some associated new and important consequences of it. These two are a pair of distinct generalizations of the same one dimensional Brownian motion process which is always an inexhaustible source of new ideas. Turning to the Wiener BM, we have the following simple property established by Wong and Zakai (1976) in the planar case: Proposition 7.3.5 Let X = {Xs , Fs , s ∈ S} ⊂ L1 (P ) be an adapted process, and the standard filtration {Fs , s ∈ S} satisfying the technical condition denoted (+) (also called F4 -in Cairoli and Walsh and elsewhere) which is a 1 2 commutativity of the projections E Fs , E Fs where Fs1 and Fs2 are presented in

310

7 Multiple Random Measures and Integrals

Definition 7.3.4 above and is equivalent to demanding that the σ-algebras Fs1 and Fs2 be conditionally independent given Fs . [One verifies without much diffi1 2 2 1 culty that E Fs (f ) = E Fs (E Fs (f ))(= E Fs (E Fs (f )) for any essentially bounded measurable f ∈ L1 (P ) and Fs1 ∩ Fs2 = Fs .] Under (+), the X process is a weak martingale if and only if it can be decomposed as Xs = Xs1 + Xs2 where {Xsi , Fsi , s ∈ S}, i = 1, 2 are i-martingales. 1 If X is a weak martingale, then condition (+) implies that X(s,t) = F(s,t) E (X(s,t) ) is a 1-martingale as defined above, and then one verifies that 2 1 X(s,t) = X(s,t) − X(s,t) has the property of a 2-martingale, so that X(s,t) = 1 2 X(s,t) − X(s,t) is the desired decomposition. The converse part that every such decomposition under (+) is a weak martingale follows easily from definition. We leave the details to the reader. The result is stated here to draw the reader’s attention to the fact that although a martingale with a onedimensional parameter set under standard conditions defines a random measure, as is well-known from earlier work, the sum which is a weak martingale, even under the restriction (+), fails to define a similar measure. Hence (by Bakry’s counterexample) one needs further restrictions, in order that the planar martingale X determines such a measure to develop a reasonable theory. In view of this (Bakry’s) example, a straight forward extension of stochastic integration to defining a random measure from a random field is not always possible. Just as in the one dimensional time, in the planar (and higher dimensional) case several types of decompositions of planar (and higher) fields exist and their use in the integrals involve new conditions and details. To get a general view, we now obtain a suitable L2,2 -boundedness condition of Bochner’s. The particular methods based on the Brownian motion – and the Wiener BM as well as the L´evy BM are different in higher dimensions. The former gives the martingale extension and the latter a definitely non BM field and here we concentrate on the L2 (P )-martingales and their extensions (thus keeping the martingale structure in view), but both will satisfy an L2,2 -boundedness prescription. Let M2T be the class of the right continuous square integrable planar martingales on T = [0, 1]2 relative to the partial ordering ‘≺’ introduced above. The following maximal inequalities play a significant role in the analysis which extends (nontrivially) the Doob one parameter result.

Theorem 7.3.6 Let X = {Xt , Ft , t ∈ T = [0, 1]2 } be a process which is separable relative to a dense denumerable collection D ⊂ T , taken for instance as the dyadic set D = {(i2−n , j2−n ), 0 ≤ i, j ≤ 2n } so that for each (t1 , t2 ) ∈ T there exists a sequence (tn1 , tn2 ) ∈ D such that lim(tn1 ,tn2 )→(t1 ,t2 ) X(tn1 ,tn2 ) = X(t1 ,t2 ) a.e. so, D is the present version of the set of separability. Also suppose that the filtration {Ft , t ∈ T } satisfies for the above technical condition (+) of Proposition 7.3.5 and using the ordering ≺ on T , consider the bounds defined for the given process X:

7.3 Multiple Random Measures, Part II:

( "

Φj (X) = sup E Φ

n−1 X i=0

F E ti (Xti+1

Noncartesian Products

311

# ) − Xti )) : ti ≺ ti+1 , ti ∈ Γi

i = 0, . . . , n − 1, j = 1, 2, (10)

and ( "

!# ) X F E t (∆g X) ΦV (X) = sup E Φ : g is a grid on T .

(11)

t∈g

If Φ(x) = x log+ (x) or = |x|p , p > 1, and for nontriviality Φj (X) < ∞, j = 1, 2 and ΦV (X) < ∞, defined in (10) and (11), then the following upper bounds hold for a given λ > 0: (a) λP sup |Xt | > λ ≤ a1 + b1 E(Φ(X(1,1) )), p = 1, (12) t∈T (13) (b) E sup |Xt |p ≤ ap + bp E(Φ(X(1,1) )), p > 1, t∈T

the constants aj , bj in (12) and (13) depend only on ΦV , Φ1 , Φ2 of (10), (11). In particular if X ∈ M2T so that X = X (1) + X (2) (hence X ∈ L1 (P )) and when again conditions (+) of Proposition 7.3.5 (conditional independence) is satisfied, then the adapted i-martingales X (2) , X (1) satisfy (12) and (13) if and only if the {Φj }2j=1 and ΦV of (10) and (11) are finite. Outline of Proof. The complete details of proof are long and are available from the papers of Brennan (1979) for (a), and Green (1997 and 1998) for (b). The basic ideas will be sketched here referring the reader for complete details to these papers. In the case of one-dimensional time parameter t, the above inequalities were established by Doob (1953) and with the separable set D [in fact any countable dense set will do as well], the result is extended nontrivially as follows. Observe that any adapted integrable sequence {Xn , Fn , n ≥ 1} can be (uniquely) decomposed into a martingale {Mn , Fn , n ≥ 1} and an integrable ‘predictable’ sequence {An , Fn , n ≥ 1} where A0 = 0, Xn = Mn + An and the Fn−1 adaptability of An which is called ‘predictable’. Indeed with A0 = 0, let An = E Fn−1 (Xn ) − Xn−1 + An−1 , n ≥ 1, and Mn = Xn − An . This is (the classical Doob decomposition) seen to be Xn = Mn + An as desired, and if {Xn , Fn , n ≥ 1}, is a submartingale then An ↑ and conversely. [The computation may be found in many textbooks (e.g. cf. Rao (1984), p. 184).] This construction in the present two-parameter case takes the following form. Let Φ(x) = x log+ x, x ≥ 0 and note that x ≤ Φ(x) + e, (e is the base of log, log e = 1) and for x, y ≥ 0, Φ(x + y) ≤ c[Φ(x) + Φ(y)] for some constant c > 0 since Φ is a “Young function”, satisfying the ∆2 -condition, Φ(2x) ≤ CΦ(x), x > 0, for this Φ as well as Φ(x) = |x|p . Let Aij = 0 if i = 0 or j = 0 or both

312

7 Multiple Random Measures and Integrals

for i, j ≥ 1. Set Fk,ℓ = F 2kn , 2ℓn for these dyadic numbers from the index set D⊂T j−1 i−1 X X E Fk,ℓ (∆X( kn , ℓn ) ), 1 ≤ i, j ≤ 2n . (14) Aij = 2

2

k=0 ℓ=0

Now define Mij = X( kn , ℓn ) − Aij analogous to the one-dimensional case. Let2 2 ting ∆Mij = Mi+1,j+1 − Mi,j+1 − Mi+1,j + Mij , the two-dimensional increment, it is seen that E Fij (∆Mij ) = 0 by the martingale property and moreover, P2n −1 P2n −1 k=0 ℓ=0 E(|∆Akℓ |) ≤ Φn + e, by (11). By another computation of these expressions, on setting Mij1 = E Fij (M2n ,j ), Mij2 = E Fij (Mi,2n ), and Mij3 = −E Fij (M2n ,2n ), one gets Mij = Mij1 + Mij2 + Mij3 . {Mij1 ,

(15)

n

Then Fij , 0 ≤ i ≤ 2 } can be verified to be a martingale for each j, 0 ≤ j ≤ 2n , and hence one has the maximal inequality from the classical results. Thus with the one-parameter (sub) martingale analysis, it can be seen that (the details are standard but not trivial, given in Brennan (1979)), using the special condition (+), which is the conditional independence of the filtration at this point, one gets: 1 P sup |Mij1 | > λ ≤ (a1 + b1 E(Φ(M2n ,2n ))). λ ij

(16)

where Φ(x) = x log+ (x), λ > 0, and the constants a1 , b1 depend only on Φ1 and Φ2 of (10). A similar analysis and consequently the bounds can be obtained for M 2 and M 3 . Since X( in , jn ) = Mij + Aij (see (14) and the next line), the 2 2 inequalities (16) (and the corresponding ones for Mij2 , Mij3 ) imply for λ > 0, n i 1 j o P sup |Xt | : t ∈ > λ ≤ (a2 + b2 E(Φ(M2n ,2n ))). , n n 2 2 λ

From this using the separability of {Xt , t ∈ T } and the fact that D ⊂ T is dense, one can finally obtain (12) for Φ(x) = x log+ x, x > 0. Next use the growth condition Φ(x + y) ≤ 2p−1 (Φ(x) + Φ(y)), p > 1, on the convex function Φ(x) = |x|p , (and x ≤ Φ(x) + 1) one can proceed in the same way and show (with the discretization D ⊂ T ) that the decomposition (15) holds. The numerous (nontrivial) details are given by Green (1998), and again using the special condition (+)—the conditional independence—of the given filtration one can establish the inequality of the type (16), to obtain (13) where again one has to use the separability of the process and density of D in T. The details are long and nontrivial, but are similar to the above. Consequently the reader may fill in the details or look up in Green’s (1998) paper for the complete account.

7.3 Multiple Random Measures, Part II:

Noncartesian Products

313

Finally, let X ∈ M2T and have the decomposition X = X (1) + X (2) and by the right continuous martingale hypothesis E Ft (∆g Xt ) = 0, a.e. for any grid g with t ∈ g. Then conditions (10) hold as verified below. By symmetry of the evaluations of Φ1 , Φ2 , it suffices to consider one, say Φ1 . X Φ1 (X) = sup E[Φ( |E Ft (∆g X)(t) | : g is a grid on T ] = 0 t∈g

since each item inside vanishes for martingales. Then n−1 X

Φ1 (X) = sup E[Φ(

i=0

|E Fti (Xti+1 − Xti )| : ti ≺ ti+1 , ti ∈ T ] = Φ1 (X (2) ), (1)

using the 1-martingale property of the decomposition of Xt . The Xt part vanishes. Interchanging 1 and 2 here shows that both the Φi (X) are finite. The converse direction is immediate. 2 Remark 7.5 1. The growth conditions on Φ should be both Φ(2x) ≤ cΦ(x), x > 0 and its derivative Φ′ (x) ↑ ∞ at least as fast as log x. Cairoli gave an example showing that for a multiple index martingale, uniform integrability is not sufficient for convergence, in contrast to the linear time dimension. Thus there are new surprises in multidimensions. 2. For the classical one parameter (sub) martingales, bounded in Lp (P ), 1 < p < ∞, the marginal inequality (13) has no constant ap (i.e. ap = 0). In the present generality the same result holds if moreover each of the Aij of (14) is bounded. More precisely m−1 X

sup E[(

i=1

|E Fti (Ati+1 − Ati )|)2 : 0 ≤ t1 < · · · ≤ tm ≤ 1, ti ∈ Γj ] < ∞ (17)

where Γj , j = 1, 2 are the lower and left boundaries of T = [0, 1]2 . Here Ati = Xti − Mti of (14), where {Mti , ti ∈ D} is the martingale part of the Xti and thus Ati is the finite variation part of the (discretized) Xt , t ∈ D, process assumed separable relative to the dyadic dense set D of T. A slightly more general result when Xt is replaced by a Cauchy sequence {Xtn , t ∈ T } for each of which the corresponding {Ant , t ∈ D} satisfies (17) is considered by Green(1998), and the details may be found there. Discussion: For continuing the analysis of a V-process X it is desirable to find conditions that the associated additive set function µX of (3) is σ-additive. This has a key role to play in showing that X induces a random measure, or equivalently it will be a stochastic integrator. The simple decomposition (14) again is seen to be important in this work, just as in the one

314

7 Multiple Random Measures and Integrals

parameter case the corresponding result was a precursor of the Doob-Meyer (sub-) martingale decomposition in the continuous parameter version. Our plan is (i) to establish σ-additivity of µX if the V -process X satisfies a modified Dirichlet condition, to be called (D′ ) here, or a local version of it (D′ L), (ii) (i) show that the {Xt , t ∈ T }, i = 1, 2, martingales satisfy a new “crossterm domination” condition, and (iii) the resulting V -process obeys an L2,2 boundedness principle of Bochner’s. That will also apply to the corresponding two-parameter (generalizable to multiparameter case thereafter) stochastic integrals. The details are many, and we shall sketch the main points referring the reader to the basic work on the subject by Cairoli-Walsh (1975), and the extension (so far the most general one known) with the L2,2 -boundedness in Green (1997), based on the prior analysis of Brennan’s (1979). Here we intend to present a birds-eye-view of the subject for further analysis later. The key concepts for introducing a successful stochastic integral are set forth by the following: Definition 7.3.7 Let {Xt , Ft , t ∈ T = [0, 1]2 } be an adapted process. Then (i) it is said to be of [extended (Dirichlet)] class D′ if given ε > 0 there exists aSδε > 0 such that for each grid g of T and sets {Bt ∈ Ft , t ∈ g} such that P ( t∈g Bt ) < δ, then we have X Z < ε. ∆ X dP (18) g t t∈g

Bt

[If T is replaced by R2+ , the positive orthant of R2 , then the process is said to be of class(D′ L), or locally of class (D′ ), if it is of class (D′ ) on every such compact sub-rectangle of R2+ . Since we consider just the compact T , this extension is not used and noted only for reference. Another way of stating this is if {Xt , Ft , t ∈ T } is right continuous, then the collection with dyadic set D⊂T   n −1  2X |E Fij (∆Xij )| : n ≥ 1 is uniformly integrable.   i,j=0

(ii) The process {Xt , Ft , t ∈ T } is cross-term dominated if there is a constant C > 0, such that   ! n X X 2 2 E  (19) αi X(si ,ti ] , αi αj X(si ,ti ] X(si ,tj ]  ≤ CE i=1 1≤i,j≤n

for all predictable simple functions of the form ti ≤ 1.

Pn

i=1

αi χ(si ,ti ] , 0 < si ≤

The above concepts are of interest in extending the B.M. (Wiener) type without assuming that its quadratic variation process is deterministic, which

7.3 Multiple Random Measures, Part II:

Noncartesian Products

315

was stipulated by Cairoli and Walsh ((1975), p. 150) in their fundamental paper to get a stochastic Green formula, and we shall obtain an analog here for the more general nondeterministic (or stochastic) variations. The preceding definition plays a key role in that work. That is of real interest here although naturally it will not give as sharp a result as with the more stringent assumption. We first have the following key assertion due to Brennan generalizing the classical Doob-Meyer decompositions on its relation to the σ-additivity of the associated set function µX . Proposition 7.3.8 Let X = {Xt , Ft , t ∈ T } be an L1 (P )-right continuous V -martingale. Then it belongs to class (D′ ) if and only if it is decomposable as X =M +A (20) where M = {Mt , Ft , t ∈ T } is a V -martingale and A = {At , Ft , t ∈ T } is a process of bounded (Vitali) variation. This holds if and only if µX , of X as defined in (3) is σ-additive. If moreover, the filtration has the conditional independence property, and X is an i-martingale i = 1, 2 then X has the cross-term domination property of Definition 7.3.7 as well. ′ Sketch of S Proof : If X is of class (D ), let Bij ∈ Fij , (i, j ∈ Dn ), such that P ( i,j Bij ) → 0, as the dyadic divisions refine when n ↑ so that by (18) Z X X Z Fij E (∆X( in , jn ) ) dP → 0 ∆X( in , jn ) dP ≤ S 2 2 2 2 Bij Bij i,j

i,j

uniformly as n → ∞. Thus µX being of finite S variation implies that |µX |(· × ·) is continuous in the second variable i.e. on Fij , and by the L1 (P )-continuity the result S holds for all t ∈ T = [0, 1]2 . This gives the σ-additivity of µX . Indeed mn (sni , tni ] × Bin ↓ ∅. Choose rin so that sni ≺≺ rin ≺≺ tni and set let Bn S = i=1 mn Cn = i=1 (rin , tni ] × Bin and to satisfy |µX |(Cn ) + 2εn > |µn |(BT n ), for given n ε > 0, which is possible by the L1 (P )-right continuity. If Fn = k=1 Ck ↓ ∅, then |µX |(Bn ) ≤ |µX |(Fn )+ε. From this a standard argument shows (assuming the opposite and get a contradiction) that limn→∞ |µX |(Bn ) = 0 implying the σ-additivity of µX on the σ-algebra P of predictable sets of the form Bn above. For the converse, let B ∈ Σ, (i.e. a measurable set) and by the standard martingale convergence theorem one shows for sn = ( 2in , 2jn ) ∈ Dn ⊂ T that limn→∞ E Fsn (χB ) exists a.e., and = YB (s, ·) where for j+1 s ∈ ( 2in , 2jn ), ( i+1 2n , 2n ) , one has YB (s, ·) = lim

n→∞

n 2X −1

E Fij (χB )(χ(a,b] (s))(·), a.e.

i,j=0

Here (a, b] is the right-closed rectangle containing ‘s’ defined above. Now using the σ-additivity of µX and some nontrivial computations (referring to Brennan’s

316

7 Multiple Random Measures and Integrals

1979 work for details) one shows the existence of a process {A(t), t ∈ T } such that A(t) = 0 a.e. on the lower boundary Γℓ of T, E(A(1, 1)) = µX ((T −Γℓ )×Ω) is finite and A is of finite variation. If µ+ X is considered, there is the corresponding A+ such that (∆A+ )(s, t] ≥ 0 for s ≺ t in T. Moreover in this case Z Z + YB dµX = ∆A+ (s, t] dP. (s,t]×Ω

B

From this relation with some additional (nontrivial) analysis one obtains the desired decomposition X = M + A, the (Doob-Meyer type) decomposition, wherein A will be of bounded variation, and one can express a V -process X which is L1 -right continuous in class D′ as, X = M + Y − Z where M is L1 right continuous V -martingale and Y, Z are nonnegative L1 -right continuous H-submartingales. The last part is obtained from this decomposition when the special conditions (+), the conditional independence of the filtration or (F4 ), is satisfied and X (i) is an i-martingale, so that X has the cross-term domination property. Indeed for rectangles (si , ti ] ∩ (sj , tj ] = ∅ one has a vertical or horizontal separating lines so that Fsi ∨sj ⊂ Fs1i or ⊂ Fs2j and that for bounded Fsi -measurable random variables αi , i ≥ 1, one has : E[αi αj X(si ,

ti ] X(sj , tj ] ]

= E[αi αj E Fsi ∨sj (X(si , = E[αi αj E

Fsi ∨sj

(E

Fs1 i

ti ] )E

Fsi ∨sj

(X(si ,

(X(sj ,

ti ]) )E

Fsj ∨sj

tj ] )]

(X(sj ,

tj ] )]

= 0. This gives the desired cross-term domination. 2 We need to consider this nontrivial detailed argument to obtain the higher dimensional stochastic integration and the related random measures. Indeed the point of the above decomposition is to show that higher order or multiple integration can be developed to random measures that are more general (hence include more applications) than Wiener B.M. only, using the generalized Bochner boundedness principle. In this way the following statement, due to M.L. Green (1997), is of interest here (we assume just the conclusion of the last part of the above proposition and not its hypothesis): Theorem 7.3.9 Let X = {Xt , Ft , t ∈ T = [0, 1]2 } be an adapted process which is a 1- (or 2-) martingaleP that is cross-term dominated. Then for each simple function of the form f = ni=1 αi χ(si , ti ] where αi is Fi -adapted is thus a predictable simple function, (si , ti ] ∩ (sj , ti ] = ∅, i, j = 1, . . . , n there is an absolute constant C > 0 and a (non-cartesian product generally) measure µ on T × Ω such that R R (21) E[( T f dX)2 ] ≤ C T ×Ω |f |2 dµ.

7.4 Random Line Integrals With Fubini and

Green-Stokes Theorems

317

In particular if X is a square integrable V -quasimartingale whose decomposition given by Proposition 7.3.8 as X = X (1) + X (2) + A

(22)

holds, where A is a predictable process of bounded variation and the X i , i = 1, 2 are i-martingales having the cross-term boundedness property (which holds if the filtration obeys conditional independence condition), then X is L2,2 -bounded so that the inequality (21) obtains. This key property is of importance in our work and uses the σ-additivity of the measure µX given in Proposition 7.3.8. It has many details to fill in. The result was established by M.L. Green (1997), pp. 83–93, providing the necessary details which are essential and the reader is referred to it. We discuss its consequences here to show its utility. Note that the integral defined above for (predictable) simple functions f ∈ L2 (T × Ω, µ) can be extended to their linear combinations and then by density to the closure of all such functions which is actually all of L2 (T ×RΩ, µ). The random measure Z : B(T ) → L2 (P ) determined by Z(A) = T χA dX or Z(B × A) = R T ×Ω χB×A dX, will thus define Z(·), and its σ-additivity in norm. This is of interest in our analysis. Now a specialization using X(·) itself will be studied in considering line integrals and related matters which will be discussed next.

7.4 Random Line Integrals With Fubini and Green-Stokes Theorems The preceding analysis leads to a study of geometric stochastic integration extending the classical Green and Stokes Theorems. For this it is necessary to obtain a form of the (stochastic) Fubini Theorem after formulating the basic line integrals in this context. The preceding section and the analysis there may be considered as a form of the stochastic surface integrals. This can be regarded as a counterpart of geometric integration of considerable interest stretching into other parts of analysis and embracing the random (linear) distributions and currents. Here and in the next section we present an outline of some of these interesting aspects, with a deeper account of geometric analysis itself to be considered in Chapter 8. Much of the classical analysis on this subject is tied to Lebesgue’s measure and the corresponding work for us is the Wiener measure as shown by Cairoli and Walsh (1975). Their basic work can be presented for more general martingale-type measures with some (even) simplifications using the Bochner boundedness principle as discussed in the preceding sections, especially with the L2,2 -boundedness aspect. This may be advanced to Lϕ1 ,ϕ2 -boundedness in future (ϕ1 , ϕ2 being a pair of increasing Young functions), but we restrict here to the simpler L2,2 -boundedness case. The idea of path or line integrals is to

318

7 Multiple Random Measures and Integrals

employ monotone paths based on sets T ⊂ Rn and then extend the resulting theory. Consider the index T = [0, 1]n ⊂ Rn where n = 2 for the present discussion but the initial definitions are valid for T = Rn+ . An oriented path Γ = γ([0, 1]) ⊂ Rn is the set {t = γ(u) ∈ Rn+ : 0 ≤ u ≤ 1}, and let Γˆ = {γ(1 − σ) : 0 ≤ σ ≤ 1}. If γ(·) is a monotone function, the path is said to be of pure type by Cairoli and Walsh. If σ ≤ σ ′ implies γ(σ) ≺ γ(σ ′ ), it is called type I, and type II if σ < σ ′ implies γ(σ) f γ(σ ′ ) where if f

n = 2, (s1 , s2 ) f (t1 , t2 ) is by definition, when s1 ≤ t1 and s2 ≥ t2 with f if there is strict inequality as in the orderings ‘≺’ and ‘≺≺’ used earlier (and likewise for n ≥ 2). With these concepts we can define polygonal and other approximations. For instance in R2 , a polygonal curve is a path whose vertices are (a1 , b1 ) ≺ (a1 , b2 ) ≺ . . . ≺ (an , bn ) or (a1 , b1 ) ≺ (a2 , b1 ) ≺ . . . ≺ (an , bn ), and would like to approximate paths of pure types with such curves. The purpose of these is to use the one-dimensional parameter processes. This type of argument is employed effectively by Cairoli and Walsh (1975) with planar BM and we would like to get analogous results for the more general Bochner L2,2 boundedness collection. The approximation of a path by polygonal curves is made precise as follows. Let Γ be a fixed path and {Γ n , n ≥ 1} a sequence of polygonal curves of the same pure types and non-crossing, all with the same initial point a1 and final point af in T . If γ : [0, 1] → R2 is a parametric representation, then a1 = γ(0), af = γ(1). Also Γ n is said to converge to Γ if the area between curves shrinks to the empty set ∅ as n → ∞, written as area (Γ n ∪ Γ )◦ → ∅. This area excludes the bounding curves Γ and Γn . This is motivated by the classical L´evy distance between a pair of distribution functions. We restrict here to R2+ without further mention. The work may be extended later to Rn . From now on let {Xt , Ft , t ∈ T } be the process which vanishes on the n coordinate axes of T , and for the curves Γ (polygonal Γ n ) let X Γ (X Γ ) be n n restrictions of X to Γ and Γ and for z ∈ Γ let z˜ ∈ Γ be the point determined by z and the horizontal or vertical line through z (meeting Γ n ). Next Γ (z) the polygonal curve of type I or II for i = 1 and type I denote by Xi or II′ for i = 2. This particular geometric definition is used to extend the one-dimensional concepts to the planar case and to introduce a line integral. We next present a “road map” for the results to be considered in this section since the full details will be many and distracting. The point is to show that several results established for the planar BM admit extensions to the (Bochner) L2,2 -bounded class which may be obtained in future for the more general case of Lϕ1 ,ϕ2 -bounded fields. If the polygonal curve Γ is of type I or Γ (z) II, the field X = {Xz , Fz , z ∈ T } restricted to Γ will be denoted by X1 Γ (z) and that to type I or II′ as X2 . Thus in detail if z = (ai , bj ) ∈ T , we have P (Xaj+1 ,bj − X(aj ,bj ) ), i = 1, Γ (z) (1) = Pupto z Xi (X aj ,bj+1 − X(aj ,bj ) ), i = 2, upto z

7.4 Random Line Integrals With Fubini and

Green-Stokes Theorems

319

where the curve Γ has the initial point z1 = (a1 , b1 ) and final point zf = (an+1 , bn ) with z = (aj , bj ) as an intermediate point. Thus Pn (X(aj+1 ,bj ) − X(aj ,bj ) ), i = 1, Γ (z ) Xi f = Pnj=1 (2) (X (aj ,bj+1 ) − X(aj ,bj ) ), i = 2, j=1

where Γ has z1 = (a1 , b1 ) as the initial point, zf = (an+1 , bn ) the fiΓ (z) nal one. If X is a planar martingale, then (XiΓ )(z) = Xi , i = 1, 2 and z ∈ Γ.

Theorem 7.4.1 Let X = {Xz , Fz , z ∈ T } be an L2,2 -bounded process with bounding measure α : B(T ) ⊗ F → R+ , and Γ, Γ ′ be a pair of polygonal curves of types I or II and I or II′ respectively both having the same initial and final points z1 and zf . Suppose that Γ lies above (respectively to the right for i = 2) and let A = area between Γ and Γ ′ without the boundaries (i.e., just interior area). Then: (i) if A¯ is the Euclidean closure of the set A we have the domination Z 2 Γ′ Γ dα, (3) χA−Γ E[(Xi (zf ) − Xi (zf )) ] ≤ ¯ T ×Ω

where A¯ is the closed set above. If in particular Γ is in a set of continuity of the in (3). This continuity hypothesis field X, then χA can be substituted for χA−Γ ¯ holds if the standard filtration is strengthened to be both left and right continuous to mean Ft = σ(∪st Fs (= Ft+ ); (ii) under the same hypothesis the following approximation also holds. There exists a sequence of polygonal curves {Γ n , n ≥ 1} such that Z Γ n (z) 2 Γ (z) ) ]≤ χA¯nz −Γ dα, (4) − Xi E[(Xi T ×Ω

where Anz is the open area set between Γ and Γn up to the vertical line Vz beΓ (z) Γ n (z) → Xi tween Γ n (z) and Γ (z) with Z ′ = inf{z : z ∈ Vz ∩ Γ n }, so that Xi 2 in L (P ), as n → ∞. If moreover X is continuous on Γ as defined above (in terms of the filtration), then Z Γ n (z ′ ) 2 Γ (z) χAnz dα, i = 1, 2. (5) ) ]≤ − Xi E[(Xi T ×Ω

This result is based on the general analysis of the random fields on T and hence many “standard” details have to be supplied. We have to omit them here, and refer the reader to a complete account given in M. Green ((1997), pp.97–104), but discuss some consequences. The above formulation is presented to introduce line integrals along these curves Γ and their derivatives to proceed further and set down the product (and repeated) integrals, introducing the

320

7 Multiple Random Measures and Integrals

(stochastic) Fubini theorem for using it in the final Green and Stokes versions which are at the root of (stochastic) differential forms. A critical part of the L2,2 -boundedness hypothesis is to obtain the bounding measure for concrete applications. The following result shows how the pure types (I, II) and (I, II′ ) of polygonal approximations have such a measure in familiar terms. Proposition 7.4.2 Let X = {Xz , Fz , z ∈ T } be an L2,2 -bounded process relative to some measure, and Γ be a (polygonal) curve of type (I) or (II) for i = 1 and similarly for type I or II′ for i = 2. Then Xi = {XiΓ (z), Fzi , z ∈ Γ }, i = 1, 2 is L2,2 -bounded with bounding measure αi = hXiΓ i(·) ⊗ P , where hXiΓ i(·) is the quadratic variation of the XiΓ field which exists. Proof. By definition (cf., (1) and (2)) we have i

XiΓ (z) = E Fz (XiΓ (zf )),

z ∈ T,

(6)

so that for the ordering ≺ [i.e., Fzi ⊂ Fzi ′ if z ≺ z ′ ], {XiΓ (z), Fzi , ≺} is a uniformly integrable martingale where for i = 1, we use ‘≺’ and i = 2, use ‘f’. Hence the orders being linear, we get for simple f adapted to Fzi , R R (7) E[( T f dXiΓ (z))2 ≤ Γ ×Ω f 2 dhX Γ iZ ⊗ dP,

from the standard martingale theory, since {[(XiΓ )2 − hXiΓ i]z , Fzi , z ∈ T }, i = 1, 2 is a martingale. 2 As an extension of the preceding ideas (with some additional work) one can establish the following consequence of L2,2 -boundedness condition. [Another class of random fields satisfying this boundedness condition is also detailed in M. Green (2001).]

Proposition 7.4.3 Let X = {Xz , Fz , z ∈ T } be an L2,2 -bounded process. If Γ is a polygonal curve of type I or II denote X1Γ , and of type I Γ (z) or II′ denote as X2Γ , then {Xi , Fzi , z ∈ ST }, i = 1, 2 each, is L2,2 S bounded where Fz1 = σ( s2 ≥0 Fs1 ,s2 ), Fz2 = σ( s1 ≥0 Fs1 ,s2 ) and where again z = (s1 , s2 ) ∈ T . Γ n (z)

→ For general curves Γ we approximate with a sequence Γ n such that Xi Γ (z) in L2 (P ), i = 1, 2 for each z. If α : B(T ) ⊗ F → R+ is the bounding Xi Γ (z) measure of X, then it also bounds Xi , i = 1, 2 and for each simple function R Γ (z) uniquely f : T → R that is Fzi -adapted, we can define the integral T f dXi which satisfies R R Γ (z) E[( T f dXi )2 ] ≤ T ×Ω f 2 dα

so that it admits a unique extension (by density) to all of L2 (α). The details here again are not entirely simple and they are given in M. Green (1997, p. 109).

7.4 Random Line Integrals With Fubini and

Green-Stokes Theorems

321

With this work if Γ is a (polygonal) curve of type I or II′ and R f is Γa R Γ f dX2 f dX and predictable simple function, then the line integrals 1 Γ ΓR R are well-defined, and they are denoted Γ f ∂1 XR and Γ f ∂2 X, to differentiate them from the general stochastic integral T f dX. Similarly if Γ is of type I′ or II, (or) I′ and II′ , we have [Γˆ denoting the opposite orientation] Z Z Z Z f ∂2 X. (8) f ∂2 X = − f ∂1 X, and f ∂1 X = − Γˆ

Γ

Γˆ

Γ

In general if Γ is of pure type we set Z Z Z f ∂X = f ∂1 X + f ∂2 X, Γ

Γ

(9)

Γ

when X is L2,2 -bounded these definitions extend to f ∈ L2 (α) where α is the bounding measure of X. That these are well-defined may be verified, and we now present the following dominated convergence type result. Proposition 7.4.4 Let X = {Xz , Fz , z ∈ T } be an L2,2 -bounded process with a bounding measure α : B(T ) ⊗ F → R+ . If {fz , Fz , z ∈ T } and n {fzn , Fzn , z ∈ T } are predictable processes fzn → fz a.e. [α], and |f(·) | ≤ g, R a.e. as n → ∞, where T ×Ω g 2 dα < ∞, then, for the same type pure curves Γ n , Γ such that Γ n approximates Γ in the sense defined above, we have Z Z fz ∂Xz (10) fzn ∂X2 → Γn

Γ

2

in the L (P )-mean. Proof. Since Γ n → Γ in the above defined “L´evy type” metric and fzn → fz pointwise a.e., by hypothesis as n → ∞, it can be considered as χΓ n fn → χΓ f a.e., by going to a subsequence if necessary. Moreover, we have Z Z Z Z fn ∂2 X fn ∂1 X + fn ∂X = χΓ n fn dX = n n Γn T Z Γ ZΓ f ∂2 X, by the earlier work, f ∂1 X + → Γ ZΓ Z = f ∂X (= χΓ f dX), Γ

T

and this is equivalent to the assertion of the proposition. The Bochner L2,2 boundedness is used together in the middle limits where the integrals are in L2 (P )-mean as in (5). 2 To present the double integral and a related Fubini theorem we have to consider first an intermediate step of an integral containing a random measure and a (non random or) scalar measure and extend. Thus if X = {Xz , Fz , z ∈ T } is L2,2 -bounded with a bounding measure α, and Γ is a (polygonal) curve

322

7 Multiple Random Measures and Integrals

of type I or II, then we have seen in Proposition 7.4.2 above that α can be taken as hX γ i ⊗ P and using the measure formalism, we may express it as dα(ω, s, t) = dµ(ω, s, t) ⊗ dP (ω). Thus on a horizontal segment Hs0 t from s0 to t (i.e. a simple curve γ) we have for any predictable bounded function f , the R integral Isf0 t = Hs t f ∂1 X is well-defined as a random variable (or process) and 0 it can be integrated relative to an increasing function F (·) using the classical Stieltjes procedure to obtain: Z t0 Z t0 Z s0 Isf0 t dF (t). (11) f ∂1 X dF (t) = 0

0

0

If f is a simple (or bounded measurable) function and the bounding measure α is as found above for the L2,2 -bounded case, then the classical Fubini theorem allows us to compute: Z Z f (ω, s, t′ ) dµ(ω, s, t′ ) ⊗ dP (ω), (12) f (ω, s, t) dα(ω, s, t) = [0,s0 ]×Ω

Hs0 t′ ×Ω

and integrating relative to F (·), interchanging the order of integrals, to get Z Z f (ω, s, t′ ) dµ(ω, s, t′ ) ⊗ dP (ω)dF (t′ ) [0,t0 ] Ω×[0,s0 ] Z Z Z f (ω, s, t′ ) dµ(ω, s, t′ ) dF (t′ )dP (ω), (13) = Ω

[0,t]

0,s0 ]

so that for Isf0 t process the bounding measure is µ(ω, s, t) ⊗ F (t′ ) ⊗ P (ω). This result may be presented formally as our intermediate: Theorem 7.4.5 Let X = {Xz , Fz , z ∈ T = [0, 1]2 } be an L2,2 -bounded separable process and let F be a positive increasing bounded function on [0, t0 ] ⊂ [0, 1]. If f is a predictable scalar function for {Fz1 , z ∈ T } such that Z Z Z |f (ω, s, t; )|dµ(ω, s, t′ ) ⊗ dF (t′ ) ⊗ dP (ω) < ∞, (14) Ω

[0,t0 ]

[0,s0 ]

where the dominating measure for X is given in terms of µ ⊗ P as in (12) (a product measure) then there exists a (measurable) process {Iz , z < z0 = (s0 , t0 ) ∈ T } such that (writing z = (s, t) for each fixed t ≤ t0 ), one has: (a) i h R P Ist = Hst f ∂X, s ≤ s0 = 1

where Hst is the horizontal line (parallel to the s-axis) through (s, t), with the 1 property that {Ist , F(s,t) , s ∈ [0, s0 ]} is a right continuous martingale whenever X itself is a 1-martingale; and (b) the L2,2 -boundedness holds in the explicit form as

7.4 Random Line Integrals With Fubini and

E

R t0 0

Green-Stokes Theorems

323

2 R Rt Rs Ist dF (t) ≤ F (t0 ) Ω 0 0 0 0 f 2 (ω, s, t′ )dµ(ω, s, t′ ) ⊗ dF (t′ ) ⊗ dP (ω)

(15)

for all Fz1 -predictable f ∈ L2 (dµ ⊗ F ⊗ P ).

Sketch of Proof : If f is a step Fz1 -predictable process, then Is0 t , 0 ≤ t ≤ t0 satisfies 2 i h R 2 i h =E f ∂ X E Iso t′ 1 Hs0 t′ R ≤ H ′ f 2 dα, by L2,2 -boundedness for an α, s t R R0 s = Ω 0 0 f 2 (ω, sz , t′ ) dα(ω, s, t′ ) dP (ω) < ∞, and this is F-integrable by (14). Hence we have since 0 < F (t0 ) < ∞ 2 2 R t0 R t0 dF (t) 2 E = F (t0 ) E 0 Is0 t dF (t) 0 Is0 t F (t0 ) ≤ F (t0 )

R t0 0

E(Is20 t ) dF (t)

by Jensen’s inequality and Tonelli’s Theorem, hR R i t s = F (t0 )E 0 0 0 0 f 2 dµ(·, s, t′ ) ⊗ dF (t′ ) , which implies (15) for simple f . For the general case we approximate f suitably in the L2 (µ ⊗ F ⊗ P )-space and obtain both (a) and (15) in the way stated. The details are standard but need some nontrivial computations. These have again been detailed in M. Green (1997) and will be omitted here. 2 The point of this result is to define the integral for products such as a pair {Xz , Fz , z ∈ T } and {Xz′ , Fz , z ∈ T } for T × T ⊂ R4+ , the four-dimensional (product of two planes!) index set, and show how it can be reduced to the iterated planar processes each of which is L2,2 -bounded. This involves some new ideas motivated by (and extending) the Cairoli-Walsh work. Here we need to use both the orderings ‘≺’ and ‘f’. These are strict if ‘≺≺’ and similarly f

‘f.’ Now for A, B ⊂ T , a pair of rectangles, A ≺ B signifies that for all z ∈ A, z ′ ∈ B one has z ≺ z ′ and similarly A f B if z f z ′ in these respective sets. Let zi = (si , ti ) ∈ T, i = 1, 2 and A = (s1 , t1 ), (s2 , t2 ) , (= (z1 , z2 )) and α : Ω → R2 be a bounded Ft1 t2 -measurable function. We consider for {Xz , Fz , z ∈ T } Yz = αX(A ∩ Rz )X(B ∩ Rz ), z ∈ T, (16)

where B = ((t1 , t2 ), (t′1 , t′2 )] so that A f B. It may be verified that if X = {Xz , Fz , z ∈ T } is a martingale, so is Y = {Yz , Fz , z ∈ T }. It can also be verified that, if X is a martingale for the order ‘≺’ then the Y process is again a martingale, as already noted by Cairoli and Walsh (1975).

324

7 Multiple Random Measures and Integrals

Proposition 7.4.6 If A = (z1 , z11 ), B = (z2 , z21 ) and Rz = [(0, 0), z] with zi = (zi1 , zi2 ) satisfying AfB and Y = {Yz = αX(A∩Rz )X(B ∩Rz ), Fz , z ∈ Tz } where α is Fz11 z2 -adapted and bounded, as in (16), and where X = {Xz , Fz , z ∈ 2 1 t} then (i) X is an i-martingale implies Y is also an i-martingale, i = 1, 2, (ii) X is a weak martingale with decomposition X = X (1) + X (2) into a sum of f

i-martingales satisfying E Fw (X (1) (C)X (2) (D)) = 0 for C f D where C, D are rectangles and w = (inf{s : (s, t) ∈ D}, inf{t : (s, t) ∈ C}) implies Y is a weak martingale. Sketch: Note that strong martingales satisfy the above conditions and hence the conclusion holds for them. Also if D = (z, z ′ ] where z = (s, t) ≺≺ z ′ = (s′ , t′ ), then one can verify that Y (D) = αX(A ∩ (Rs′ t′ − Rs′ t ))X(B ∩ (Rs′ t′ − Rst′ ))

(17)

from which it is found after using various standard properties of conditioning that (18) E[Y (D) | Fz1 ] = 0 and E[Y (D) | Fz ] = 0. Hence Y is a weak martingale as desired. 2 The conditional independence hypothesis for Fz1 and Fz2 , also called F4 , about the filtration {Fz , z ∈ T }which was not used thus far will be of particular interest in considering X · X, the product. This is needed to define the desired product on T × T ⊂ R4+ and to obtain a repeated integral pair on T, and T separately which then gives the corresponding (stochastic) Fubini theorem. Proposition 7.4.7 Let {Fz , z ∈ T = [0, t]2 } be a right continuous (or standard) filtration satisfying the conditional independence (or F4 ) hypothesis on (Ω, F, P ). If X is Fz1 adapted and Y is Fz2 adapted, then E Fz (X.Y ) = E Fz (X)E Fz (Y )

a.e. [P ],

(19)

so that if A f B and w = (inf{s : (s, t) ∈ B}, inf{t : (s, t) ∈ A}), then X (1) (A), X (2) (B) satisfy the equation E Fw [X (1) (A)X (2) (B)] = E Fw [X (1) (A)]E Fw [X (2) (B)], Proof. By hypothesis and the F4 condition we have

a.e. [P ].

(20)

7.4 Random Line Integrals With Fubini and

E Fz (XY ) = E

Fz2

[E

Fz1

(XY )] = E

Fz2

[XE

Fz1

2

1

Green-Stokes Theorems

325

(Y )], since X is Fz1 − adapted, 2

= E Fz [XE Fz (E Fz (Y ))], since Y is Fz2 − adapted, 2

= E Fz [XE Fz (Y )], by the F4 hypothesis, 2

= E Fz (Y )E Fz (X) since Fz2 ⊃ Fz 1

2

= E Fz [E Fz (X)]E Fz (Y )

= E Fz (X)E Fz (Y ) by F4 again. The rest of the conclusion is now immediate. 2 We next define the product integral first for simple functions and then extend it as follows. Consider the dyadic rationals of T = [0, 1]2 as before and let zij = ( 2in , 2jn ), 0 ≤ i, j ≤ 2n and set ∆ij = (zij , zi+1,j+1 ) a subrectangle of T. Define the step function Ψij,kl (ξη) = αχ∆ij (ξ)χ∆kl (η),

0 < i < k ≤ 2n − 1, 0 < l < j ≤ 2n − 1, (21)

for α which is Fzkj -adapted, ∆ij f ∆kl . For an L2,2 -bounded process X define the product integral as Z Ψij,kl dX dX = αX(∆ij ∩ Rz )X(∆kl ∩ Rz ). (22) Rz ×Rz

Then (X ·X)(ξ, η) = X(ξ)X(η), ξ, η ∈ T is called adapted if X(ξ)X(η) is Fξ∧η measurable for the filtration {Fz , z ∈ T } and L2,2 -bounded, where ξ ∧ η stands for the point with smaller of the first and correspondingly second coordinates of ξ and η of R2 . Now if X = {Xt , Ft , t ∈ T } is L2,2 - bounded relative to a bounding measure dµ(ω, z) dP (ω), then we see that E Fkj (X(∆ij )) is bounded by µ(∆ij ), with probability 1. In fact if F ∈ Fk and Fij ⊃ Fkj , we have Z Z X(∆ij )2 dP E Fkj (X(∆ij ))2 dPkj = F

F

=

Z

χF

Ω

Z

T

χ∆ij dX

2

dP

R = E[( T χ∆ij×F dX)2 ] Z Z ≤ χ∆ij×F dµ dP, Ω

T

by the L2,2 -boundedness condition, Z = µ(∆ij ) dP, F

Fkj

2

so that E (X(∆ij ) ) ≤ µ(∆ij ), a.e, for ∆ij f ∆kj . Now we can introduce the corresponding key domination property for the product.

326

7 Multiple Random Measures and Integrals

Definition 7.4.8 Let X = {Xz , Fz , z ∈ T } be an adapted process in L4 (P ). Then X · X is said toP obey the cross-term domination condition if for each n m simple function ψ = m=1 ψij,kl , and a fixed constant K ≥ 0, X

p p m E[αm αp X(∆m ij )X(∆kl )X(∆ij )X(∆kl )]

1≤m
≤K

n X

2 2 E[α2m X(∆m ij ) X(∆kl ) ].

(23)

m=1

If the X process is a strong martingale with four moments then it is easily seen that X · X satisfies the cross-term domination condition (23) so that the planar BM is included. An interesting and useful consequence for us here is that the L2,2 -boundedness condition extends for cross-term dominated processes if each obeys it before their multiplication. More generally we have: Theorem 7.4.9 If X = {Xz , Fz , z ∈ T } is L2,2 -bounded with bounding measure dµ(ω, z) dP (ω), Xz ∈ L4 (P ) and X ·X is cross-term dominated, then X ·X is also L2,2 -bounded with the bounding measure dµ(ω, ξ, z) dµ(ω, η, z ′ )dP (ω). m (η) with = αm χ∆m (ξ)χ∆m Proof. Let ψ be a simple function given by ψij,kl ij kl m m f ∆ disjoint is T × T and ψ = αm as Fkj -adapted and bounded, ∆ ij kl R Pn m ψ dX dX is well-defined. To verify the ψ . Then using (22), m=1 ij,kl T ×T L2,2 -boundedness condition, we use (23) and simplify the L2,2 -expression as follows: 2 R E T ×T ψ dX dX X 2 2 m 2 =E αm X(∆m ij ) X(∆kl ) h X i p p m + 2E αm αp X(∆m ij )X(∆ij )X(∆kl )X(∆kl ) 1≤p<m≤n n X

≤ 2(K + 1)

2 n 2 E[α2m X(∆m ij ) X(∆kl ) ],

m=1

by the cross-term domination of X(·), n X 1 2 2 m 2 = 2(K + 1) E{E Fkj [α2m E Fkj (X(∆m ij ) X(∆kl ) )]}, m=1

using the conditional independence (F4 )-condition, n X 2 2 Fkl = 2(K + 1) E{α2m E Fkj (X(∆m (X(∆m ij ) E kl ) )} m=1

by Proposition 7.4.7,

7.4 Random Line Integrals With Fubini and

≤ 2(K + 1)

n X

Green-Stokes Theorems

327

m E[α2m µξ (∆m ij )µη (∆kl )]

m=1

= 2(K + 1)E

R

T ×T

ψ 2 dµξ dµη ,

which implies the desired assertion. 2 Let D be the σ-algebra generated by the simple functions ψ defined in the above proof, on T × T × Ω to be called a predictable σ- algebra. The function ψ(ξ, η) is then adapted to Fξfη if it is measurable relative to it. Let L2XX (z0 ) be the class of processes ψ such that (i) ψ is predictable (D), f

(ii) ψ(ξ, η) = 0 unless ξ f η, and R (iii) T ×T ×ω dµξ dµη dP < ∞.

With this basic material we can present the desired Fubini type product measure result: Theorem 7.4.10 (Stochastic Fubini) Let X = {Xz , Fz , z ∈ T } be an L2,2 bounded separable process such that X · X isRcross-term dominated and ψ ∈ L2XX (z0 ). Then ψ(ξ, η) is Fη1 -predictable and T ×Ω ψ 2 (ξ, η) dα < ∞ for a.a. ξ with X having the bounding measure α = µ ⊗ P . Moreover Z ψ(ξ, η) dXη (24) I(ξ) = T

is defined for (µξ ) a.a. ξ, and we have the repeated integrals verifying: Z Z Z I(ξ) dXξ ψ(ξ, η) dXξ dXη = T

T

T

=

Z

I(η) dXη

T

=

Z Z T

ψ (ξ, η) dXη

T

dXξ .

(25)

Proof. The method is to establish (25) for simple functions ψ of the form used in (23), and then extend the result by the L2 -approximation with the L2,2 boundedness hypothesis. A sketch of the argument follows. Since a simple function here is of the form (for suitably bounded α’s) ψ=

n X

m=1

m m , ψij,kl , ψij,kl = αm χ∆m χ∆m ij kl

328

7 Multiple Random Measures and Integrals

m with ∆m ij f ∆kl , α’s are Fkl -adapted, then for a step ψ, let

I(η) = αχ∆ij X(∆kl ), η ∈ ∆ij , 2 I(η) is Fη2 -adapted for all η ∈ ∆ij so that Fkj ⊂ Fη2 . We then obtain Z Z αχ∆ij (η)X(∆kl ) dXη I(η) dXη = T ZTZ = ψij,kl (η, ξ) dXη dXξ Z ZT ×T = ψij,kl (η, ξ) dXξ dXη .

(26)

T ×T

This equation extends to the finite sums or to ψ, a simple function as above. From this it is not difficult to derive, utilizing the L2,2 -boundedness condition, i Z Z hZ 2 dµξ dµη dP, ψij,kl E ( I(η) dXη )2 ≤ Ω

T

T ×T

which extends to all simple ψ. In the general case, let ψ ∈ L2XX (z0 ). Then we can find a sequence of predictable simple (relative to its dyadic Dn ) ψn such that ψn → ψ pointwise and in mean, Z Z (ψn − ψ)2 dµξ ⊗ dµη ⊗ dP → 0. (27) Ω

T ×T

R (η, ·) dX, (and = 0 if this does not hold). UsNext define I(η) R= limn→∞ T ψnRR ing the fact that T In (η) dXη = T ×T ψn (η, ξ) dXξ dXη , and then that I(η) = R L2,2 -boundedness hypothesis conclude that T In (η)dXη → Rlimn→∞ In (η), with I(η) dXη in L2 (P )-mean. Similarly for I(ξ), and next using (27) it can be T shown without much difficulty that Z Z Z I(η)dXη = ψ(η, ξ)dXξ dXη . (28) T

T ×T

Interchanging η and ξ in this, one gets a similar result from which one may conclude (25). The routine computation is left to the reader. 2 The preceding result is of interest in describing the product (or surface) integral in terms of a repeated integral for predictable integrands. Motivated by the classical analysis connecting the line and surface integrals through a Gauss-Green theorem (and its extension to Stokes result), we now present a stochastic version of it. This however requires the introduction of a special measure to be denoted JX associated with ∂1 X∂2 X, since one should anticipate some new hurdles to obtain stochastic versions of the classical results. The notation JX was introduced in the work by Cairoli and Walsh (1975) for a similar purpose and the same is used by others in the subject. Recall that Rz = [0, z] is a rectangle with 0 = (0, 0) as the lower vertex, and z ∈ T . For

7.4 Random Line Integrals With Fubini and

Green-Stokes Theorems

329

f

X ∈ L4 (P ), define, with Ψ (η, ξ) = 1 if η f ξ holds, and = 0 otherwise, and consider the mapping: ZZ Ψ dXη dXξ , z ≺ z0 . (29) JX (z) = Rz ×Rz

This function JX is extended to B(T ) ⊗ B(T ) for all L2,2 -bounding processes X ∈ L4 (P ) and use it in obtaining the stochastic Green theorem. The details of computation in showing that JX is extendible as desired are nontrivial. These were given in M.Green (1997) following and generalizing the work of Cairoli and Walsh, and we state them here referring the reader to this paper. Thus if T is divided into dyadic squares with vertices at zij = ( zin , zjn ) to have the rectangles (= squares now) as ∆ij = (zij , zi+1,j+1 ), consider δij = (z0j , zi,j+1 ), εij = (zi0 , zi+1,j ) and set n Jij (z) = X(δij ∩ Rz )X(εij ∩ Rz ),

f

δij f εij

where Rz is the rectangle with diagonal (0, 0), (z1 , z2 ) as defined before. Setting P2n −1 n n (z) = i,j=0 Jij (z), Ψijn (η, ξ) = χδij (η)χεij (ξ) and JX n JX (z)

=

ZZ

n 2X −1

χδij (η)χεij (ξ) dXη dXξ .

(30)

Rz ×Rz i,j=0

n n It then follows that JX (z) → JX (z) a.e. as n → ∞, for each z. But JX (z) = X(εij )X(δij ) and hence JXR will be related to ∂1 X∂2 X. The relation is obtained after defining the integral T X dX by approximating it through sequences for each z ∈ ∆ij as n 2X −1 n Xzij χ∆ij (z) Xz = i,j=0

which are Fz -predictable and vanish on the axes. This implies on simplification Z

T

n

X dX =

n 2X −1

i=0

Xi,2n X(εizn ) −

n 2X −1

i,j=0

X(δij )X(εij ) −

n 2X −1

X(δij )X(∆ij ).

i,j=0

Each of the terms on the right can be simplified putting into integral forms, the first one as a line integral on the horizontal line Hz0 through z0 . After some computations (the details are again in M.Green (1997, Sec 2.6)) one obtains the following: Theorem 7.4.11 Let X = {Xz , Fz , z ∈ T = [0, 1]2 } be a separable right continuous L2,2 -bounded process with Xz ∈ L4 (P ), z ∈ T and obeying the maximal inequality of Theorem 7.3.6(b) with Φ(x) = x4 . If α(·), the bounding measure of the Xz -process, is locally bounded in the sense that dα(ω, z) = dµ(ω, z) ⊗ dP (ω)

330

7 Multiple Random Measures and Integrals

R

where Ω |µ(ω, T )|2 dP (ω) < ∞ and if {zij , 1 ≤ i, j ≤ 2n } ⊂ T , such that for zij ≺ z ∈ T 4 (31) E sup(Xzi,j+1 − Xzi,j ) → 0 as n → ∞, i,j

m n 2 E(JX − JX ) 2,2

then obeys an L

n → 0 as n, m → ∞ so that the mean limit JX = limn JX also -boundedness condition relative to the measure β : Z h Z 2 i E ≤ K0 f 2 dβ (32) f dJX Ω×T

T

for a fixed constant 0 < K0 < ∞ and a bounding measure β so that f ∈ L2 (β). The details of proof of this result, being routine but long, will be omitted. They are described in M. Green (1957, pp. 137–142) again where the composition of β (depending on α) can also be found. For space reasons the reproduction of these will not be undertaken here. To present the stochastic version of the classical Green theorem, it is also necessary to state the partial derivatives in the planar context. Recall that an L2,2 -bounded A-quasimartingale X = {Xz , F, z ∈ T } admits the Doob-Meyer decomposition in the horizontal and vertical directions. Thus for fixed t˜, in the s-direction the horizontal paths for X are denoted Xs,t˜ = Mst˜ + Ast˜ where {Mst˜, Fs,t˜, 0 ≤ s ≤ 1} is a right continuous martingale and Ast˜ process is a predictable process of finite variation, so that the classical Itˆ o change of variable formula corresponds to f (x) = x, gives Z s Z s ∂f ∂f (Xut˜) dMut˜ + (Xut˜) dAut˜ f (Xst˜) − f (X0t˜) = ∂x 0 ∂x 0 since the second derivative term vanishes (and X = 0 on the axes). Hence in differential form we have ( ∂f ∂x = 1 is also used now) df (Xst˜) = dXst˜ = dMst˜ + dAst˜

(33)

so that ∂1 Xst˜ = ∂1 Mst˜ + ∂1 Ast˜ for the ‘line derivatives’ or stochastic partial derivations. Following the classical deterministic case, if a process Y = {Ys,t , F(s,t) , (s, t) ∈ T )} is representable as Z Z g dt (34) f ∂2 X + Ys,t = Ys,0 + Vst

Vst

where f and g are {Fs,t , (s, t) ∈ T } predictable and if Y is L2,2 - bounded Rt relative to a bounding measure β on T × Ω, also 0 f 2 dβ exists a.e, and Rt 0 |g|dt < ∞ a.e, then f, g are called the stochastic partials of Y relative to (X, t) along Vs,t0 and partials relative to (X, s) are similarly defined along the horizontal line through (s, t), denoted Hs0 ,t . These concepts reduce to

7.4 Random Line Integrals With Fubini and

Green-Stokes Theorems

331

the classical case if Ω is a single point. Using these new notions of stochastic partial derivative, it is possible to extend the classical Green theorem to the stochastic context, but only under a restriction that the Bochner bounding measures are deterministic which however cover the important case of Brownian motion. It is desirable to relax and (even remove) this condition, but it is still an open problem. The original case of Cairoli and Walsh was extended to the L2,2 -bounded case by M.L. Green, and the thus obtained result is given as follows which therefore is a restricted version of the stochastic Green formula for rectangles. A sketch of proof is given to explain the issues involved. Theorem 7.4.12 Let X = {Xz , Fz , z ∈ T = [0, 1]2 } be a process having four moments and satisfy an L2,2 -boundedness condition with bounding measure α of the form dα = dµ ⊗ dP . Let f, g be Fz -predictable processes satisfying (34) so that they are stochastic partials along Vst0 relative to (X, t) Rfor a.a.(Lebesgue) 0 ≤ t ≤ t0 ≤ 1, 0 ≤ s ≤ s0 ≤ 1, and for a process Y, Ω×T Ys20 dα < ∞. Moreover, for a simple process γn , when the bounding measure α is of the restricted form dα(ω, z) = dX(z) ⊗ dP (ω), γn , and predictable simple functions ϕn , ψn suppose also that, on the vertical and horizontal lines Vst and Hst , the following two conditions hold: h R 2 R 2 i hR i R 2 2 E γ ∂ X + γ ∂ X ≤ E γ dλ , (35) γ dλ + n 1 n 2 Hst Vst Vst n Hst n and for (s0 , t0 ) ≺ (0, 1) we have hR R i Rs Rt s t E 0 0 0 0 ϕ2n (u, v) dλ(s0 , u) dλ(v, t0 ) + 0 0 0 0 ψn (u, v) dv dλ(u, to ) < ∞. (36) Then for any rectangle B ⊂ T one has (∂B denoting the boundary of B) Z Z Z ZZ Y ∂1 X = Y dX + ϕ dJX + ψζ ∂1 Xdt (37) ∂B

B

B

B

where the integral is in the clockwise direction of B. Outline of Proof : It may be assumed that Y = 0 on the lower edge of B since Yst = Yst1 + (Yst − Yst1 ) for B = (z1 , z2 ], z1 = (s1 , s2 ) ≺≺ z2 = (s2 , t2 ), and we find that Z s2 Z Yst1 d(Xst2 − Xst1 ) Yst1 dXst = s B Z s2 Z 1s2 Yst1 (∂1 Mst1 + ∂1 Ast1 ) Yst1 (∂1 Mst2 + ∂1 Ast2 ) − = s s Z 1 Z 1s2 Yst1 ∂1 X. ∂Yst1 (∂Xst2 − ∂Xst1 ) = = s1

∂B

Hence Snone may restrict to Yst − Yst1 . If ϕ, ψ are simple functions, then express A = i=1 Bi , Bi is a rectangle on which ϕ as well as ψ are constant valued so

332

7 Multiple Random Measures and Integrals

that since Ys0 = 0 and ϕn , ψn are constants on the vertical line Vst with values ϕ1 , ψ1 , one has Z Z ψ1 ∂2 dt = ϕ1 Xst + ψ1 t. ϕ1 ∂2 X + Yst = 0 + Vst

Vst

Consequently writing B = Rs2 t2 − Rs2 t1 − Rs1 t1 + Rs1 t1 , and JX (Rs2 t2 ) = JX (s2 , t2 ) from (30) after the limit, and X = 0 on the axes. Similarly, Z Z ϕ1 dJX = ϕ1 JX (B), ϕ dJX = B

B

on substitution for B and simplifying, one gets for the left side Z Z Z ϕ1 (Xst − Xst1 ) dX ϕ1 (Xst − Xst1 ) ∂1 X − ϕ dJX =

(38)

B

∂B

B

and similarly (Itˆo’s formula for semimartingales intervenes) Z

B

ψ1 (t − t1 )dXst =

Z

∂B

ψ1 (t − t1 )∂1 X −

Z

t2 t1

Z

s2

ψ1 ds Xst dt.

(39)

s1

Also one has Yst = ϕ1 (Xst − Xst1 ) + ψ1 (t − t1 ).

(40)

Integrating (40) over B and substituting from (38) and (39) we get on using the integral for Yst above, Z Z Z Z Ystn ∂1 X = ϕn1 dJX + Ystn dX + ψ1n ds Xst dt. (41) ∂B

B

B

B

So far the conditions (35) and (36) have not been used and these are now employed in order to let n → ∞ in (41) with the L2,2 -boundedness crucially. From (32) and (15) with F (t) = t there, we see that the following bounds hold: h R 2 i R n ϕ dJ ≤ Ω×T ϕ2 dβ E X T

and

E

hR R s0 t0 0

0

ψ n ds Xst dt

2 i

≤ t0

R

Ω×T (ψ

n 2

) dα ⊗ dt ⊗ dP.

Now we need to consider bounding of the following four types of integrals: R R (i) T ( Vst ϕn ∂2 X) dX, R R (ii) T ( Vst ψn dt) dX, R R (iii) Hst ( Vst ϕn ∂2 X)∂1 X, R R (iv) Hst ( Vst ψn dt)∂1 X.

7.4 Random Line Integrals With Fubini and

Green-Stokes Theorems

333

R

R

The assumptions imply that Vst ψn dt and Vst ϕn ∂2 X are predictable, and since γn is Fz1 -adapted, we have the following inequalities: "Z # Z 2

E

≤

γn ∂i X

Hst

Ω×T

γn2 dµ ⊗ dP,

i = 1, 2.

Using (35) and (36) we find that since µ is dominated by the Lebesgue measure λ: "Z # Z Z Z 2

E

≤

γn ∂1 X

Hst

Ω

Hst

s0

γn2 dλ dP ≤

0

E(γn2 ) dλ.

With the hypothesis on bounds, one can, after some standard computations, conclude that the limits in (i)–(iv) can be effected as ϕn → ϕ, ψn → ψ in L2 (P ) and one gets, Z Z Z ZZ Y ∂1 X = ϕ dJX + Y dX + ψ dXst dt ∂B

B

B

B

giving (37). It is also possible to consider alternative conditions instead of (35) and (36), and may even relax to have piecewise pure boundaries for B so that X will vanish there. These conditions and the many omitted details can be found in the classical case generalizing the hypothesis that implies L2,2 -boundedness implying Cairoli and Walsh (1975), and the extension to the stated general boundedness case in M. Green (1997). This completes our outline. 2 With stochastic versions of the classical Green (also called Gauss) theorem in the above, it is natural to ask for the related Stokes theorem in three dimensional rectangles. Although the technical conditional independence (through F4 ) is not essential for much of the work, as already observed by Zakai in 1982, it is convenient and that played an important role in the above proofs. The same line of thought leads us to formulate the corresponding result using conditional independence for the Stokes theorem, and it is formulated by Dozzi (1989) as follows. Here the usual filtration relative to the partial order (coordinate wise) of {Fz , z ∈ T˜ = [0, 1]3 } : [T˜ can be replaced by R3+ here] takes the following form. For all subsets C1 ⊂ C2 ⊂ {1, 2, 3} we have (F˜4 ) again called conditional independence to be satisfied for bounded Borel functions f : Ω → R in the form: F ′ E FC2 (f ) = E C2 [E FC1 (f )], a.e., (42) where Fz1 ⊂ Fz2 for z1 ≺ z2 are complete σ-algebras from (Ω, Σ, P ), F2 = ∩z≺z′ Fz′ and FC = σ(Ft , t ∈ C), C2′ = {1, 2, 3} − C2 . [This is a simple generalization of the case of T (⊂ R2+ ).] Thus Ra process {Xz , Fz , z ∈ T = [0, 1]3 } = X is one such that the stochastic integral (0,t] ϕ dXz is defined for all “pure type” paths (i.e., increasing along all oriented curves zi = γi (t), 0 ≤ t ≤ R1 as beP3 fore and i = 1, 2, 3); it is called path independent if I(C) = i=1 C ϕ ∂i X on [0, t0 ] ⊂ T , for all curves C starting at the initial point a ∈ T with the

334

7 Multiple Random Measures and Integrals

final point b ∈ T , remaining the same. This is a strong condition, but in cases of strong L2 (P )-martingales it often holds, and the strong martingale condition may very well be equivalent to path independence. This is an implied conjecture of Cairoli and Walsh ((1975), p.165). In any case, the following form of (stochastic) Stokes theorem can be given. While for the GaussGreen theorem the integration relates the (oriented) curve that bounds a (simply connected) plane region (rectangle in the above) the integration for the Stokes theorem connects the boundary of an oriented surface in the threedimensional space which is thus an extension. There are many possibilities and problems also now. In Stokes theorem the analysis is from a smooth surface to a finite number of simple regions, and there is the intermediate region satisfying more restrictions, known as Gauss’s (or divergence) theorem. In the classical (nonstochastic) situation the Gauss theorem uses the “divergence operation” which is scalar, while the Stokes result is related to the “curl operation” which is vector, valued both being extensions of the basic Green theorem. We present a version of Stokes’ theorem to round out our discussion as follows: Theorem 7.4.13 Let B ⊂ R3+ be a differentiable surface, oriented and possesses a normal, at each point in the same “octant”, so that the surface may be termed a pure type (an extension of our earlier concept of pure type curve). Suppose the filtration {Fz , z ∈ [0, 1]3 } of X = {Xz , Fz , z ∈ [0, 1]3 } is standard and satisfies the conditional independence (i.e., F˜4 ) restriction. It also is L2,2 -bounded E(Xz4 ) < ∞, z ∈ [0, 1]3 and the bounding measure is scalar. Then the measure JX of X as defined earlier exists in this context, is L2,2 -bounded, , i = 1, 2, 3 the following and for a bounded ϕ having the stochastic partials ∂∂ϕ iX representation holds:

i=1

Z

∂ϕ ∂ϕ ∂1 JX − ϕ ∂i X = ∂2 X ∂3 X B ∂B Z Z ∂ϕ ∂ϕ ∂ϕ ∂ϕ ∂2 JX + ∂3 JX . + − − ∂3 X ∂1 X ∂1 X ∂2 X B B (43)

3 Z X

Sketch. Since B is a finite union (say k) of pure surfaces of the type considered in Green’s theorem above, the result holds on each rectangle Bk whose union is B. Then on Bk the result follows from (37) and then adding the finite collection of terms, we deduce (43) for simple ϕ. But the integrals exist by the L2,2 -boundedness condition satisfied by JX . Then we can extend the result for ϕ ∈ L2 (P ), by the same argument as before. This gives (43) in complete generality as asserted. 2 The preceding statements on Green, Gauss and Stokes serve as important special forms and motivations to formulating the theorem on orientable nmanifolds, instead of T ⊂ R3+ which can be extended if T = R2+ or = R3+ .

7.5 Random Measures on Partially Ordered Sets

335

The direct coordinate approach is clearly complicated, and a more standardized version must involve an extension of the exterior forms starting with the analysis, for instance, in Abraham, Marsden and Ratiu ((1988), Chapter 7). The work in this section will be a motivation and perhaps serves as a leading example to such a desired extension. At present this is not available. It will be of considerable interest to study this aspect of the stochastic theory and the above work should be a step in the direction of such an analysis. To aid this view, we shall indicate briefly another, a combinatorial, method for extensions into general partially ordered spaces, since Rn+ with a coordinate-wise extension is more involved although may also be useful as a concrete model (cf. Dozzi (1989)).

7.5 Random Measures on Partially Ordered Sets After seeing the spaces (Rn+ , ≺) partially ordered by the coordinate-wise inequalities, denoted by ‘≺’, one may consider an abstractly given partially ordered set, denoted (V, ≤) where ‘≤’ is the given partial order symbol. If (Ω, Σ, P ) is a probability space it is desired to introduce a random measure Z : V → Lp (P ), in such a way that it reduces to the case considered in the preceding section. This is found to depend on a set of combinatorial relations induced by ‘≤’. The most important of these is the ‘M¨ obius function’, introduced to satisfy the following conditions. We present a brief exposition to explain the ideas. Along with ‘≤’, there exist the lattice relations ‘∨, ∧’ corresponding to ‘max, min’ ¯ = x and a closure operation x → x ¯, (x ∈ V ) such that x ≤ x¯, x ¯ and x ≤ y implies x¯ ≤ y¯, with x = x ¯ indicating that the element is ‘closed’. For x, y ∈ V the segment (or interval) is the set {z : x ≤ z ≤ y}, and V is locally finite if every segment is finite. An incidence algebra of V is the set of real functions f : V × V → R of two variables, such that f (x, y) = 0 if x ≤ y is not valid, and if f, g are two such functions their linear combination is an incidence function. Their (noncommutative) product h = f g (also denoted f ∗ g) is defined as X h(x, y) = f (x, z)g(z, y) (1) x≤z≤y

and then these form an algebra having the identity, δ(·, ·) as the Kronecker delta, so that δ(x, y) = 1 when x = y and = 0 if x 6= y. [It is denoted I(V ) instead of I(V × V, R).] A mapping µ : V × V → R is called the M¨ obius function if µ(x, x) = 1, x ∈ V and if µ is defined on the (open) segment [x, y) for y ∈ V by the equation X µ(x, y) = − µ(x, z), µ(x, x) = 1, x ∈ V. (2) x≤z
The following simple but fundamental property of M¨ obius inversion is essential for this work. If V is a locally finite partially ordered set (or poset )

336

7 Multiple Random Measures and Integrals

and f P : V → R such that for some p ∈ V, f (x) = 0 unless x ≥ p, and if g(x) = y≤x f (y), then one has the useful inversion formula X f (x) = g(y)µ(y, x) (3) y≤x

where µ is the M¨ obius function of V , given by (2) (cf., Rota (1964), p. 344). If V ∗ is the dual of a poset V locally finite with inverted order, then its M¨ obius function µ∗ and that of V namely µ are related as µ∗ (x, y) = µ(y, x) and the M¨ obius function of a segment [x, y] ⊂ V is simply the restriction of µ from V. Similarly if V1 is the Boolean algebra of all subsets of a finite set of n elements (with inclusion as partial order) then its M¨ obius function µ satisfies µ((x, y)) = (−1)k(y)−k(x) , y ≥ x (4)

where k(x) is the number of elements of the set x. (Details of these statements and several related results and some applications are again in Rota (1964).) We shall use these in introducing the multiple stochastic integral on V , when V is a σ-ring of an abstract set (or Rn+ or R∞ + etc). First we need to state the concept of filtration of a family of σ-subalgebras. Our treatment here follows essentially H¨ urzeler (1984, 1985) and Rota-Wallstrom (1997), somewhat restricted.

Definition 7.5.1 Let {V, ≤} be a poset for the ‘≤’ (called the lower ordering) and suppose there is an increasing sequence {Kn , n ≥ 1} of finite subsets of V (for ≤) with µi as a M¨ obius function on Ki so that (3) gives a difference operator Dn operating on functions f of Kn → X by the relation: X (Dn f )(x) = µn (y, x)f (y), x ∈ Kn , n ≥ 1 (5) y∈Kn

where X is a real vector space. Then (a) (i) (ii) (iii)

V is called separable (for the order ‘≤’) if S V = ∞ K n , Kn -finite n=1 for each x ∈ V there is a sequence xn ∈ V with x = inf n xn , and for each pair (z1 , z2 ) in Kn there exists a y ∈ Kn and integers (depending on y) such that the indicator function χ(·) satisfies: X χ[x≤z1 ]∩[x≤z2 ] = cj χ[x≤y] ; ([x ≤ zi ] ⊂ V, i = 1, 2) .

(b) if (Ω, Σ, P ) is a probability space and {Fs , s ∈ V } are σ-subalgebras of Σ such that (i) s ≤ t implies Fs ⊂ Ft , (ii) for each s ∈ V if V is separable in the T sense of (a) so that s = inf n sn for = a sequence then F = lim F s n→∞ sn n≥1 Fsn , (and A ⊂ B ⊂ V ⇒ FA ⊂ S FB = σ( i∈B Fi )),

7.5 Random Measures on Partially Ordered Sets

337

(iii) if µn is the M¨ obius function of Kn (using the separability of V ), then [ FAns ⊂ Ft , t ∈ Kn with µn (t, s) 6= 0, ( Kn = V ) (6) n

S where Ans is an atom of Kn , i.e., Ans = {x ∈ V : x ≤ s} − t∈Kn {x ∈ V : x ≤ t < s}, and the σ-algebra of V × Ω is given by (iv) P = σ{Ans × F : F ∈ FAns , s ∈ Kn , n ≥ 1} is predictable on the set V × Ω. A family {Fs , s ∈ V } satisfying conditions (i) and (ii) of (b) above is the standard filtration (of the previous section) and the family {Kn , n ≥ 1} is termed a separating sequence of the filtration. An additional condition on the filtration which is often used in our work is the conditional independence (or F4 ) and is introduced as follows (again for s∧t, s, t ∈ V we set Fs∧t = Fs∩t = Fs ∩Ft and similarly for s∨t by remembering the fact that V is a poset here): Definition 7.5.2 Let {Fs , s ∈ V } be a standard filtration of a probability space (Ω, Σ, P ) where (V, ≤) is a poset. Then it is said to possess the conditional independence property, if for s, t ∈ V, Fs and Ft are conditionally independent given Fs∩t , i.e., in the familiar notation: E Fs∧t (f ) = E Fs (E Ft (f )),

a.e., s, t ∈ V

(7)

for any bounded measurable f , so that f ∈ L∞ (P ). We the concept of a stochastic integral for simple S∞ functions Pℓ Pnow introduce i i , where αF f = s∈Kn i=1 αF s χ An s ∈ R, Fi ∈ Σ, V = n=1 Kn , and s ×Fi let X = {Xs , Fs , s ∈ V } be an adapted process for a standard filtration {Fs , s ∈ V }, ℓ ≥ 1, n ≥ 1. If Dns is the difference operator (defined with M¨ obius function µ(·, ·) of Kn ) on Kn as in (5) the integral of f relative to X is defined as: Z ∞ ℓ [ X X s i D X, V = Kn . (8) χ αF f dX = Fi n s V

s∈Kn i=1

n=1

This is an abstraction of the concepts of the preceding section to a general poset space, but even a more general one is in H¨ urzeler’s work. It can be seen that the integral does not depend on the representation of f and is well-defined for simple functions, and it is also (finitely) additive. A process is termed an integrator if the integral can be extended to all predictable functions. Without further restrictions this cannot be done. A general condition for such an extension is a form of the L2,2 -boundedness principle which we now state in this context. Thus a process X = {Xs , Fs , s ∈ V } as above, satisfies the L2,2 -boundedness condition if there exists a (σ-finite) measure ν on B(V ) × Σ → R+ such that for each simple f with its integral as given in (8), then we have:

338

7 Multiple Random Measures and Integrals

E

h R

V

f dX

2 i

R

≤ C0

V ×Ω

|f |2 dν

(9) R

for some absolute constant C0 > 0. Since the mapping T : f 7→ V f dX is linear and by (9), T : L2 (ν) → L2 (P ) is bounded, and has an extension uniquely to the closure of all simple functions on L2 (ν), which is obviously the whole space, and thus X determines a (σ-additive) random measure, as desired. We now need to show the existence of classes of random processes X that are L2,2 -bounded. Note that for such a measure the dominated convergence statement holds. This condition was the basic assumption for H¨ urzeler’s (1985) work and similarly Rota–Wallstrom ((1997), p. 1264) assume that the clearly finitely additive X of (8) has a σ-additive extension to some ring containing the semi-ring of rectangles on V into L2 (P ). For the L2,2 -bounded class both these assumptions are fulfilled, and this will be the basic property, as in the preceding work, for our treatment here. We show quickly that the cases considered in the preceding sections are included now. (Note that the ‘integrals’ on V are really sums in the above discussion.) An adapted process X = {Xs , Fs , s ∈ V } ⊂ L1 (P ) is, as usual, termed a martingale if for each s ≤ t (s, t, ∈ V ), E Fs (Xt ) = Xs a.e, and a submartingale if ‘=’ isSreplaced by ‘≤’ here, or equivalently using the difference operator ∞ Dsi if V = i=1 Ki , Ki finite, compatible for intersections and each s ∈ V is to satisfy s = inf n≥1 P sn , sn ∈ Kn some n. Moreover, Dsi X(= (Di X)(s)) i as in (5) is given as = t∈Ki µ (t, s)Xt for s ∈ Ki . Here we use the difference operator as defined in (5) through the M¨ obius function µ(·, ·). Then for X ∈ L2 (P ) a martingale, its square X 2 is shown to be a submartingale as follows: Lemma 7.5.3 Let X = {Xs , Fs ∈ V } be a square integrable martingale. Then X 2 is a submartingale when the filtration has the conditional independence property (or condition F4 ). Proof. By Definition 7.5.1(b) above the filtration has the property that for S t ∈ Ki (V = i=1 Ki ), Ais ⊂ Ki we have FAis ⊂ Ft whenever µi (t, s) 6= 0. Using the conditional independence hypothesis (cf. Definition 7.5.2), for q, t one has µ(q, s) 6= 0, µ(t, s) 6= 0 and E

FAi

s

(Xt Xq ) = E

FAi

s

(Xt∧q ), a.e.

(10)

Hence 0≤E

FAi

s

(Dsi X)2

=E

FAi

s

X

µ(q, s)µ(t, s)Xt Xs

q∈Ki r∈Ki

=E

FAi

s

X

q∈Ki

µ(q, s)

X

u∈Ki t∈Ki t∧q=u

µ(t, s)Xu2

7.5 Random Measures on Partially Ordered Sets

=E

FAi

=E

FAi

s

X

q∈Ki

X µ(q, s) µ(u, s)Xu2 χ[q≥s] u∈Ki

339

i 2 s (D X ), s

since µ(q, s) = 0 when q 6≤ s and=1 for q = s so X 2 is a submartingale, as asserted. 2 It is of interest to observe that the M¨ obius function is an indispensable tool in these calculations when the usual order structure is lacking. The above result allows us to obtain the following important conclusion first established by H¨ urzeler (1985) with a related analysis. We state it in our form to use: Theorem 7.5.4 Let X = {Xt , Ft , t ∈ V } be a square integrable martingale where the filtration is standard and has the conditional independence property. Then it satisfies the L2,2 -boundedness hypothesis and hence is a stochastic integrator for a suitable (bounded) measure ν on V × Ω integrating in particular all predictable ν-integrable functions. Let us also recall the Brownian motion in this context and observe that it satisfies the measurability conditions but is not a simple consequence of the above theorem. In fact in the higher dimensional index (say Rn , n ≥ 2), as was already noted, there are two distinct types of Brownian motions, based on the definitions of Wiener and L´evy the former will be a martingale but the latter is not. So separate S∞ treatments are needed for both these extensions. Recall that f : V (= i=1 Ki ) → R is of bounded variation if X |Dsi f | < ∞ (11) var (f ) = sup i≥1

Dsi f

P

i

i

s∈Ki

obius function of Ki and = where t∈Ki µ (t, s)f (t), µ being the M¨ f is increasing if Dsi ≥ 0 for s ∈ Ki , i ≥ 1. It can be verified that a (scalar) function on V is of bounded variation if and only if it is expressible as the difference of two increasing functions, and a Lebesgue type integral is defined relative to f by observing that such an f : V → R gives a (signed) S measure P f : B(V ) → R by the equation P f [x : x ≤ s] = f (s) for s ∈ V = i≥1 Ki with B(V ) as the ring generated by all such rectangles. Then one of g : V → R relative to f in the standard way, and R an integral R defines f g dP . Since there is no natural length for the intervals (s, t) ⊂ V , g df = V V take f : V → R+ as an increasing function in the above sense. Then a Gaussian process X = {Xs , s ∈ V } in L2 (P ) is termed a (Wiener) BM relative to an increasing (martingale) function f on R if E(Xs ) = 0 and if Ais , Ajs are disjoint measurable subsets of V with E[(Dsi X)(Dtj X)] = P f (Ais ∩ Ajt ) = 0. As usual, X is Gaussian means that its finite-dimensional distributions are Gaussian. The existence of such a process can be proved with the classical Kolmogorov theorem. We now show directly that the BM defined above and some others obey the L2,2 -boundedness condition. The process of showing that

340

7 Multiple Random Measures and Integrals

this BM is a martingale involves verifying various conditions on the filtration and then deducing it from the above theorem which will be left to the reader. Proposition 7.5.5 Let X = {Xs , Fs , s ∈ V } be an L2 (P )-valued process whose differences (or ‘increments’) satisfy the following (McShane type) bounds relative to a function g : V → SR+ which is increasing, locally bounded T∞ and that ∞ Dsi g ≥ 0, for s ∈ Ki , V = i=1 Ki and Fs = limn→∞ Fsn = n=1 Fsn , sn ↓ s: ¯i

(i) |E Fs (Dsi X)| ≤ Dsi g, s ∈ Ki , i ≥ 1, ¯i (ii) E Fs [(Dsi X)2 ] ≤ Dsi g, s ∈ Ki , i ≥ 1. S S T i ¯ i = σ(S ¯i Here F s t∈Ki ,t6≥s σ((Dt X) s6
E

≤ c(g, i)

f dX

Ki

Ki ×Ω

|f |2 (s, ω) dg(s) dP (ω), i ≥ 1,

(12)

where c(g, i) > 0 is an absolute constant, and supn c(g, n) = c(g) < ∞ if X is integrable on V × Ω itself. [Since for a BM X, defined above relative to g, has two moments and is centered, this result applies to it at once.] P Proof. For a simple predictable f = s∈Ki αs χAis , Ais are Ki atoms, the αs are FAis -adapted for all s ∈ Ki , and one has the following: E

Z

V

f dX

2

=E

hX

α2s (Dsi X)2 +

s∈Ki

=

X

E[α2s E

s∈Ki

+

s,t∈Ki s6=t FAi

s

αs αt Dsi XDti X

i

(Dsi X)2 ]

X

h i F i ∧F i ∧σ{Dsi X} E αs αt Dsi XE As At (Dsi X)

X

h i F ∧F ∧σDti X E αt αs Dti XE Ais Ais (Dti X) .

s,t∈Ki t6≤s

+

X

s,t∈Ki t<s

(13)

For the BM , X has independent increments, implied by our definition, Dsi X and Dti X for t 6> s are independent and also independent of FAit as well as FAis so that in the above expressions of (13) on the right all but the first term vanish. Hence we get

7.5 Random Measures on Partially Ordered Sets

E

"Z

V

f dX

2 #

=

X

E(α2s Dsi g) =

s∈Ki

Z

V ×Ω

|f |2 (s, ω) dg(s) dP (ω).

341

(14)

This shows by the density of such functions in L2 (dP ⊗ dg), that L2,2 boundedness property holds for the BM . With this (12) can be obtained using the triangle and the CBS-inequalities in Hilbert space, as follows. Since each Ki , is finite, let s1 , . . . , sm be an enumeration of Ki such that for j < k, sj 6≥ sk . Then for a simple function f (use the fact that Dsi g ≥ 0): ! 12

X

X X

i i 2 i αs Ds X ≤ 2 kαs k2 Ds g + kαs k2 Ds g

s∈Ki

2

s∈Ki

≤2

X

s∈Ki

h ≤ 2

s∈Ki

1 1 [kαs k2 Dsi g] 2 (Dsi g) 2

2

X

Dsi g

s∈Ki

! 21

+1

"

i X

+

s∈Ki

hX 1 = c(g, i) kαs k22 Dsi g 2 ,

"

X

s∈Ki

2 kαs k2

# 21

(kαs k22 Dsi g)

Dsi g

# 21 (15)

s∈Ki

where c(g, i) is the first factor in (15), and thus extends to all f in the space L2 (dP ⊗ dg) as before by density. 2 The preceding account shows how the random measure analysis can be formulated and studied if (R+ )2 of Sthe last section is generalized to partially ∞ ordered sets (V, ≤) where V = n=1 Kn each Kn being a finite set. This type of analysis has an immediate extension to projective (also called inverse) limits of spaces starting with the Kn ’s, and identifying V as a subspace of lim← {Kn , πn,m }, the limit space, where πn,m : Km → Kn , m ≤ n, are connecting mappings. There is however another interesting type when the poset V is specialized to a Boolean algebra or particularly to a σ-algebra of some abstract set, with inclusion among sets as the partial ordering relation which also generalizes the classical work (of the preceding section) but allows the range space to be a cartesian or tensor product of spaces of random elements. Because of its interest, extending the Wiener-Itˆ o multiple integrals, we shall now include a brief discussion of it in the manner of Rota and Wallstrom (1997). This will contrast the special cases of concrete and abstract views that are supporting each other. The necessary (new) concepts of posets in the present context will be given first where the old (V, ≤) is replaced by the Boolean analogs. Thus let (S, S) be a measurable space, and b be a finite subset of the integers N. If Sb denotes the set of all (measurable) rectangles from S b , let S⊗b fin be the (finite product set) Boolean algebra generated by Sb and similarly the product σalgebra be denoted by S⊗ whose members are the measurable sets of the

342

7 Multiple Random Measures and Integrals

space S b . If b = {1, 2}, this is the familiar object of the standard measure theory. The set Ab = A × · · · × A (b-terms) is also called a cube, and if the Ai are disjoint for i ∈ b, then the product ×i∈b Ai is termed a triangular set. Recall that Z : S → L2 (P ) is a random measure if Z(·) is σ-additive either in probability or in the norm of L2 (P ) so that in the latter case ν(A) = E(Z(A)) is defined and ν(·) is a signed measure in the classical sense. However, as we have already seen in Chapter 2, that there are Z(·) with ν(A, B) = E(Z(A)Z(B)) = (Z(A), Z(B))L2 (P ) defining a bimeasure that is not σ-additive, i.e., cannot be written as ν(A, B) = m(A × B) for a signed measure m on the product ring S × S, but only finitely additive. The same finite additivity result is seen to be true if Z : S → Lp (P ), p ≥ n, and ν(A1 , . . . , An ) = E(Z(A1 )Z(A2 ) · · · Z(An )). The problem is to find conditions in order that ν(·, . . . , ·) is σ-additive and hence we can have ⊗b Z(·) as a multiple or multi-random measure and ν(·, . . . , ·) as its multi(or poly) measure. It will allow us to define a multiple stochastic integral, and this type of product measure can be constructed through using the M¨ obius function. This can be explained as follows. Let b = {1, 2, . . . , n} be the set of integers, written as [n], and consider it as b = {(1, 2, . . . , n), ((1, 2), (3, 4, 5), . . . , (n)), · · · } consisting of distinct arrangements of disjoint (nonempty) subsets, called blocks, whose union is [n], and if σ, π are two such, let σ ≺ π(or σ ≤ π) if and only if the partition of σ is finer than that of π in the sense that every block of σ is included in some block of π. This is a partial order. A segment [σ, π] is the set of all partitions ρ such that σ ≺ ρ ≺ π, and the minimal one ˆ0 and the maximal one ˆ 1 are such that ˆ0 consists of one element sets (the finest) and ˆ1 contains the whole set as a block, the crude one, so that ˆ0 ≺ σ ≺ ˆ1 for all partitions σ. If A ∈ S⊗b let Aπ be the product set also called the diagonal, whose constituents belong to the partition π and A(≻π) the collection of all sets of S⊗b whose partitions satisfy σ ≻ π. The set Aˆ1 is a pure diagonal of S [n] , and if ‘w’ is a permutation of [n], then w(Aπ ) is the set with permuted constituents, as for instance w(A1 × · · · × An ) = Aw(1) × · · · × Aw(n) where (w(1), . . . , w(n)) is a permutation of (1, . . . , n). The set A is symmetric if w(A) = A ∈ S⊗b , and the collection of all sets of S⊗b fin is a subalgebra, denoted by S⊗b , and the collection of all symmetric sets in S⊗b will be denoted fin, sym o’s multiple stochastic by S⊗b sym . These distinctions are needed in representing Itˆ S integrals. It is simple, but important, to note that A(≥π) = σ≻π Aσ , as a disjoint union. With this long-winded but necessary notations and concepts, we define product random measures in two forms and relate them using the M¨ obius function. [Note that ‘≻’ is opposite to the usual refinement order.] Definition 7.5.6 Let (S, S) be a measurable space and Z : S → Lp (P ), p ≥ 2 be sufficiently large, and A ∈ S⊗b , the tensor product σ-algebra. Then for any partition π of b we denote (if i, j are in the same block, then we iden-

7.5 Random Measures on Partially Ordered Sets

343

tify as i ∼π j, and A≻π = {(A1 , . . . , An ) ∈ A : Ai ∼π Aj } for bookkeeping): Zπb (A) = (⊗i∈b Z)(A≻π )

(16)

Z˜πb (A) = (⊗i∈b Z)(Aπ ).

(17)

and similarly, define another function Z˜πb as:

π.]

[Since A≻ˆ0 = A, we also have Zˆ0b = ⊗i∈b Z, and Zπb , Z˜πb are only defined for The set functions Zπb , Z˜πb have the following key relationships:

ˆ one Proposition 7.5.7 For partitions σ ≻ π, defined above ˆ0 ≺ σ, π ≺ 1, has: X X Zπb = Z˜σb , and Z˜πb = µ(π, σ)Zσb (18) σ≻π

σ≻π

where µ(·, ·) is the M¨ obius function on the sets [n] × [n]. In particular, if π = ˆ0 with b = [n], we have X [n] Z˜ˆ0 = (19) µ(ˆ0, σ)Zσ[n] σ≻ˆ 0

and for a scalar measurable simple function f : S [n] → R, one has: Z f (s1 , . . . , sn )Z˜ [n] (ds1 , . . . , dsn ),

(20)

Sn

the n-fold (or nth order multiple) stochastic integral of f relative to Z˜ [n] . The result follows from the definition of Aπ and the fact that the integral can be seen as well-defined without depending on the representation of the simple function f. However, we cannot extend this to more general measurable functions without knowing that Z, Z˜ are σ-additive. By our previous work this σ-additivity obtains for generalized Poisson and Wiener-BM measures. We discuss this further for convenience. First let us consider a few properties of the measures Zπ (·) and Z˜π (·) both being finitely additive in general. However in the cases of BM and certain Poisson types of Z(·), the σ-additivity of multimeasures obtains, and this fact will also be indicated here. In the above discussion we have introduced the triangular sets A1 ×· · ·×An ∈ S[n] if the Ai ∈ S are disjoint. If w denotes the permutation, then there exist n! ⊗[n] of Aw(1) × · · · × Aw(n) ∈ Sfin . If perm (A1 × · · · × An ) is this set, we have the following easy result: Lemma 7.5.8 The σ-algebra generated by sets perm (A1 × · · · × An ) where ⊗[n] A1 × · · · × An are triangular, coincides with the σ-algebra Ssym , the σ-algebra [n] of all symmetric sets of S0 . Moreover, for a random measure Z : S → Lp (P ),

344

7 Multiple Random Measures and Integrals

[n] p ≥ 2 is large, there exists uniquely a measure Z˜ˆ0 on the σ-algebra of symmet[n]

ric sets of Sˆ0 , satisfying:

[n] Z˜ˆ0 (perm(A1 × · · · × An )) = n!Z(A1 ) . . . Z(An ),

Ai ∈ S.

[Here p ≥ n makes the right side well-defined.] Proof. Indeed for Z(·) with independent values on disjoint sets, and b = [n] we have X [n] µ(ˆ0, π)Zπ[n] Zˆ0 = π

since (A1 × · · · × An )π 6= ∅ only if π = ˆ0 so that the right side simplifies to [n] Zπ (A1 × · · · × An ) = 0 except when π = ˆ0. In this case [n] Z˜ˆ0 (perm(A1 , . . . , An )) = n!Z(A1 )Z(A2 ) . . . Z(An ),

by the fact that Z(·) is independently valued for triangular systems, and the rest follows. 2 Just as the diagonal set, we also define the nth diagonal measure of a random measure Z : S → Lp (P ), for n ≥ 1, as follows which is again a random measure: [n] [n] (21) ∆n (A) = Zˆ1 (A[n] ) = Z˜ˆ1 (A[n] ), and also a collection of “binomial type” random set functions ψn (A) = [n] Z˜ˆ0 (A[n] ) for any measurable A ⊂ S, (i.e. A ∈ S). One may then verify the following recursion, using (21): ψn (A) =

n X

k=1

(−1)k−1 (n − 1)k−1 ∆k (A)ψn−k (A),

(22)

where nk = n(n − 1) · · · (n − k + 1) for k ≤ n, the truncated factorial. At the end of the preceding chapter we briefly considered the multiple Wiener-Itˆ o integrals by using Hermite polynomials in BM, and the related random measure. In view of the preceding generalized combinatorial methods it will be of interest to consider the corresponding representation. This will be indicated here, again following the methods of Rota and Wallstrom (1997) which use the diagonal measure and generalized Hermite polynomials. Recall that the generating function of the (classical) Hermite polynomials is x2

obtained from the generating function G defined by G(t, x) = etx− 2 , using the standard Gaussian density function N (0, 1), i.e., the density with mean zero and unit variance. For our purpose here, the variance v > 0 is needed, N (0, v). (v) If Hn (x) denotes the corresponding polynomial of nth degree, then one gets the recurrence relation for u, v > 0 as:

7.5 Random Measures on Partially Ordered Sets

X n (u) (v) (u+v) Hn (x + y) = Hk (x)Hk (y), k k≥0

345

(u)

dHn (u) (x) = nHn−1 (x), dx

(23)

which is obtained from the original one by a suitably modified computation. Also if Z : S → L2 (P ) is a centered measure, and is such that (E(Z(A)) = 0), E(Z(A)2 ) = v(A) > 0, which is the variance, and if Z(A) is normal (i.e. Z(A) ∼ N (0, v(A))), then in this case (22) reduces (after a little computation) to the following useful relation (v(·) is a measure, hereafter also assumed to be nonatomic): ψn (A) = Hn(vA ) (Z(A)), vA = v(A), (24) (v )

Hn A (x) being the Hermite polynomial in x and a variance parameter vA . Then (23) reduces to (writing u, v for u(A), v(A) for fixed A ∈ S of finite measures) X n (u) (v) Hn(u+v) (x + y) = )Hk (x)Hk (y). (25) k k≥0

Further the nth diagonal measure ∆n of (21) reduces, when Z(·) is centered Gaussian, as is the case here, to the relations (to be established later): E(∆2 (A)) = v(A), E(∆n (A)) = 0, (v)

and this shows that Hn

for n > 2,

(26)

must satisfy: (v)

(v)

Hn(v) (x) = xHn−1 (x) − (n − 1)vHn−2 (x),

H0 (x) = 1.

(27)

Thus on a further computation (with (2n)!! for even factorials and { n2 } denoting the integral part ): Hn(v) (x)

=

{n 2}

X j=0

xn−2j (−v)j

n! . (n − 2j)!(2j)!!

(28)

We outline (26) and then the (27)–(28) follow from (22) and thus the recurrence relations (27) as well as some classical Hermite function relations. First the following nontrivial facts about diagonal stochastic measures are needed. A measure Z : S → Lp (P ) is independently valued if for each disjoint sequence Ai ∈ S, i = 1, . . . , n, the Z(Ai ), i = 1, . . . , n are mutually independent. Then for every n ≥ 1, and partition π of [n](= (1, . . . , n)) the product random measures satisfy (expectation of the product being equal to the product of expectations of individual integrable independent random variables) the following: For any partition π of [n], if it is expressed as π = (b, bc ) ⊂ [n] where b is a subset and bc is its complement in [n] then

346

7 Multiple Random Measures and Integrals

O E( Z˜ˆ0b ) = b∈π

X

O

E(Z˜ˆ1b )

(29)

{σ:σ∧π=ˆ 0} b∈σ

for partitions π, σ of [n] where Z, Z˜ˆ0 , Z˜ˆ1 were defined earlier. Note that for a random measure Z, ν(·) = E(Z(·)) is a scalar measure and this is the meaning given to both sides of (29) above. If now σ is a partition of [n], σ = {b, bc }, and if π0 = σ ∧ τ for a partition τ , and π0 is the zero partition, then τ cannot have more than one element of b and bc . But if τ has any blocks with only one element from [n], then since Z˜ [n] is multiplicative, the expectation of that element (by independence) is a zero measure and so [n] E(Z˜τ ) = 0 (again a zero measure). So we can have a nonzero measure here only from partitions containing exactly one element and E(Z(·)) = 0 (zero measure). Thus if |b| 6= |bc | (meaning the cardinalities |b| and |bc | are different), then there can be no non-trivial expectation measure, and if |b| = |bc | there will be one such non-zero measure for every bijection of b and bc . This may be stated, usually called a Feynman observation as the following: Proposition 7.5.9 If Z : S → L2 (P ) is such that E(Z(·)) = 0, the trivial ˜ one has: signed measure, and b ⊂ [n] is a subset, then for Z(·) X O c {i,f (i)} E(Z˜0ˆb ⊗ Z˜0ˆb ) = ), = 0 otherwise, (30) E(Z˜ˆ1 f ∈[b,bc ] i∈b

c

where [b, b ] is the set of all bijections f : b → bc . In particular if |b| > 1 (i.e, b has more than one element), |bc | ≥ 1 and E(Z(·)) = 0, then we always have c Z˜ˆ0b ⊗ Zˆ1b = 0.

The fact that Z : S → L2 (P ) should have the product or multiplicative property may be made explicit as follows. For every integer n > 0, and every [n] partition π of [n] one should have for Z˜π , the following relation between their expectations: O E(Z˜π[n] (·)) = E(Z˜ˆ1b (·)), (31) b∈π

where the right side tensor product is between the scalar measures. One observes that the multiplicative property (31) holds if and only if the diagonal measures ∆n are such that the scalar set functions E(Dn (·)) are nonatomic for n ≥ 1. To see this consider, for a rectangle A = A1 × · · · × An , the mean measures that satisfy the relation, for any partition π of [n]: Q E(Z˜π[n] (A)) = E(Z˜π[n] (Aπ )) = E[ b∈π Z˜ˆ1b [(×i∈b Ai )] O Y T = E(Z˜ˆ1b (Aπ )). (32) E[∆|b| ( i∈b Ai )] = b∈π

N

N

b∈π

Now b∈π E[Z˜ˆ1b (A)] = b∈π E[Z˜ˆ1b (A≥π )], because for A(>π) = A(≥π) − A(>π) N ˜ ˜ b (Aσ )] = N and for σ > π, one has b∈π E(∆|b| (Aσ )) for each b∈π E[Zˆ 1

7.5 Random Measures on Partially Ordered Sets

347

A˜ ∈ S[r] , r = |π| and σ ′ > ˆ0, a partition of [r]. But N this vanishes if and only if the diagonal measures are nonatomic so that b∈π E(Z˜ˆ1b (A(>π) )) = 0. Hence the measure Z(·) is multiplicative on rectangles and then generally, if and only if E(∆n (·)) all vanish for n ≥ 1. This implies all the above assertions. The next result is a modification of the classical Kolmogorov two series theorem on the convergence of sums of independent random variables, modified to be useful here for certain applications. Proposition 7.5.10 Let Z : S → L2 (P ) be a random measure taking independent values on disjoint sets and let ∆n be its diagonal measure for each n, given by (21). Suppose that the mean and variance are dominated by a nonatomic measure ν(·), so that |E(∆n (A))| ≤ dn ν(A), and Var∆n (A) ≤ cn ν(A).

(33)

Then, for each A and N , if {AiN , i = 1, . . . , N } is a decomposition (or partiSN tion) of A(= i=1 AiN ), we have for k > n, the following L2 (P )-limit ∆k (A) = lim

N →∞

N X

(Z(AiN ))k ,

i=1

existing for all such measurable sets A. Sketch of Proof : Observe that the (stochastic) diagonal measure ∆n (·) gives N X S [n] [n] ∆n (A) = ∆n ( N [(Z(AiN )n − Z˜(<ˆ1) (AiN )] A ) = iN i=1 i=1

where we set

[k] [k] Z˜(<ˆ1) (AiN )

||∆k (A) −

N X i=1

=

P

˜ [k] [k] ˆ 0≤π<ˆ 1 Zπ (AiN ).

(Z(AiN ))n ||22 = || =

N X

Rewriting this as:

[n] [n] Z˜(<ˆ1) (AiN )||2

i=1

N X i=1

+

[n]

[n]

||Z˜(<ˆ1) (AiN )||22 X

[n] [n] [n] [n] E[Z˜<ˆ1 (AiN )Z˜(<ˆ1) (AjN )], (34)

1≤i6=j≤N

we show that the right side terms each tends to zero as N → ∞ under the conditions given in (33), where k = n. All the hypothesis is utilized in this simplification and evaluation. Rota and Wallstrom (1997) show (with work) that the term is bounded by 1st term of (34) ≤

1 ck v(A)2 N

348

7 Multiple Random Measures and Integrals

and that after computation the 2nd term of (34) ≤

c˜k v(A)4 , N2

where ck and c˜k are certain absolute constants depending only on k, and the order of the moment of Z(·). Hence as N → ∞, (34) implies that ∆k (·) is σ-additive in L2 (P ). 2 With the above discussion and analysis, we can now verify (26), and then consider the (multiple) Wiener-Itˆ o integration as a consequence. Since for the centered Gaussian (random) measure Z : S → L2 (P ), the variance measure v(·) is nonatomic and satisfies (33) with dn = 0 and cn is a finite positive numPN ber, we have for k > n, ∆k (A) = limN →∞ i=1 (Z(AiN ))k holding in L2 (P ). But X N N √ X 1 1 (Z(AiN ))k = k ( N Z(AiN )k , N 2 −1 N i=1 i=1 √ and each N Z(AiN ) is a centered Gaussian random variable with variance v(A) and hence the right side converges to zero in L2 (P ) for k > 2 and to v(A) for k = 2, by the (weak) law of large numbers. Hence (26) is valid by Proposition 7.5.10. So (26)–(28) are obtained. Further ψn (·) defined as [n] ψn (A) = Zˆ0 (A[n] ), A ∈ Σ of finite v(·)- measure, when Z(·) takes independent values on disjoint sets, then one finds that E(ψn (A)ψm (A)) = δmn (n!(E(∆2 (A)))n ), (δmn is Kronecker delta) and E(ψk (A)∆m (A)) = 0, for k > 1 and m ≥ 1. If Z : S → L2 (P ) is a centered Gaussian measure with variance E(Z(A)2 ) = ν(A) ν(A), then ψn (A) = Hn (Z(A)), defined through the Hermite nth order polynomial with Z(A), and using(22), can be expressed as follows. For any partition π of [n] with blocks bi of sizes ni , i = 1, · · · , k so that n = n1 + · · · + nk , one obtains the product moments of ψni (A) : X Q E(ψn1 (A) · · · ψnt (A)) = E(∆|b| (A)) {σ : σ∧π=ˆ 0}

=

X

O

{σ : σ∧π=ˆ 0} b∈σ

E(∆|b| (A))

(35)

and E(∆|b| (A)) = 0 unless |b| = 2 in which case E(∆2 (A)) = ν(A). Thus if ℓ is the number of partitions of σ of two element sets with σ ∧ π = ˆ0, then one can see that (36) E(ψn1 (A) · · · ψnt (A)) = ℓσ(A)n and ℓ = 0 if n is odd. These computations are needed in Feynman diagrams which are usually calculated using different methods.The point of this discussion and sketch is that many of these computations can be obtained with combinatorial methods (using M¨ obius function) in a unified way. But the above details are not yet completed. They appear however to give more general results and useful for other processes also.

7.5 Random Measures on Partially Ordered Sets

349

The preceding discussion is to show that when the parameter set of a random process is multidimensional, then a coordinate-wise study becomes unwieldy and an overall comprehension will often be lost. A way of generalizing the univariate analysis for higher dimensions is to employ the M¨ obius transform when a partial order can be found in the indexing space. This was emphasized by G.-C. Rota in a number of publications. A possible application of this view is given in the present section which shows that a detailed analysis of the special results will also aid the general formulation of the subject. In this way the discrete case (Poisson-Charlier using orthogonal polynomials) of interest may be studied as modifications of the conditions which include (or be applicable) to the continuous parameter case. These are suggestive ideas but have not yet been completed in the literature. We thus conclude this general discussion at this stage. It was noted early that most of the work in higher dimensions is either originated or motivated by the Wiener-BM which instigated martingale measures and we remarked that there is another one called the L´evy-BM which is not a martingale in higher (≥ 2) dimensions. This will be discussed briefly for comparison and a possible future study. A process (or a field)X = {Xt , t ∈ T = Rk+ , k > 1} is a L´evy-Brownian motion if X0 = 0, and is a real Gaussian field with covariance given by: rX (t, t′ ) = E(Xt Xt′ ) =

1 (ktk + kt′ k − kt − t′ k) 2

(37)

where k · k is the metric of T . It is not at all obvious that (37) is positive definite for t, t′ ∈ T which is necessary to call it a covariance. However L´evy has verified this using a special method. Note that a Euclidean space T here has a general (semi)-group structure and rX of (37) is now called a L´evy-Schoenberg (positive definite) kennel after some deep results on the subject by Schoenberg. Replacing T by a locally compact group, a general structure of these kennels was investigated by Gangolli (1967) generalizing the initial (special) work by L´evy (1948) as well as Schoenberg (1938). This could be utilized in extending the analysis indicated here, since we will again show that X satisfies an L2,2 -boundedness condition. Thus it is used to define random measures and integrals relative to it. We have the following initial result which is motivated by the preceding discussion: Proposition 7.5.11 Let T be a Hilbert space and r(·, ·) be defined by (31). Then there exists a probability triple (Ω, Σ, P ) on which a L´evy-BM X = {Xt , t ∈ T } can be defined which is centered (i.e has mean zero), is Gaussian (i.e, finite-dimensional distributions are all Gaussian) with covariance rX given by (37) so that it is positive definite. L´evy constructed such a process (cf. his monograph (1948)) and deduced that its covariance rX satisfies (37) so that it is positive definite. A shorter proof of the above result was later given by P. Cartier, which is interesting

350

7 Multiple Random Measures and Integrals

and direct. It was included in the present author’s book (Rao (1995), pp. 482– 484) and will be omitted here. The method of proof presented there yields the following nontrivial and useful integral representation: Corollary 7.5.12 If T = Rk , k ≥ 1, then the L´evy BM X = {Xt , t ∈ T } is representable as a vector (or Dunford–Schwartz) integral relative to a Z : B0 (T ) → L2 (P ), on the δ-ring of bounded Borel sets with E(Z(A)) = 0, E(Z(A)Z(B)) = λ(A ∩ B) where λ is the Lebesgue (or translation-invariant) measure on the δ-ring such that Z ft (u) dZ(u), t ∈ T, (38) Xt = T

where ft (u) = given by

−1 ck 2 ht (u),

ht (u) = kuk− and ck satisfies

d+1 2

with ht (·) and the constant ck > 0 respectively are

(1 − ei(t,u) ), (t, u) =

k X i=1

ti · ui , u ∈ T − {0}, u ∈ T,

(39)

kht − ht′ k2T = ck kt − t′ kT .

For k = 1, one has the familiar formula Z 1 Xt = √ |u|−1 (1 − eiut ) dZ(u) 2π R

(40)

so that Xt = Z([0, t]) or = −Z([t, 0]) according as t ≥ 0 or t ≤ 0. For T = R (i.e, k = 1) both Wiener and L´evy definitions coincide, and (40) is a form of the classical Fourier representation of the BM on R. Since the random measure Z(·) of (38) is simply a “white noise” measure it satisfies the L2,2 -boundedness condition which implies, since the kernel ft (·) is nonstochastic, X also satisfies an L2,2 -condition. Hence one can define a different class of integrals and (nontrivially) extend the earlier work by interested workers. [For another proof of this corollary, see a detailed sketch in Exercise 5 below in the Complements Section 7.]

7.6 Multiple Random Integrals Using White Noise Methods Motivated by the last representation of a (not necessarily martingale type) process with white noise measure, we now outline a method of obtaining a multivariate extension with similar measures proposed by T. Hida (1975) and later generalized by him and his associates to become a powerful tool. Since the BM X = {Xt , t ∈ I ⊂ R+ } does not have finite Vitali variation on

7.6 Multiple Random Integrals Using White

Noise Methods

351

any open or non-degenerate interval I, the Wiener-Itˆ R o integral, for any predictable f ∈ L2 (I × Ω, dt ⊗ dP ), denoted Jf (I) = I f (t)dXt , exists one can ask whether Jf (I) may not be represented R as a weak (or the Schwartz distributional) sense integral redefined to be I (∂ ∗ f )(t)dt. Here ∂ ∗ is a “weak differential operator” acting on a random function f taking it into an integrable (random) function. The mapping ∂ ∗ : L2 (I × Ω, dt ⊗ dP ) → L2 (I × Ω, dt ⊗ dP ) is formally defined as an unbounded linear operator (in some weak sense as an ‘adjoint’ of a differential operator). All of this will be made precise here using the so-called white noise approach. In fact 5.5.5 we R R in Exercise have sketched a case in which the Wiener integral I f dZ = I f Y dt, f ∈ L2 (I, dt) is well-defined for a vector Y , where the symbol on the left is a stochastic integral and that on the right is in the sense of Bochner. But in the case of multidimensions in which Z can be a product of BM’s, the right side will be more involved. Now an important extension was made by T. Hida at this point, generalizing the very basic role of Schwartz theory of distributions, well-known in partial differential equations, to stochastic integration making the mapping ∂ ∗ : L2 (dt ⊗ dP ) → L2 (P ) rigorous and meaningful. This will be sketched here refining the work of Exercise 5.5.5 recalled above. It was already seen in Chapters 2 and 4 that Schwartz (distributions and) spaces provide very detailed and useful analysis. Thus let S(Rk ) ⊂ L2 (Rk , dt) be the Schwartz space of rapidly decreasing continuously (infinitely often) differentiable functions. Since the space of compactly supported Cc∞ (Rk ) ⊂ S(Rk ) ⊂ L2 (Rk , dt) is embedded and Cc∞ (Rk ) is dense in L2 (Rk , dt), hence also is S(Rk ), and each of these spaces have stronger topologies of their own—the so-called Schwartz topologies—it follows that each is a complete locally convex linear topological space. Moreover their duals contain (L2 (Rk , dt))∗ with a continuous embedding. This will be simply denoted as: S(Rk ) ⊂ L2 (Rk ) ∼ = (L2 (Rk ))∗ ⊂ ∗ k S (R ), where the star (∗) above the brackets denotes the dual spaces. Identifying (L2 (Rk , dt))∗ with L2 (Rk , dt) canonically, we have the continuously embedded spaces, called Gel’fand triples as: S(Rk ) ⊂ L2 (Rk , dt) ⊂ S∗ (Rk ).

(1) k

The significance of this continuous inclusion is that the space S(R ), and even its subspace Cc∞ (Rk ), is dense in the norm topology of L2 (Rk , dt). But when we give the Schwartz topology, they are still infinite dimensional and each bounded set in it generates a finite-dimensional subspace in the norm topology of L2 (Rk , dt). In fact every closed bounded set is compact in the Schwartz topology while this is false for L2 (Rk , dt), since a bounded set in the norm topology (e.g. the unit ball) is non-compact. A decisive consequence is that a great many results known in finite dimensions have close analogs in infinite dimensions, and the topology which makes this possible is termed countably normed, and its abstraction (by Grothendieck) is termed a nuclear space, briefly discussed below. These ideas will also be seen useful in Chapter 8 as well.

352

7 Multiple Random Measures and Integrals

A vector space H with a sequence of inner products {(·, ·)′n }, which can ′ ′ be assumed p increasing, (for instance, take k · kn = maxk≤n (·, ·)k ) so that kf kn = (f, f )n ≤ kf kn+1 , n ≥ 1 and if Hn is the completion of H under k · kn , then Hn ⊃ Hn+1 ⊃ H all n). The space H is given the topology in terms of k · kn , i.e., the neighborhood basis is obtained by the system for each n ≥ 1 and ε > 0, Un,ε = {x ∈ V : kxkn < ε}. If H is complete for this neighborhood system, it is called a countably Hilbert space and then one has H = ∩∞ n=1 Hn . An example of such an H is the space of infinitely differentiable real functions on R with compact supports and for f, g ∈ H with derivations f (k) , g (k) , (and Lebesgue measure): n Z X f (k) (x)g (k) (x)dx, n ≥ 0. (f, g)n = k=0

R

This is an example of a Schwartz space. Such spaces have been recalled and also used in Chapter 2 (cf. e.g. Section 2.3). A standard reference for nuclear spaces is Gel’fand and Vilenkin (1964). An important result that was employed in Chapter 2, is the Bochner–Schwartz theorem on the integral representation of positive definite generalized functions on a Schwartz space. Another general result on an abstract nuclear space that contains the preceding (concrete) case is the Bochner-Minlos theorem (conjectured by Gel’fand and established by Minlos (1959; English translation 1963)), and is stated for reference and later use as follows: Theorem 7.6.1 Let H be a nuclear space and ϕ : H × H → C, a positive definite function. Then there exists uniquely a probability measure ν on the Borel algebra B of the adjoint space H∗ such that Z ei<x,h> dν(x), h ∈ H (2) ϕ(h) = H∗

if and only if ϕ(0) = 1 and ϕ(·) is continuous in the (nuclear) topology of H. This is again a generalization of Bochner’s theorem in Rn as well as its extension to LCA groups, since H is not locally compact (but nuclear) and infinite dimensional. The proof is given both in the Gel’fand–Vilenkin volume (1964, Section IV.4) and Minlos (1963, Section 4). We shall state some of its consequences in our work. Here ϕ(·) is termed a characteristic functional. By specializing H and ϕ(·) suitably we can obtain different types of stochastic integrals (including multiple ones), and here are a few applications. Let H be the Schwartz space of real rapidly decreasing functions on R, so that (by definition), |xn f (k) (x)| → 0 as |x| → ∞ for each f ∈ H, f (k) beingRthe k th derivative and n, k ≥ 0, with norms {k · kn,k , n, k ≥ 0} where kf k2n,k = R kxn f (k) (x)k2 dx in terms of which it is nuclear, H ⊂ L2 (R)(≃ L2 (R)∗ ) ⊂ H∗ and for each f ∈ H, ϕ(·) defined by

7.6 Multiple Random Integrals Using White 1

2

ϕ(f ) = e− 2 kf k2 , kf k22 =

Z

R

Noise Methods

|f |2 (x) dx,

353

(3)

is continuous on H and positive definite. So by (2) there is a unique probability measure µ on B(H∗ ) such that (writing µ in place of ν there) Z ∗ eihf,h i dµ(h∗ ), f ∈ H. (4) ϕ(f ) = H∗

∗

∗

The triple (H , B(H ), µ) is called the White noise space, and corresponds to the Gaussian case. Other important examples include Poisson processes for which H will be a suitable nuclear sequence space and which will have a triple as above, termed Poisson White noise. A few more examples will be included in the complements section. In the above cases the characteristic functional ϕ(·) has the property that ϕ(f + g) = ϕ(f )ϕ(g) for f, g of disjoint supports (i.e (f · g)(x) = 0, all x). Equivalently taking a suitable branch of the logarithm and setting L(f ) = log ϕ(f ), f ∈ H, one finds that L(f + g) = L(f ) + L(g) for f · g = 0, i.e, L(·) is an additive functional on disjointly supported functions, but is not linear (take L(h) = h2 for an example). A characterization of L : H → C was raised by Gel’fand and Vilenkin ((1964), p. 275) and several extensions of the problem on various function spaces other than H (or the Schwartz space) were considered by many authors. (E.g., Mizel (1970) and the author, cf. Rao (1968, 1971) among others.) The point of this analysis is to show how multiple stochastic integrals appear naturally in different contexts. Observe that the concept of independent values, of a generalized random process F : H → L2 (P ), on disjointly supported functions of H is just an abstraction of the BM process having (stationary) independent increments. Also a general method of constructing classes of nuclear spaces, starting with any Hilbert space, that are contained densely in the latter can be given. See also Section 4.4 on this type of analysis for additive functionals with constraints, also called ‘local functionals’. We first introduce the multiple Wiener–Itˆo integral, using a somewhat different procedure than that given in Sections 1 and 2 above, namely an approach through martingale theory starting, for instance, as in Hida (1980, p.159) but applying the Bochner L2,2 -boundedness principle appropriately (also related to Theorem 2.7 of this chapter). If {Bt , t ≥ 0} is the BM and {Ft , t ≥ 0} is a standard filtration of (Ω, Σ, P ) such that Bt is Ft -measurable and f : R+ ×Ω → R is a function which is B(R+ )×B(Bt , t ≥ 0)-adapted where f (t, ·) is Ft -measurable Rt and 0 f 2 (s, ω)2 ds < ∞ for a.a.(ω). Let f1 : R+ × Ω → R be a bounded function that is continuous for a.a.(ω) instead of being square integrable (a stronger condition). Now f1 being bounded on [0, t] for almost all ω, so that f1 f satisfies the earlier condition and since {B(t), t ≥ 0} is a square integrable Rt martingale for each t > 0, 0 (f1 f )(s) dBs is well-defined for t > 0. If we take Rt f1 (t, ·) = 0 f (s, ·) dBs , then it is continuous as well as a martingale and satisfies the previous condition. Hence

354

7 Multiple Random Measures and Integrals

f2 (t, ·) =

Z

0

t

(f1 f )(s, ·) dB(s) =

Z tZ 0

s

f (v)f (s) dB(v) dB(s)

(5)

0

is well-defined, since f2 satisfies the earlier conditions and we now apply inducRt tion to define fn (t, ·) = 0 fn−1 (s)f (s) dB(s), so that this may be expressed as a multiple (not repeated) integral: Z sn −1 Z t Z s1 f (s1 )f (s2 ) · · · f (sn )dB(s1 ) · · · dB(sn ). (6) ··· fn (t) = 0

0

0

Next consider the nuclear space H of Theorem 7.6.1 above based on the Schwartz space S on Rk , and identify the BM as one determined by an element of the adjoint space S∗ of S. Thus we can view (6) differently as follows. First consider k = 1. Then (3) and (4) give on considering a sequence fm ∈ S such that fm → f t on [0, t], where f t = χ[0,t] , in L2 (µ), so that 1

2

t 2

1

t2

ϕ(fn ) = Eµ (eihfm ,hi ) = e− 2 kfm k2 → e− 2 kf k2 = e− 2 , by the bounded convergence theorem on L2 (S∗ , µ). It follows that the last defines the characteristic t functional of Bt , t > 0 where {Bt , t ≥ 0} is the BM. Thus R Bt = h·, f i is det 2 fined for all f ∈ L (R, dx), such that for any f ∈ S, R f dBt is the Wiener integral. If one uses integration by parts, one has with f ∈ S, B0 = 0, a.e and f (±∞) = 0) Z Z W (f ) = f (t) dBt = − Bt f ′ (t)dt = (−f ′ , B). (7) R

R

So W (·) is a continuous linear functional on S and setting (−f ′ , B) = (f, B ′ ) where B ′ is the distributional derivative we find that W = B ′ on S. Thus t on extension of W (·) to L2 (R, dt), we can set Wt = W (f t ) = dB dt with 2 ′ Bt ∈ L (µ), and B = W termed the White noise so that Wt (f ) is integrable, for each f ∈ S, W (f ) is normally distributed with mean zero and variance kf k22 . Thus W is a continuous linear functional on S and belongs to a space that contains L2 (R, dt), denoted S∗ , the inclusion being continuous, and the image dense. A characterization of S∗ , the adjoint space of S(= S(R)) is useful. Thus the defining equation of B ′ = W is (7) which is unambiguous by (3) and (4), i.e., by the Bochner–Minlos theorem, and thus S becomes our test space. It is possible to describe the Schwartz space as a countably Hilbert space in different ways. The following method, although unmotivated, has a useful application in a direct sum decomposition of the Wiener integral and introducing a multiplication in S∗ . Thus let us consider, without motivation, the differential d2 2 operator A = − dx 2 + x + 1 on L (R, dt) which is (unbounded but) densely defined. It is found to have applications in studies of (quantum) physics. Its eigen functions are related to Hermite polynomials with positive integral (even integers) eigen values, such that Aen = (2n + 2)en ,

n = 0, 1, 2, . . . ,

(8)

7.6 Multiple Random Integrals Using White 1

n

1

in which en (x) = π − 4 2− 2 (n!)− 2 e

Noise Methods

2 − x2

Hn (x), with the Hermitian n −x2 ) n x2 d (e (−1) e . These coefficients dxn

355

polynomial of

degree n so that Hn (x) = differ slightly from the definition of Hn in the preceding section but the change is made only for convenience of computation. Although A is closed (and unbounded) on L2 (R, dt), the inverse, A−1 exists and is bounded with operator norm kA−1 k = 12 . Now for any p > 21 the bounded A−p is actually Hilbert–Schmidt on L2 (R, dt) with norm k · kHS which satisfies kA−p k2HS =

∞ X

(2n + 2)−2p .

If we set Sp = {f ∈ L2 (R, dt) : kf kp < ∞}, where kf kp =

∞ hX

(9)

n=0

(2n + 2)2p (f, en )2L2 (R)

n=0

i 12

,

p > 0,

T then (Sp , k · kp ) is a Hilbert space and S = p≥0 Sp with k · k0 as the L2 (R, dt)norm (= S2 ). But then S is (countably) normed as well as nuclear since it can be verified that Sp+1 is included in Sp , the inclusion mapping being Hilbert– Schmidt. It can be verified that S∗p ⊃ L2 (R, dt) with norm kf k−p = kA−1 f k, and S∗p contains L2 (R, dt) densely. Although this definition is based on the operator A, S itself is not dependent on A. There are many other ways of obtaining it, each being useful for different applications. Some of these are described here. If S is the Schwartz space on R with S∗ as its dual, thus for each positive definite continuous ϕ : S → C, ϕ(0) = 1, then as seen in (4) by Minlos representation that there is a unique probability µ on the Borel σ-algebra σ(S∗ ) such that 2 1 ϕ has a (unique) Fourier representation for µ. Now if ϕ(f ) = e− 2 kf k2 , f ∈ S, then µ is a Gaussian measure with mean zero, and has all moments so that for any f1 , . . . , fn from S, and P (·) a polynomial in n variables, then the function F (·) = P (h·, f1 , i, · · · h·, fn i) : S∗ → C is integrable for µ where fi ∈ S, i = 1, . . . , n, are any elements. Consequently the Gaussian property also implies if P is replaced by an exponential function Fα = eαh·,f i for any α ∈ C and f ∈ S, that Fα ∈ L2 (S∗ , B(S∗ ), µ)(= L2 (µ)). In fact if P is the algebra of polynomials of E = {eih·,f i , f ∈ S} the algebra spanned by these functions, we can extend the classical results on Gaussian measures and obtain the following: Proposition 7.6.2 The classes E and P are dense in the (separable) Hilbert space L2 (µ), defined on (S∗ , B(S∗ ), µ), µ being Gaussian determined by (4). Moreover, if Hn = sp{P ∈ P : P is a polynomial of the form Const. Qkj h·,fj i √ ), fj ∈ S} ⊂ L2 (µ), where Hnj (·) is the Hermite polynomial j=1 Hnj ( 2 of degree nj , then L2 (µ) admits an orthogonal decomposition as : L2 (µ) =

∞ M n=0

Hn ,

(F0 = R).

(10)

356

7 Multiple Random Measures and Integrals

Proof. The facts that µ is Gaussian and elements of E are spanned by the random variables of the form Xf = eαh·,f i , f ∈ S, α ∈ C and |Xf | = |exh·,f i |, α = x + iy with h·, f i as N (0, kf k22), imply that |Xf |k is integrable for k = 1, 2, . . . . It follows that the algebra E ⊂ L2 (µ). To see this is norm Qdense, we obn 2 ih·,fk i serve = k=1 e P that any F ∈ L (µ) Pn with F ⊥ E must vanish. Since i n h·,f i i h·,f i 2 k k e k=1 , let F ⊥ e k=1 , and note that L (µ) is separable and so B(S∗ ) is generated by Bn = σ((h·, f1 i, . . . , h·, fn i), fj ∈ S, j = 1, . . . , n). Thus 0 = E(F ei

Pn

k=1

uk h·,fk i

Pn

) = E[E Bn (F ei k=1 kh·,fk i )]] Pn = E[ei k=1 uk h·,fk i E Bn (F )], n ≥ 1,

where E Bn (·) is the conditional expectation operator. By the classical Fourier Bn transform S theory, E (F ) = 0, a.e. Now since n ≥ 1 is arbitrary, and F is B = σ( n≥1 Bn )-measurable, this implies that F = 0, a.e. Also P is uniformly dense in E and consequently it is again dense in L2 (µ). Moreover, if f1 , . . . , fn ∈ S have disjoint supports then h·, fj i, j = 1, . . . , n are independent for µ, and so from the first part of this proof, we may now infer that finite products of such random variables form a dense set in S and hence in L2 (µ) as well. The classical theory of orthogonal polynomials in L2 (R, dt) shows that Hermite polynomials form a complete orthogonal set and the fact extends to Q h·,f i L2 (µ) as follows. Let g = c j Hnj ( √2j ), where c ∈ R is a constant and the product is finite, the {fn , n ≥ 1} ⊂ S will be a complete orthonormal set of (smooth) functions, fn ∈ L2 (R, dt), n ≥ 1. It may be verified by use of the Gaussian integral that (along with some Hermite integral identities) Z Y |g|2 dµ = |c|2 (nj !)2nj E(|g|2 ) = S∗

j

Q 1 so that if we take c = ( j (nj )!2nj ) 2 in the definition of g and arrange them in a sequence {g1 , g2 , . . .} then it forms a complete orthonormal set in the (new) space L2 (µ). This is the key connecting link of BM and the set {gn , n ≥ 1} to be called the Fourier–Hermite Polynomial system. If Hn ’s are subspaces of L2 (µ) determined by the above polynomials of degree exactly n, then Hm ⊥ Hn for m 6= n. Now the completeness of {gn , n ≥ 1} ⊂ L2 (µ) along with the above result, give the desired decomposition (10). 2 Remark 7.6 The use of Fourier–Hermite Polynomials was first noted by Cameron and Martin (1947), and was extended by Itˆ o (1957) employing the available important work on the subject by Kakutani (1950). The members of Hn are termed multiple Wiener integrals. The result can be restated which we present without detail as: for each f ∈ L2 (µ),

7.6 Multiple Random Integrals Using White

f (ω) =

∞ X

n=0

Noise Methods

357

fn (ω), ω ∈ S∗ , fn ∈ Hn ,

the series converging in mean and fn ⊥ fm , n 6= m. Alternatively one can express each fn as a multiple integral of the type (6). An earlier (slightly specialized) discussion appeared in Exercise 6.5.7, and the preceding proposition expands it to a large extent. We now introduce another operation, due to G.C. Wick (1950) in connection with a (new) matrix multiplication which in our context makes S∗ an algebra and which also admits a generalization of Itˆ o stochastic integrals of BM whose integrand now is not necessarily ‘predictable’ or non-anticipative. This R allows us to define R f (t, ·) dBt for all f (t, ·) ∈ L2 (dt ⊗ dP ) through the L2,2 boundedness property, which may be used to define a convolution operation for random measures. First we need to present some basics, of a combinatorial nature, on this new multiplication, to be called a the Wick product, and then obtain some consequences. Definition 7.6.3 Let S and S∗ be the Schwartz space and its dual. If f, g ∈ S∗ , with a series representation using Hermite polynomials X X b j Hj , a j , b j ∈ Rk , (11) a j Hj , g = f= j

j

then their (Wick) product is the function h, denoted h = f ⋄ g, given by X (ai , bj )Hi+j (x), x ∈ Rk , h(x) = (f ⋄ g)( x) =

(12)

i,j

where (ai , bj ) is the inner product of R (or Rk , k ≥ 1). Since both the sums in (11) and (12) are finite, and define elements of S∗ , as linear functionals on S, they are meaningful. It should however be shown that the product operation ‘⋄’ is unambiguous in the sense that it does not depend on the representation of f and g where f, g, Hi are (can be) regarded as elements of L2 (Rk , dx). This involves a detailed computation using some properties of Hermite polynomials which can be found in (Holden et al (1996), Appendix) and is referred to. The product ‘⋄’ is not the ordinary pointwise operation except in very simple cases, e.g., if f = c ∈ R (a constant) and g is general as in (11) or both f, g are scalar nonrandom variables then f ⋄ g = c · g (or f · g) the pointwise product. If f, g are general random variables the product computation is not so simple to describe. The following result exemplifies the situation: Proposition 7.6.4 Let f, g ∈ L2 (µ) where µ is as in Proposition 7.6.2, so that they can be given multiple Wiener–Itˆ o representations given by (10):

358

7 Multiple Random Measures and Integrals

f=

∞ Z X

fn dB

⊗n

, g=

Rn

n=0

∞ Z X

m=0

gn dB ⊗m

(13)

Rn

where B ⊗k is the k-fold product of BM, as in (6). Then the new product f ⋄ g defined by (12), if square integrable for µ, is of the form: f ⋄g =

∞ Z X

n=0

X

Rn i+j=n

ˆ gj dB ⊗n fi ⊗

(14)

ˆ is the symmetrized tensor product of fi , gi so that f ⊗ g = f · where f ⊗g ˆ g = 1 (f ∗ g + g ∗ f ) etc. (with ‘∗’ for g (pointwise product) and f ⊗ 2! convolution). Outline of Proof: The Hermite polynomials Hα in Rα with α = (1, 2, . . . , k) are the products of one-dimensional functions, namely: Y hj (h·, fj i), fj ∈ S(Rα ), Hα (·) = j∈α

the product extends to all subsets of α, and the Hα are still orthogonal. Then Hα (ω) is representable as a multiple Wiener-Itˆ o integral so that Z ˆ ˆ ˆ β⊗β ) dB ⊗|α+β| (x, ω), fα ∈ S (15) Hα+β (ω) = (fα⊗α ⊗f Rα+β

where we used the symmetric tensor product notation for the multiple WienerItˆ o integral in terms of Hermite polynomials defined earlier. With this notation the (Wick) product of (12) becomes (i, j are integer indexes) X (f ⋄ g)(ω) = aα bβ Hα+β (ω) α,β

=

∞ X

X

aα b β

n=0 |α+β|=n

=

∞ Z X

n=0

Rn

X

i+j=n

Z

Rn

ˆ ˆ ˆ gβ⊗β fα⊗α ⊗ dB ⊗n

ˆ gj dB ⊗n , fi ⊗

after reshuffling the tensor product and writing the sum |α| = i, |β| = j, this gives (14). There are more details in Holden et al (1996), with these extensions and may be consulted. 2 A simple example of the above is to consider f = χ[0,1] = g. Then one finds B(t) ⋄ B(t) = B(t)2 − t so that the Wick product generally is quite different. One of the reasons that we would like to know about this operation is to study some of the consequences of the orthogonal decomposition given by (10) of L2 (Sk , B(S∗ ), µ)(= L2 (µ)). Indeed every element f ∈ L2 (µ) admits a Wiener–Itˆ o series representation as:

7.6 Multiple Random Integrals Using White

f (t) =

∞ Z X

n=0

Noise Methods

hn (s1 , . . . , sn , t)dB1 (s1 ) . . . dBn (sn ),

359

(16)

Rn

ˆ 2 (Rn , dλ), n = 0, 1, . . . and t ∈ R (the symmetrized with hn (·, . . . , t) ∈ L Lebesgue space). HoweverRthe random function f (t) need not be Bt = P σ(Bs , 0 ≤ ∞ o integrable even if (∗) n=0 (n+ s ≤ t)-adapted and hence R f (t)dB(t) is not Itˆ 1)!kfˆn kL2 (Rn+1 ) < ∞ where fˆ is the symmetrization of f . Such f ’s were termed predictable for the (standard) filtration {Bt , t ≥ 0}, only when f (·, t) is Bt adapted which generally need not happen. However, if the above condition (∗) holds for an f (·, t), then M.Hitsuda (1972) and R ∗ A.V.Skorokhod (1975) have (independently) defined an extended integral R f dB, by the following equation using symmetrization of fn ∈ Hn , Z ∗ ∞ Z X fˆn (s1 , . . . , sn+1 )dB ⊗n+1 (s1 , . . . , sn+1 ), (17) f (t)dB(t) = R

n=0

Rn+1

where these fˆn satisfy the above growth condition (∗), and moreover the integral is shown to reduce to the Itˆ o integral if f (·, t) is Bt -adapted. Thus the generalized concept of (17) is termed the R ∗ Hitsuda–Skorokhod integral. There is an alternative representation of this R f dB, using the Wick products, for which reason we introduced the latter. This depends on the fact that B(·) has a weak derivative Wt dt = dBt taking its values in the dual space S∗ of S (not just in L2 (µ)), and the Wt is called the white noise process, and this will be explained. It is now necessary to define an integral for the white noise process, i.e., for functions W defined on R with values on S∗ . The situation is confounded by the fact that (S∗ , B(S∗ ), µ) is the measure space with µ being a Gaussian measure, produced by the Minlos theorem, and W as a vector function on R with values in the adjoint space S∗ which is not reflexive and R hence taking for R, the Lebesgue space (R, B(R), dt), one has to define R ϕ(t)W (t)dt as a vector integral, valued in the non-reflexive Fr´echet (not Banach) space S∗ . Even in the (separable) Banach spaces case, the integral of hϕ, W i(·) can be defined as a Dunford–Gel’fand integral with values in the second adjoint of S∗ , i.e., S∗∗∗ , which will not be very useful. But now S is a nuclear space and that comes to the rescue and in our case the integral fortunately will take values in S∗ itself (not the very large complicated S∗∗∗ ), as an extended Pettis integral. When once this is defined, then we can use Wick products and replace ϕ(t)W (t) by (ϕ ⋄ W )(t), and obtain the Hitsuda-Skorokhod integral in this alternative manner. It should be emphasized that the weak integral used here is actually a weak*integral (weaker than the Pettis integral) and its properties (not to speak about its characterization) which correspond to the material of Sections 3.7 and 3.8 of Hille–Phillips (1957) in the context of nuclear Fr´echet spaces is needed for our study. This is not yet available [beyond a few elementary results in Kuo (1996)] in the literature as far as is known to this author, and a functional analysis

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7 Multiple Random Measures and Integrals

study seems very desirable. An extensive Banach space study of Pettis integrals was presented by Talagrand (1984) and it needs to be extended to include our specifications. The analysis in Huang and Yan (2000, Chap IV) may be useful here. With the preceding remarks (and reservations), the following result may be given connecting the Hitsuda–Skorokhod integral and the (weak* Pettis type) Wick product integral: Proposition 7.6.5 Let f ∈ L2 (µ) be an S∗ -valued random function representable as the following series (Wiener–Itˆ o chaos expansion): X f (·, t) = cα (t)Hα (18) α

Ra where the Hα are Hermite polynomials and 0 E(f 2 (·, t)) dt < ∞, a > 0. If ∗ t {W (t, ·)(= dB dt ), t ≥ 0} is the white noise process in S , then we have: Z Z ∗ f (t) ⋄ W (t) dt, a > 0. (19) f (·, t) dBt = (0,a)

(0,a)

The proof involves several properties of the Wick product identities and will be skipped here referring the reader to Holden et al (1996). This is presented here to indicate another aspect of the white noise theory, a large part of which from a slightly different point of view is given in Kuo (1996). In this connection see also Huang and Yan (2000, Sec. IV.3). There are two closely related but different types of transformations on L2 (S∗ , B(S∗ ), µ) into functional spaces L(S) which are useful in the analysis of multiple Wiener-Itˆ o integrals. It will be interesting to introduce them as a final item of this section (and chapter). The first is an analog of the Fourier transform and the other corresponds to a convolution type. Thus let J : L2 (µ) → L(S) be defined as: Z [J(ϕ)](f ) = eih·,f i ϕ(h) dµ(h), f ∈ S, ϕ ∈ L2 (µ). (20) S∗

The following property of the mapping J is fundamental:

Proposition 7.6.6 Let L0 (S)(⊂ L(S)) be the image of J on L2 (µ). Then J is one-to-one and there is an inner product on L0 (S) relative to which it is a Hilbert space isomorphic with L2 (µ). Proof. Since J is an integral operator given by (20), it is linear and if (Jϕ)(f ) = 0, ∀f , then by Proposition 7.6.2 above (P and E being dense in L2 (µ)), ϕ = 0 a.e., (µ) so that J is one-to-one and we need to show the existence of an inner product in L0 (S). But µ is defined from a given positive definite continuous functional on S with value ‘one’ at ‘0’. Let Cµ (·) be the functional (uniquely) determining µ [by the Bochner–Minlos theorem]. Then Cµ : S × S → C defined by Cµ (f − g) for f, g ∈ S is positive definite. So we can associate an

7.6 Multiple Random Integrals Using White

Noise Methods

361

Aronszajn (or reproducing kernel inner product) space HCµ by Cµ such that (Cµ (· − f ), Cµ (· − g)) = Cµ (f, g) and Cµ (f − ·), f ∈ S} is dense in HCµ for the inner product (·, ·) which has the natural reproducing property that (g, Cµ (· − ξ)) = g(ξ), for all g ∈ L0 (S). Then Hµ = sp{L0 (S), (·, ·)} is the desired space with Cµ as its (unique) reproducing kernel. It follows that J : L2 (µ) → Hµ is an isometric isomorphism. This J is the analog of the Plancherel’s mapping on the L2 (R), but L2 (µ) and Hµ are distinct function spaces. 2 The second operation noted above is a shift (or translation) on the space ˜ L2 (µ) as the mapping S : L2 (µ) → L(S) given by: Z (Sf )(u) = f (h + u) dµ(h), u ∈ S ⊂ L2 (R) ⊂ S∗ , (21) S∗

∗

so that h + u ∈ S and the integral is well-defined for all f ∈ L2 (µ). Here S∗ is used in two senses, one as a base space for the probability µ, and also having h, u as part of the Gel’fand elements in it since S ⊂ S∗ in the Gel’fand triple! Now with the direct sum decomposition of L2 (µ) given in (10), it is possible to obtain using the representations of Hn given there, generalizing (21) in the following form, when R is replaced by Rk repeated k-times:

Proposition 7.6.7 Using the decomposition (10), and if f ∈ Hn , which on Rk can be represented as, with ui ∈ Rk , i = 1, . . . , n, Z Z fˆ(u1 , . . . , un )dBx (u1 ) · · · dBx (un ), (22) ϕ(x) = · · · (Rk )n

then the shift operator S of (21) is now representable as: Z Z fˆ(u1 , . . . , un )h(u1 ) · · · h(un )du1 · · · dun , (Sϕ)(h) = · · ·

(23)

(Rk )n

for ϕ ∈ L2 (S∗ (Rk ), µ) and symmetrized f ∈ L2 (Rn , dt1 , . . . , dtn ). This is an n-dimensional version of the earlier case of one dimension (or variable), i.e., on Rk . There is the notational complexity, and the corresponding multidimensional analogs of the results on white noise have been detailed by Redfern (1991). We present this result which is a n-dimensional representation of the earlier result, but several other n-dimensional forms of Hida’s results are possible and some were also considered by Redfern. The reader may wish to pursue this line of work and its extensions from the point of view of multiple integrals. This gives us a reasonable and general view of the multidimensional stochastic integration complementing the previous sections. Each part can be studied for further analysis and even infinite dimensional extensions are possible. Their use as well as applications for stochastic differential equations in infinite dimensions were discussed by Itˆ o (1984) himself. We terminate this work with some complements to end the chapter.

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7 Multiple Random Measures and Integrals

7.7 Complements and Exercises 1. Let X = {Xn , Fn , n ≥ 1} be an integrable adapted sequence of random variables on a probability space (Ω, Σ, P ) and Fn ↑⊂ Σ be σ-algebras. Suppose V = {Vn , Fn , n ≥ 1} ⊂ L∞ (P ). An integrable sequence defined Pn−1 as (V X)0 = V0 X0 , and for n ≥ 1, (V X)n = V0 X0 + k=0 Vk (Xk+1 − Xk ) be given so that the new integrable composition {(V X)n , Fn , n ≥ 1} is of integrable elements, called a transform of X by V which need not be linear. It is also termed a martingale transform of X by V if X is a martingale. Using the general Doob decomposition of any adapted integrable sequence as X, into a martingale and a sequence of bounded variation, show that (i) {(V X)n , Fn , n ≥ 1} is a martingale or a quasi-martingale with variation bounded by K if {Xn , Fn , n ≥ 1} is a quasi-martingale with variation bound K and |Vn | ≤ 1, (uniformly bounded by 1) and (ii) P [sup1≤k≤n |(V X)k ] > λ| ≤ Kλ1 [E(|Xn |) + K]. [Hints: Use Burkholder’s earlier known inequality that for X as a martingale {(V X)n , Fn , n ≥ 1} is also a martingale and P [supk≤n |(V X)k | > λ] ≤ 18 λ E(|Xn |), λ > 0. We can take K1 < 30.] 2. The stochastic integral of Section 1 for c` agl` ad processes (i.e., for those which are left continuous with right limits) are not the same as those defined by P.A. Meyer, whose definition is as follows: for each square integrable process H = {Ht , F0 , t ≥ 0} relative to an L2 (P )-martingale Rt M = {Mt , Ft , t ≥ 0} given by Lt = (H · M )t = 0 Hs dMs , such that for all square R ∞integrable martingales N = {Nt , Ft , t ≥ 0}, one has E(L∞ N∞ ) = E( 0 Hs d[H, N ]s ). The existence of such an element was shown by Meyer (1976, p.274, Definition 32) who also presented some properties. We assert that Rthis is different from the one given in Sect tion 1, denoted simply as 0 f (s)dMs . Consider the following example: Let(Ω, Σ, P ) be the triple with Ω = {A, B}, Σ, the four elements algebra of Ω, P (A) = P (B) = 12 , the probability measure on Σ. Let the filtration be F0 = {∅, Ω}, F1 = Σ, Ft = F0 for 0 ≤ t < 1 and Ft = Σ, for (1 ≤ t < ∞). Then Mt = 0χ(0,1) (t) + 1χ(1,0) (t)[χA − χB ]. So {Mt , Ft , t ≥ 0} is a bounded martingale and any other martingale adapted R to {Ft , t ≥ 0} is simply a constant multiple of M . Also it is seen that R+ Mt dMt = 1 · χ[1,0) which is the R classical Stieltjes pathwise integral. But Meyer’s integral gives the value ∗ R+ Mt dMt = 0, seen as follows. In Meyer’s condition for any martingale R∞ N on this Ft -filtration of (Ω, Σ, P ), one has, for L∞ = ∗ 0 Ms dMs , the following: i h R R∞ E ∗ R+ M dM N∞ = E 0 Ms d[M, N ]s , {[M, N ]s , s ≥ 0} (+) is the covariation process. Now in this example one has

7.7 Complements and Exercises

[M, N ]t = lim

n→∞

P2n

i+1 i=0 [M ( 2n

363

∧ t) − M ( 2in ∧ t)]

i × [N ( i+1 2n ∧ t) − N ( 2n ∧ t)] = bχ(1,∞) (t)

for some b ∈ R such that N = bM by the earlier R t comment. Now the right side sum is a Stieltjes approximation of ∗ 0 Ms dM (s) = a0 Mt , for a constant aR0 ∈ R. If N = M in (+) above, then the left side is ∞ a0 E(M1 )2 = E[ 0 Ms d[M, M ]s ] = 0, since the variation is a constant. Rt Hence a0 = 0 and ∗ 0 M dM = 0 while our definition gives the value as 1 · χ[1,0] . Thus these two are seen to be different integrals. [This example is due to Brennan (1978) whose thesis was read by Meyer later in that year.] 3. Let Ci : S → C be positive definite continuous functionals with Ci (0) = 1, i = 1, 2. Let µi be the corresponding probability measures on (S∗ , B(S∗ )) given by the Bochner–Minlos theorem. By considering different forms of Ci (·), verify that the resulting µi can be measures that are nonatomic as well as having atoms. Discuss the cases relative to the Gaussian as well as Poisson functionals. [Note that the random functions h·, fn i on S⋆ are independent for an orthonormal sequence {fn , n ≥ 1}, (complete or not) which are independent for the Gaussian case, and still orthogonal in the non Gaussian case (with Pn two moments) and by the (strong) law of large numbers the average n1 k=1 h·, fk iβ → a constant a.e., in the Gaussian case β = 1, 2, and in the non-Gaussian case with two moments finite so that h·, fn i and h·, fm i are just uncorrelated, then again the same conclusion can be deduced using the Rajchman’s form of the strong law. (On the latter, see the author’s probability text ((1984), p.59).) In the Gaussian case if the Ci correspond to variances σi2 (σ12 6= σ22 , and means = 0), then [µ1 ⊥ µ2 ] so the measures are orthogonal. 4. In Proposition 6 of Section 6 we defined a mapping J : L2 (µ) → L(S) by the integral of (20) and mentioned that it is one-to-one. If we restrict it to Hn , the Hermite subspace of L2 (µ) where µ is the Gaussian measure cor2 1 responding to C(f ) = e− 2 kf k2 on S, then its relation with Lˆ2 (Rn , dt), the ˆ 2 (Rn , dt) means that symmetrized Hilbert space, is given thus: Now Fˆ ∈ L P 1 ˆ F (t , . . . , t ) = F (t , . . . , t ) for any permutation π : (1, . . . , n) 1 n π(1) π(n) π n! into itself, where F ∈ L2 (Rn , dt). Thus with the same symbol J|Hn for J on L2 (µ), we have, using the characteristic functional C on S given by the exponential (in the Bochner–Minlos theorem), for any random variable ϕ ∈ Hn there exists a unique (symmetric) F ∈ L2 (Rn , dt) such that for f ∈ S(R) of the type f˜(t1 , . . . , tn ) = f (t1 ) · f (t2 ) · · · f (tn ), f˜ ∈ S(Rn ), Z Z n ˜ ˜ Fˆ (t1 , . . . , tn )f (t1 ) · · · f (tn )dt1 · · · dtn Jϕ(f ) = i C(f ) · · · Rn

and the correspondence ϕ ↔ Fˆ is bijective. The norms in the spaces L2 (µ) on (S∗ , B(S∗ ), µ) and L2 (Rn , B(Rn ), dt1 · · · dtn ) are related as;

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7 Multiple Random Measures and Integrals

kϕkL2 (µ) =

√ n! kFˆ kL2 (Rn ) ,

ϕ ∈ Hn

which explains how the spaces L2 (S∗ , µ) and the sequence {L2 (Rn , dt)}n1 , are associated. These subspaces {Hn , n ≥ 0} “fill in” L2 (µ) as a direct sum and so the corresponding direct sum of the Lebesgue spaces can be given by the above correspondence. This can be represented as, writing L2 (µ) for L2 (S∗ , µ) and Lˆ2 (Rn ) for symmetrized L2 (Rn , dt): L2 (µ) ∼ =

∞ ∞ √ M M n! Lˆ2 (Rn ) (= Hn ) n=0

n=0

where on both sides for n = 0 one has just the constants R. [This representation is of interest in quantum mechanics, and was introduced differently by the physicist V.A. Fock, and is often called a Fock space representation.] RDetails of this representation are obtained by using the fact that (Jϕ)(f ) = S∗ eihf,hi ϕ(h) dµ(h), of Minlos theorem on the space S∗ taking elements into S (as well as the basic probability space is again (S∗ , B(S∗ ), µ)), and the fact that exponential functions are µ-integrable. [For complete details related to this result, the reader is referred to a nice treatment in Hida (1980), pp. 140–141.] 5. The following is a simplified approach to the L´evy–BM which coincides with the Wiener–BM only in the one-dimensional time. Let d ≥ 1, and consider a random process or field {Xt , t ∈ Rd } such that X0 = 0 and E(Xt − Xs )2 = |s − t|, E(Xt ) = 0, s, t ∈ Rd . The existence of such a process with d-parameters, follows from Kolmogorov’s fundamental theorem if it is shown that the covariance is given by cov (Xs , Xt ) = E(Xs Xt ) = 12 [|s| + |t| − |s − t|] which implies that the latter is positive definite. Verify that the needed representation of E(Xs Xt ) can be obtained directly using the following trick. Let ϕs : Rd → C be a function given by (and this has no motivation except a good familiarity with complex analysis) ϕs (x) = Cd |x|−

d+1 2

(ei(s,x) − 1), s, x ∈ Rd ,

(s, x) =

d X

si xi .

i=1

Note that the complex functions ϕs ∈ L2C (Rd , dx), and these form a dense subset in this Hilbert space as we vary s ∈ Rd . Clearly ϕ0 = 0, and a direct verification shows that Z Z dx |ϕs − ϕt |2 (x) dx = c2d |ei(s,x) − ei(t,x) |2 d+1 = |s − t| |x| Rd Rd for an appropriate choice of cd , a fixed real constant. Now let {Bu , u ∈ C} be a complex Wiener BM, which exists from the Wiener-Itˆ o work. Next √ define a real Gaussian random function t 7→ Xt = 2Re(Bϕt ), t ∈ Rd .

7.7 Complements and Exercises

365

Since the above constructed ϕt is a family of complex continuous functions, {Xt , t ∈ Rd } is well-defined, X0 = 0, and for any pair s, t ∈ Rd we have E(Xs − Xt )2 = E(Re (Bϕs − Bϕt )2 ) = E[|Bϕs − Bϕt |2 ] Z |ϕs (x) − ϕt (x)|2 dx = |s − t|, = kϕs − ϕt k22 = Rd

and expanding the left side (real) process we get |s| + |t| − 2E(Xs Xt ). Hence cov(Xs , Xt ) = 21 [|s| + |t| − |s − t|], as asserted. [This shows that the d-dimensional L´evy-BM and Wiener-BM are in fact also related for d > 1 (d = 1 being identical)although Wiener-BM is a martingale and L´evy-BM is not. This is an interesting connection, and is given in Neveu (1968) which shows that the latter is a nonlinear functional of the former, and reinforces the importance of the Wiener-Itˆ o (complex) Brownian motion, and strengthens the motivation for the study of BM included in Section 6 in particular. [See also Gangolli (1967) on several of its important extensions.] 6. Let {Xt , t ∈ R} be a centered process with pairwise nontrivial Gaussian distributions (i.e. so for any s, t, s 6= t in R, cov(Xt , Xs ) 6= 0) process that is stationary and Markovian. This means that E(Xt ) = 0 and E(Xs Xt ) = r(s−t), as well as E Fs (Xt ) = Xs for any s < t where Fs = σ(Xr , r ≤ s) and E Fs (·) is the conditional expectation. Verify that the process then is actually Gaussian (i.e. all finite-dimensional distributions are Gaussian) with the covariance given by r(s, t) = βe−α|s−t| for some α > 0, β > 0, as observed by Doob (1942). Moreover, the process admits an integral representation relative to the BM as: Z t Xt = e−λ(t−u) dBu , λ > 0. −∞

This X-process is also called the Ornstein-Uhlenbeck process, which was formally derived first by the physicists after whom the process is named. [Hints: By stationarity r(s) = E(Xs+t Xt ) for all t. The joint (Gaussian) density of the pair Xs and Xt is then 1

[2π(1 − r2 (s)]− 2 exp{−(Xt − r(s)Xs ))2 /2(1 − r2 (s))}. The case of t1 < t2 < · · · < tn of Xt1 , . . . , Xtn is, using the Markovian property, the corresponding density ft1 ,...,tn (x1 , . . . , xn ) = [

Qn

2 − 12 i=1 (2π)(1−rj )]

exp{−

x21 1 2 −2

Pn

j=1

(xj+1 −rj Xj )2 } 1−rj2

where rj = r(tj ) and Xj goes with Xtj . Since r(t3 − t1 ) = E(Xt1 Xt3 ), we find with easy calculation of covariance from the density to be r(t3 − t1 ) = r(t3 )r(t1 ) which, written in fall, gives the key relation

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7 Multiple Random Measures and Integrals

r(t3 − t1 ) = r(t2 − t1 )r(t3 − t2 )

(+)

and the square of the covariance is bounded by the product of (the constant) variances. Hence (+) gives the functional equation r(s+ t) = r(s)r(t), where r(·) is bounded and measurable. Then the solution of this wellknown equation is r(s) = a0 e−αs , for some α > 0 and a0 > 0 where a0 = var Xt , all t. Taking a0 = β > 0, the result obtains. For the representation, note that Tt Xs = Xt+s , with s = 0, and generally implies that {Tt , −∞ < t < ∞} is a shift, so kTt Xs k22 = kXt+s k22 = a0 = kXs k22 by stationarity and that {Tt , t ∈ R} is a unitary operator family on L2 (P ) which is strongly continuous. Then by the classical spectral theoRt rem, Tt X0 = ( −∞ e−λ(t−u) dZ(u))X0 , ({Z(u), u ∈ R} being the resolution of the identity I.) If we take Z(u)X0 = Bu , u ∈ R, λ > 0 it is easily seen that {Bt , t ∈ R} is a process with orthogonal and stationary increments, being Gaussian we can easily conclude that it is BM. See also Doob (1942) for more on this process.] 7. In Exercise 4 above we have defined a mapping J connecting L2 (µ) and L2 (Rm , dt). Analogously we can define a sort of ‘convolution’ of µ with elements of L2 (µ), by the relation : Z ϕ(x + h) dµ(x), h ∈ S(⊂ L2 (µ) ⊂ S∗ ). (τ ϕ)(h) = S∗ Z ϕ(x) dµ(x − h) = S∗

and µ(·) = µ(· − h) ≪ µ holds if (and only if) h ∈ L2 (µ) by a well-known reh sult on “measurable norms”, and this is satisfied here so that dµ dµ exists and R R 2 2 1 1 (τ ϕ)(h) = S∗ ϕ(x)ehx,hi− 2 khk2 dµ(x) since S∗ ei(x,h) dµ(v) = e− 2 khk2 (by 2 1 h·,hi− 21 khk22 h Minlos’s theorem) and so dµ . Let w(eh·,hi ) = eh·,hi− 2 khk2 , to dµ = e get a Wiener-Itˆ o decomposition of w(eh·,hi ) by expending the exponential as ∞ X 1 w(ehx,hi ) = hw(x⊗n ), h⊗n i. n! n=0 Using these relations one defines another mapping, called an S-transform, on (still infinite-dimensional but smaller than) S-spaces defined by (Sp )β = {f ∈ L2 (µ) : kf kp,β < ∞}, 0 ≤ β < 1 and p ≥ 0 with the norms defined as ∞ X 1 kf kp,β = ( (n!)1+β k(Ap )⊗n fn k22 ) 2 n=0

d2 − du 2

2

+ u + 1 is the second quantization operation introduced where A = earlier and one can show that with (S)β as the projective limit of (Sp )β with p → ∞, (S)β ⊂ (Sp )β ⊂ L2 (µ) ⊂ (Sp )∗β , p ≥ 0. Now define the S-transform, with the duality mapping (h·, ·i) between (S)∗β and (S)β , by the following equation:

7.7 Complements and Exercises

S : (S)∗β → C, (Sϕ)(f ) = ((ϕ, w(eh·,f i ))),

367

f ∈ Sc

where In particular, if ϕ(x) P∞ Sc denotes the complexification of S. P ∞ = n=0 hw(x⊗n ), Fn i), ϕ ∈ (S)∗β , then (Sϕ)(f ) = n=0 hFn , f ⊗n i, f ∈ Sc , is the representation giving the S-transform. 8. This problem continues the ideas of the preceding one wherein the pairs of spaces (Sp )∗β ⊃ L2 (µ) ⊃ (Sp )β constructed concretely, admitting an abstract extension: Let H = L2 (µ), F = L2 (ν) where µ, ν are finite measures on the same measurable space. Consider in place of (Sp )β a pair of spaces F+ and H+ so that we generalize the triples F+ ⊂ F ⊂ F− (F− = (F)∗ ) and similarly consider H+ ⊂ H ⊂ H− (H− = (H)∗ , which are also called ‘rigged Hilbert spaces’ in Gel’fand-Vilenkin (1964, p.103) with a key supplement by Gould (1968) wherein the existence of a (not necessarily continuous) linear lifting is discussed in the space L2 (µ), since a continuous linear lifting need not exist on L2 (µ) in general. Define the mappings U± : F± → H± , and τF : F− → F+ . Similarly define τH , and τH : H− → H+ the mappings satisfying hU− y, U+ xiH = hy, xiH , ∀y ∈ F− , ∀x ∈ F+ so that the pair is meaningfully termed biunitary since in case F+ = F, H+ = H then U− = U+ becomes unitary between F and H. Show that the operators U− , U+ intertwine through τF , τH . [This extension was considered by Berezansky (1998) wherein the measures need not be Gaussian or Poisson, and the work has interest in such studies as generalized Fock spaces in Mathematical Physics and perhaps elsewhere.] 9. Here we present an opening to stochastic variational calculus. Recall that if f : D ⊂ Rn → R is a continuously differentiable function, then its gradient ∂f and the directional (∇f )(x) is the vector (fx1 , · · · , fxn )(x) where fx = ∂x i n derivative in the direction of a unit vector u ∈ R is ∂f ∂u = (∇f )(x) · u, the dot product of Rn . Here the domain D of f is an open set. To obtain a Hilbert space extension, consider the BM process X(·) on C0 (I), I = [0, 1], of continuous functions vanishing at 0, with the Wiener measure on its subspace H of C0 (I) whose members have square integrable derivatives (for Leb. measure) on I so that C0∗ (I) ⊂ H ⊂ C0 (I), continuous embedding. To obtain an infinite dimensional extension, we replace D by a differentiable manifold and the range space Rn by a C ∞ -topological vector space. [To have some generality, take I = G, a compact Lie group with a distinguished point 0, and Rn is replaced by G, the Lie algebra of G with its invariant metric, which is finite dimensional. (An elementary but a nice exposition of these matters is in Hausner and Schwartz (1968) which is sufficient for us.) If H is the space of absolutely continuous G-valued functions vanishing at 0, and C(G), consider (eh g)(t) = exp h(t) · g(t), t ∈ G, h ∈ H. We take G = I = [0, 1], the unit interval for concreteness.] Then the gradient of h f : G → R at h ∈ H is given by ((∇f )(g), h)H = (∂h f )(g) = df dt (e g)|t=0 whenever this exists. Let H0 ⊂ H be the class of functions that vanish at t = 1 also so that its elements vanish at both ends of I(= G). Since

368

7 Multiple Random Measures and Integrals

traditionally H is the space on which the Wiener measure µ is defined, consider the conditional measure µ1 determined by µ and the conditioning σ-subalgebra σ(X(1)). Note that it is regular so that µ1 can be treated as a measure and L2 (µ1 ) as a function space. If ν is the standard Gaussian product measure on Rn , then Gross (1975) has shown that for any f with a weak gradient (i.e., in the weak topology) for f ∈ L2 (ν) the following inequality is valid: Z Z 2 |∇f |2 (x)dν(x) + kf k22,ν log kf k2,ν , f (x) log |f (x)|dν(x) ≤ Rn

Rn

called the logarithmic Sobolev inequality which plays an important role in mathematical physics. Now ∇ being a differential operator is only closed and the above inequality implies that it is bounded from below. But ∇∗ ∇ is a self-adjoint positive definite operator and the finite dimensionality of the space we are working on is needed here. Motivated by the above inequality, a self-adjoint operator A on a more general Hilbert space L2 (µ1 ) is called a general log-Sobolev generator on it if the next inequality holds for all f ∈ dom(A): Z |f (ω)|2 log |f (ω)|dµ1 (ω) ≤ (Af, f ) + kf k22,µ1 . Ω

Verify the following three properties of such an operator A: (i) A is a positive (self-adjoint) operator using the fact that |x| log |x| is convex. (ii) If f : Ω → R such that ke−f k2,µ1 < ∞, then (the operator) A + f I satisfies the operator inequality A + f I ≥ −[log ke−f kµ1 ], ∀f ∈ L2 (µ1 ), and in fact this inequality characterizes ‘log-Sobolev operators’. [Here and in (iii) below I denotes the identity operator.] (iii) If A is such an operator between a pair of spaces L2 (Ωj , µj ), j=1,2, then A ⊗ I + I ⊗ A is again a log-Sobolev operator on L2 (Ω1 × Ω2 , µ1 ⊗ µ2 ). [For a discussion and extensions of the subject, see Gross (1991) and the references therein. This is a first step in extending the work to infinite dimensional Banach-Lie groups and their Lie algebras. For a survey of the latter see de la Harpe (1972).] Regarding Problems 7 and 8, with some work on Wick functions w(·), the analysis can be taken further. But it leads to specialized results, and we refer the reader to H.H.Kuo’s (1996) monograph, and related work to Holden et al (1996) for related discussion and details and we conclude the general treatment of the chapter for now.

Bibliographical Notes

369

Bibliographical Notes The material in this chapter has focused on several aspects of multiple random measures and applications, partly to motivate the necessary new concepts and mostly to exhibit the individual nature and power of these vectorvalued set functions and their integrals. Both demand new techniques as well as showing the inadequacy of the classical vector measure analysis methods. Here the first section is devoted to quasi-martingales and their integrals. The analysis is for c` adl` ag processes. The treatment centers around the basic study given by Brennan (1978) which is partly a generalization of the fundamental treatment in Cairoli and Walsh (1975) for the planar Brownian motion. The extension is not simple and straight forward. Comparisons with the approach by McShane (1976) and the classical point of view of Stieltjes integration (without stopping times being used by L.C.Young (1974)) are compared and contrasted. The basic study following Meyer’s school appears here perhaps for the first time in a relatively advanced volume such as this one. Much of the available literature has been discussed in the textual treatment. The integration is nonabsolute, and often is non-Lebesgue type. The preceding analysis is then continued for the multiple random measures in the rest of the chapter. Sections 2 and 3 are devoted to cartesian products in which Brownian motion is prominent. But Poisson processes are also treated in detail. In both cases one finds the key role played by Hermite polynomials and the Poisson–Charlier class respectively for the continuous and discrete classes. They both use several important properties of orthogonal polynomials in this study. The work by Engel (1982), directed by S. Kakutani, has been very important for this analysis. When non-cartesian products are treated, as in Section 3, there are several new questions, and Brennan’s work has initiated this study. The related extended work on integration is based on the analysis by M. L. Green (1997). A direct extension fails as shown by an example of Bakry (1981). Hence a more detailed study and a new approach appear to be essential. Our treatment is primarily based on Bochner’s boundedness principle for which one has to find (nontrivially) a bounding measure. Using this and constructing the bounding (product) measure, the (non-cartesian) multiple stochastic integral is presented in the somewhat long Section 3. Since the technical details and computations are many, sometimes the main ideas are sketched here and in other places, so that the reader will gain a fair idea of the subject, but may need to consult the original sources on fine points. The Bochner boundedness principle is perhaps treated in this generality and specificity only in this book. I have also discussed some key aspects of this principle in my earlier volume (1995). It is contrasted here in the L2 (P )-context with the Grothendieck’s fundamental theorem of metric geometry, and concluded that both these complement each other in many respects. [See also Sections 9.2 and 9.3 later for further discussion.] After this, a natural application to stochastic Gauss-Green-Stokes theorems is inviting and so is included. This raises the question of a further

370

7 Multiple Random Measures and Integrals

study of these measures and integrals for future applications and the analysis of differential forms. It would seem that the work of Noltie (1986) on chains and measures in spaces isomorphic to Hilbert spaces given in the next chapter could be used in extending this analysis as well as Itˆ o’s theory of random currents. This will in fact be discussed there. Dozzi’s (1989) work is of interest here although it is from a somewhat different angle. See also Bichteler and Jacod (1985) for analysis of specialized classes of random measures. Also Kussmaul (1977) should be noted in this context. One of the difficulties in higher dimensions is the lack of a linear order. So one has to settle for partial orders and recall properties of posets in analysis. This point of view was pursued by H¨ urzeler (1984, 1985). His work has been of basic interest here and shows how combinatorial analysis, especially M¨ obius function, plays a key role. We have included some of his results, and also because of continuous emphasis on this aspect by G.C. Rota in several of his publications on the use of combinatorial methods in probability, we have included a large part of the recent study by Rota and Wallstrom (1997) in Section 5. Several of their results and the main ideas of the arguments are discussed. This gives a new perspective on higher-order stochastic integrals of noncartesian type which no other book seems to have discussed in detail. Again we see that the L2,2 -boundedness principle plays a basic role here also. The preceding work brings us back to Brownian motion and its use in multiple integration especially considering it as a process which is studied for generalized fields and on Schwartz spaces. This allows one to differentiate the BM in the weak (or Schwartz) sense and study its derivative process, called white noise. This was originated by Hida (1975) and his admirable book on Brownian motion (1980). See also the work by Hida and Hitsuda (1993) which is important and useful for our presentation. Section 6 has a considerable analysis both specialized (Wiener–Itˆ o) multiple series expansions, the key role played by Hermite polynomials etc. and the generalized study (Fourier–Wiener representations) where Hida’s work is utilized in. The analysis presented here has been of special interest to many researchers (cf. Kuo (1996) and also Holden, Øksendal, Ubøe and Zhang (1996)) for further study and applications to (stochastic) differential equations. But we did not go into the latter area which is expanding rapidly in both directions of ordinary, and partial differential equations with specializations. Also see an interesting general early exposition on this topic presenting an overview in McKean (1973). It is seen that in the first two sections we also included the integration of c´ agl´ ad processes which did not yet get into the main stream. This was developed by Brennan (1980), and it is studied here with many details, and presented an example in the problems section to show that the earlier brief discussion by Meyer (1976) has not been the same as the one in the text. In fact the definition by Brennan is more general and appears to be better oriented to applications as it is based on an extension of the Riemann-Stieltjes concept. Since most of the

Bibliographical Notes

371

references are included in the text, further detailed discussion will be curtailed here. The Gel’fand triples with extensions as well as indications of stochastic log-Sobolev inequality, due to Gross (1991) in the Exercises section, are worthy of note. It may be observed that the use of Bochner’s boundedness principle gives an abstract framework for the theory. In particular applications, the methods of verification of the hypotheses will often be nontrivial. This is well exemplified by the work of Kuo and Russek (1988) where it is shown how the anticipative and the general integrands even for the BM need much more work in connecting them. Of course this is also true of all abstract integrations in infinite dimensions, irrespective of stochastic analysis. With this extensive (multiple) random integral and measure analysis of this somewhat long chapter, we proceed to give a brief account of classical (i.e., usually restricted to Banach space valued) vector measures in the final two chapters.

8 Vector Measures and Integrals

In the preceding chapters we have discussed at length different types of integrals relative to random measures which take values in vector spaces based on a general probability space. Here we consider vector spaces that are normed but not necessarily related to any underlying probability or other measure spaces and the vector measures will be general enough that they have unbounded variation. The spaces of integrable functions relative to such (vector) measures, some aspects of suitable convergence theorems and the structures of the resulting linear spaces will be discussed. Analogs of Lebesgue spaces for them are also considered. Vector measure integrals are of interest in studies of spectral analysis of certain operators and in classes of random fields as well as the joint analysis of problems involving aspects of tensor products of such spaces. They will be studied. Some of these matters lead to bimeasures and more generally multi- (or poly-) measures, naturally involving different types of variations (Fr´echet, Hardy, Vitali etc.). The latter subject will again be taken up for more detailed analysis in the next (and final) chapter of this book, but the introductory and motivational aspects of these will be discussed here in the last part. One of the main purposes is to compare and contrast how distinct differences between the random and purely vector measure points of view show up both in the types of results considered and the importance attached to their individual inferences and applications that are detailed. Naturally in all these cases finite (Vitali) variation of measures cannot be demanded.

8.1 Vector Measures of Nonfinite Variation Motivated by the analysis of the preceding chapter, it is desirable to consider vector measures and their integrals not be restricted by finiteness conditions on the variations so that the work can also be used for classes of stochastic functions. The primary analysis (and integration) on this subject

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8 Vector Measures and Integrals

seems to be that due to Bartle, Dunford and Schwartz (1955) with scalar integrand, and extended to the vector (or operator) functions by Bartle (1956) followed by Dinculeanu (1957). In these cases the vector integrands as well as integrators are special in the sense that hypotheses do not depend on filtrations (except for the trivial ones namely the same fixed measure space that is in the background), and this distinguishes the stochastic and non stochastic or classical cases. We specialize the latter in the following analysis which will be of interest in later work. The basic formulation and analysis of Morse-Transue (or MT-) integration with its necessary restatements (and some suitable strengthening) play a key role in bimeasure (or bilinear) integrations which will take the center stage here. This has been extended to multi- (or poly-) measures and integrals by Dobrakov (1970–90) which are streamlined and reworked by Panchapagesan (2008), and it will be compared with our view here. All this work is pointed towards specialized analysis than that of the earlier chapters. We first recall vector measure variations: Definition 8.1.1 Let (S, S) be a pair with S as a δ-ring of subsets of S and µ : S → X be σ-additive in the norm topology of a Banach space X. Then µ has finite semi-variation on A denoted by kµk(A) < ∞, if for each A ∈ S, ai ∈ C, |ai | ≤ 1 n X µ(Ai )kX : Ai ∈ S(A), disjoint, n ≥ 1 , kµk(A) = sup k

(1)

i=1

where S(A) = {A ∩ B ∈ S; B ∈ S} and µ is said to haveP finite (Vitali) variation on A, denoted |µ|(A), if the sum in (1) is replaced by ni=1 kµ(Ai )kX with all the other conditions remaining in force.

It is clear that kµk(A) ≤ |µ|(A), often with strict inequality. For instance R if X = {Xt , t ≥ 0} is Brownian motion or BM, and µf (A) = A f dX for a bounded measurable f : R+ → R, then |µf |(A) = ∞ for all nondegenerate intervals A, but kµf k(A) < ∞. However, as shown in Theorem 1.3.4, a vector measure with values in a Banach space has a controlling measure. Here we present a somewhat more general result, useful for many other applications. For extending Theorem 1.3.4 to multi-dimensions, the following concept discussed by Dinculeanu (2000) will be useful. Definition 8.1.2 Let M = {µ : S → X} be a set of σ-additive X-valued measures µ on the ring S of S. The set M will be called (uniformly) σ-additive if for every sequence An ∈ S, An ↓ ∅ one has limn→∞ µ(An ) = 0 uniformly in µ ∈ M. It may be verified that the above condition is equivalent to σ-additivity of the (vector) measures µ satisfying P∞ For any sequence Bn ∈ S, disS∞the condition: S∞ joint with n=1 Bn ∈ S, then µ( n=1 An ) = n=1 µ(An ), the series converging uniformly in µ. The desired extension of 1.3.4 is given by:

8.1 Vector Measures of Nonfinite Variation

375

Theorem 8.1.3 If M is any set of X-valued uniformly σ-additive (vector) measures then the set M is uniformly dominated by a positive finite (or controlling) measure, λ : S → R+ so that µ ∈ M ⇒ µ ≪ λ and λ(A) ≤ sup sup kµ(B)k

(2)

µ∈M B⊂A

for all measurable A, B(∈ S, taken as a σ-algebra). Proof. Since S is a σ-ring and M is a set of vector measures, for each x⋆ εX⋆ and µ ∈ M, the associated scalar collection {x⋆ ◦ µ : x⋆ εX⋆ } is a class of signed measures on S. But a real or complex measure on S is bounded, and the above set is pointwise bounded, i.e. it is bounded for each x⋆ in X⋆ . Since the latter space is complete, the uniform boundedness theorem implies that sup{|x⋆ ◦ µ| : kx⋆ k ≤ 1} < ∞. This is usually stated for S a σ-algebra and the variations |x⋆ ◦ µ|(S) are bounded). In view of Kluv´anek’s observations (cf. Propositions 1.3.1), both are equivalent statements. But there is a controlling measure λµ : S → R+ for the vector measure µ by Theorem 1.3.4. From this using the uniform σ- additivity of the set M, we next deduce the desired single λ that dominates (or controls) the entire collection of the family M. ˜ = {x⋆ ◦ µ : µ ∈ M, x⋆ εX⋆ } which is a (uniformly) bounded Consider M 1 set of scalar measures on S where X⋆1 is the ball of functionals from the adjoint space X⋆ each with unit norm. If we show that there is a controlling measure ˜ such that λ for M λ(A) ≤

sup sup{|(x⋆ ◦ µ)(B)| : B ∈ S(A)}, A ∈ S,

(3)

˜ x⋆ ◦µ∈M

on the σ-algebra S(A) = {B ∩ A : B ∈ S}, the trace of S over A, then the main result of the assertion follows. Indeed by the permissible exchange of the ‘sups’ in (3), the right side of (3) is seen to be exactly sup{kµ(B)k : µ ∈ M} for B ∈ S(A) which gives (2). Thus it suffices to establish: (i) there is ˜ (the scalar class) and (ii) the a finite measure λ : S → R+ , dominating M inequality (3) is obtained from it. The necessary argument for this is essentially the same as that of the proof of Theorem 1.3.4 which is an indirect one, a somewhat simplified form, due to Gould (1965) from that given in Dunford and Schwartz ((1958), Theorem IV.9.2). It will be outlined here for completeness and emphasis. (i) The key observation is to show that there exist, for each ε > 0, a finite subset Jε of indices and a δ > 0 such that |νj |(A) = 0, j ∈ Jε im˜ where νi = x⋆ ◦ µ(∈ M) ˜ inplies |νi |(A) < δ for all i, i.e., all νi ∈ M dexed by I. An indirect short argument was employed in Step 4 of the proof of Theorem I.3.4 to establish this fact, and it is a key part. This means that there exists a countable collection of finite Jk ⊂ I indices such that

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8 Vector Measures and Integrals

|νj |(A) = 0, j ∈ Jk ⇒ |νi (A)| < ε where ε = k1 and i ∈ I. This key observation P |νi | (proved there) implies that λk = ki=1 2−i 1+|ν , is a finite measure (in fact i |(S) P∞ −n it is bounded by 1). Then λ = n=1 2 λn is similarly a finite measure and each νi is λ-continuous for all i ∈ I uniformly, as shown in the earlier result. This establishes (i). (ii) Returning to M, the relations between λ, λi and νi introduced above give for each A ∈ S, (νi = x⋆ ◦ ν, kx⋆ k ≤ 1) and by the uniform σ-additivity of the µ ∈ M, one has the following: X λ(A) ≤ sup λk (A) 2−n ≤ sup |νi |(A) k

i∈I

n≥1

⋆

= sup |x ◦ µ|(A) ≤ sup kx⋆ k≤1

kx⋆ k≤1

µ∈M

µ∈M

sup kx⋆ ◦ µk

B∈S(A)

= 4 sup kµ(B)k. B∈S(A)

µ∈M

This gives (ii), and the theorem follows. 2 The point of this result is an extension of Theorem 1.3.4 from one vector measure to a family of such measures, denoted by M here, under the (additional) uniform σ-additivity condition. As the proof of the key earlier result shows that the indirect argument is used in different (nontrivial) ways. Also the fact that (S, S) is a measurable space with S ∈ S was used in the construction of λk in the above proof where λk (·) =

k X i=1

2−i

|νi (·) 1 + |νi |(S)

for a finite set of indexes Jk . By the uniform σ-additivity the set {νi , i ∈ I} is bounded. Thus 1 + |νi |(S), can be replaced by 1 + supA∈S |νi |(A) < ∞ for each i (by Kluv´anek’s observation), Proportion I.3.1. Consequently λk is a finite measure and the rest of the argument holds if S is a δ-ring instead of a σ-algebra. It is hence only necessary that supA∈S |νi |(A) < ∞. Then the result extends to δ-rings as in Theorem 1.3.4 and the λ will be localizable, instead of finiteness. This extension is given below for reference as, (see Rao (2004), p. 73ff, on localizability): Corollary 8.1.4 Let (S, S) be a measurable space with S as a σ-ring and µ : S → X, (a Banach space) a vector measure. If M is a collection of uniformly σ-additive (vector) measures µ on S to X, then there exists a finite measure λ : S → R+ relative to which M is uniformly λ-continuous and in fact satisfies an inequality of the type given in (2). If S is only a δ-ring, and S S = n≥1 Sn is a countable union where Sn ∈ S, then again there exists a σ˜ that dominates M. Without the finite and hence an equivalent finite measure λ ¯: S → R ¯ + that is localizable and dominates last restriction on S, there exists a λ M.

8.1 Vector Measures of Nonfinite Variation

377

The preceding discussion also has an interesting consequence for operator valued measures that are σ-additive in the strong (not uniform) topology and it has some applicational potential. Proposition 8.1.5 Let S be a σ-ring of a set S and X a separable Banach space. If µ : S → B(X) is an operator valued strongly (not in uniform norm) σ-additive, then there exists a measure λ : S → R+ such that µ is λ-continuous. Moreover, if µn : S → B(X) is a sequence of strongly σ-additive measures, as above, and µn (A) → µ(A), in the strong (operator) topology for each A ∈ S, then µ is strongly σ-additive and the sequence µn is uniformly strongly σ-additive. Proof. By hypothesis for each x ∈ X, µ(·)x : S → X is σ- additive and is a vector measure (into the Banach space X). Hence there exists a dominating measure λx : S → R+ for µ(·)x. Since X is separable, there exists a dense denumerable collection {xi , i ≥ 1} ⊂ X and λxi being the corresponding finite measure, let λ : S → R+ be defined by λA (·) =

∞ X λxi (·) 1 : S(A) → R+ i 2 1 + λ x i (A) i=1

for each A ∈ S, so that S(A) is a σ-algebra. By the preceding Corollary 8.1.4, there actually exists a finite measure that dominates each {µ(·)xi , i ≥ 1}. Since {xi , i ≥ 1} ⊂ X is dense, this easily implies that λ dominates {µ(·)x, x ∈ X}. Finally if µn (·)x : S → X converges to µ(·)x for each x ∈ X then by the vector extension (due to Nikod´ ym) of the Vitali-Hahn-Saks theorem µ(·)x : S → X is a vector measure and that the convergence is uniform in x. So by Theorem 8.1.2, and the first part, the inequality (2) holds for the set {µn (·)x, x ∈ X, n} and it is bounded. This easily implies all the assertions of the proposition with strong uniform convergence of µn (·), towards µ(·)x for each x ∈ X, as desired. 2 If S is a δ-ring and S = ∪∞ n=1 Sn , Sn ∈ S, then the result extends to this case as before without difficulty since S(Sn ) is a σ- algebra for each n because a sequence {λn , n ≥ 1} of finite measures determines an equivalent (finite) measure λ on S that dominates all {µ(·)x, x ∈ X}. These observations will be of interest in considering (vector) integrals relative to the operator measure µ considered above. Consequently we devote the next two sections for (vector) integrals relative to such (vector or operator) measures without demanding that they have finite variations and presenting some limit theorems and differentiation results.

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8 Vector Measures and Integrals

8.2 Vector Integration with Measures of Finite Semivariation, Part I In the preceding section some properties of vector and strong operator measures with finite semi-variation have been detailed for use in integration of suitable functions relative to them. Let (S, S) be a measurable space where S is a δ-ring of subsets of S, and Z be either a Banach space or Z = B(X), the B-algebra of all bounded linear operators on a separable Banach space X. If µ : S → Z is a strongly σ-additive function (to be again termed a vector measure), where in the case Z = B(X) it is strongly σadditive. Let λ be a dominating measure λ : S → R+ provided by Proposition 8.1.5 or Corollary 8.1.4 of the preceding section. There it is moreover assumed that S = ∪∞ n=1 Sn , Sn ∈ S for each n. Of course λ is not unique, but this is not of concern at this point. The semi-variation kµk(·) of µ is given by (this is stronger than Definition 1.1 above since Z is too large!) kµk(A) = sup{k

n X i=1

µ(Ai )zi k : Ai ⊂ A, Ai ∈ S, kzi kZ ≤ 1, 1 ≤ i ≤ n}, A ∈ S,

(1) which in case Z is a Banach space, or zi are scalars, |zi | ≤ 1. A simple function f : S → Z is given by f=

n X i=1

zi χAi , Ai ∈ S,

zi ∈ Z.

Here the Ai are disjoint and the integral is as usual defined by Z n X µ(Ai ∩ A)zi , A ∈ S. f dµ = A

(2)

(3)

i=1

It is a standard computation to see that the integral is well-defined and does not depend on the representations of f. In general a vector function f : S → Z is said to be µ-integrable if there is a sequence of simple functions fn of the form (2) such that fn → f, µ a.e. (meaning a.e.R λ where the latter is a controlling measure for µ) and such that limn→∞ ARfn dµ exists in theR norm of Z for all A ∈ S. Here the limit is denoted by A f dµ(= limn→∞ A fn dµ). That the limit exists independently of the sequence {fn , n ≥ 1} is a nontrivial fact, and it depends on the availability of a controlling (positive) measure λ. If Z is a standard Banach space, this was established by Dunford and Schwartz (1958, IV.10.8) and the case that Z = B(X) is discussed by Dobrakov (1970, p.521). The essential point being the existence of λ in both cases, as established in Corollary 8.1.4 and Proposition 8.1.5 in order to invoke the Vitali-Hahn-Saks theorem and its Nikod´ ym extension for the vector valued functions. We now present the precise result with essential detail for a convenient reference.

8.2 Vector Integration with Measures of Finite

Semivariation, Part I

379

Theorem 8.2.1 Let (S, S) be a measurable space, S being a σ-ring (or a δring but S = ∪∞ n=1 Sn , Sn ∈ S) and suppose ν : S → Z is a vector measure. Then there exist a pair (ϕ, λ) where ϕ : R → R+ is a convex increasing function ϕ(x) + is a measure such that (ϕ(0) = 0, ϕ(·) |x| ր ∞ as x → ∞, and λ : S → R symmetric can be demanded) n X kν(Ai )k 1 ϕ k λ(Ai ) : Ai ∈ S(A)] , kνkϕ (A) = inf k > 0 : [1 + k λ(Ai ) i=1

(4)

the Ai are disjoint, (ϕ, λ) being determined by ν and X only, λ dominating ν. Moreover, if {fn , n ≥ 1} is a sequence of ν-integrable functions, fn : S → Z, with Z = C or R, and |fn | ≤ g where g is ν-integrable and fn → f pointwise on S − N, kνkϕ (N ) = 0, then f is also ν-integrable and further(k · kϕ of (4) is a norm for ν) Z Z f dν,

fn dν =

lim

n→∞

A

A

A ∈ S.

(5)

Proof. Since µ : S → Z is σ-additive, then for disjoint An ∈ S with A = S ∞ ⋆ ⋆ n=1 An ∈ S, we have on writing Z1 for the unit sphere of the adjoint space Z of Z, that 0 = lim kµ(A) − n→∞

n X

k=1

µ(Ak )k = lim sup |z ⋆ ◦ µ(A) − n→∞ z ⋆ ∈Z⋆ 1

n X

k=1

z ⋆ ◦ µ(Ak )|

so that the set of scalar measures {z ⋆ ◦ µ, z ⋆ ∈ Z⋆1 } is uniformly σ-additive. Then the preceding discussion shows that there exists a finite positive measure ⋆ λ : S → R+ which dominates the above set of scalar measures and gz⋆ = d(zdµ◦µ) exists. Moreover Z (6) gz⋆ dλ = lim |z ⋆ ◦ µ(B)| = 0 lim λ(B)→0

B

λ(B)→0

uniformly in z ⋆ ∈ Z⋆1 so that the set {gz⋆ : z ⋆ ∈ Z⋆1 } ⊂ L1 (S, S, λ) is uniformly integrable for λ. Hence by a classical de la Vall´ee Poussin theorem (cf.,e.g., Rao (1981), pp.17–20) there exists a convex function ϕ ≥ 0, (can and will be taken symmetric with ϕ(0) = 0) such that ϕ(x) |x| ր ∞ as in the statement and (4) holds. The properties of ϕ and the ‘ϕ-variation of µ’ relative to λ are classical and have been described in the above reference, as well as in many other places on Orlicz spaces which are familiar and need not be repeated. [The relevant analysis of Orlicz spaces leading to the explicit representation of the norm can be obtained easily from Rao and Ren (1991, p.22).] R Now let fn be ν-integrable such that if νfn (·) = (·) fn dν, n ≥ 1 which exists on S by hypothesis, suppose that limkνk(A)→0 kνfn (A)k = 0 uniformly in n. This holds automatically if |fn | ≤ g where g is ν-integrable. We assert that (5) is

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8 Vector Measures and Integrals

true and when fn → f on S − N pointwise where kνk(N ) = 0,R then f is also νintegrable. Now from (1)-(3), it is seen that kνkϕ (A) = sup{k A f dνk : kf kψ ≤ 1}, where ψ(x) = sup{|x|y − ϕ(y) : y ≥ 0}, called the complementary Young R function, is convex and kf kψ = inf{k > 0 : S ψ( fk )dλ ≤ 1} with ν ≪ λ, to mean that ν is λ-continuous, and kνfn (A)k ≤ kνfn k(A) ≤ 4 kνfn (A)k ≤ 4kfn k∞ kνk(A). Moreover with the variation ‘| · |’ of scalar measures, kνfn (A)k ≤ sup {|x⋆ ◦ νfn |(A) ≤ sup |x⋆ ◦ ν|fn | |(A).

(7)

kx⋆ k≤1

kx⋆ k≤1

With the definition of ν-integrability of fn , there exist simple gn,k such that gn,k outside a set of kνk(·)-measure zero, and R R k → ∞, pointwise R → fn , as k A gnk dν − A gnk′ dνk = k A (gnk − gnk′ )dνk → 0 for A ∈ S, as k, k ′ → ∞. 1 ) ≤ 2−n , Hence for each n, there is a kn such that kνk(ω : kgnkn − fn k(ω) > 2n and Z Z fn dνk < 2−n . (8) gnkn dν − k A

A

Thus for each ε > 0 one must have

ε ε } ∪ {ω : |fn − f |(ω) > }. 2 2 R Since gnkn → f in kνk(·) measure, and by hypothesis the integrals { A fn dν, n ≥ 1} are uniformly absolutely continuous for kνk(·), (8) implies that Z

Z

1

(9) dν g ≤ k fn dνk + n , A ∈ S.

nkn 2 A A {ω : |gnkn − f |(ω) ≥ ε} ⊂ {ω : |gnkn − fn |(ω) >

Hence from (8) and (9) we conclude that f is ν-integrable and Z Z f dν = lim fn dν, A ∈ S. A

n→∞

(10)

A

Now in general if fn → f pointwise on S − N where kνk(N ) = 0, and |fn | ≤ g with g being ν-integrable, then the fn are uniformly absolutely integrable relative to ν. So the integrals satisfy the hypothesis of (8)- (10) using (7). Hence fn → f in kνk(·) measure and the collection {fn , n ≥ 1} is uniformly integrable for ν so that f is also, and what is more, (10) holds. This implies that (5) is valid as asserted. 2 The following consequence, contained in the above proof is worth separating, as being an analog of the dominated convergence. Corollary 8.2.2 (Dominated convergence for vector measures) Let (S, S) be a measurable space and µ : S → Z be a vector measure. If fn : S → C is a sequence of measurable C-valued functions such that fn → f pointwise on S − N with kµk(N ) = 0, and if |fn | ≤ |g| where g is µ-integrable, then fn and f are µ-integrable so that the limit relation (5) holds.

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Semivariation, Part I

381

In the above work, the case that Z = B(X), and µ : S → Z is strongly σadditive (operator) measure, has not been discussed. But it was already noted that, if Z is as above, in Proposition 1.5 (i.e., of Section 1), then for separable X, the strongly σ-additive µ has a controlling measure. Then starting with simple functions and extending via Dunford-Schwartz integration,R and the VitaliHahn-Saks theorem (along with Nikod´ ym’s extension) that A f dµ, A ∈ S, is well-defined and the corresponding vector analysis extends. An interesting question is to allow f to be an operator along with µ. Since in the theory of Dunford-Schwartz integration of (scalar) functions relative to a vector measure µ is through a controlling measure λ which is positive and finite, one can define the exceptional (null) sets through the use of λ (hence the classical Lebesgue theory is usable). But since there exist many such controlling λ’s, it is sometimes convenient to designate the exceptional sets through the semi-variation kµk(·) of µ itself, which has the σ-subadditivity property. For convenience we state the desired concept as: Definition 8.2.3 Let µ : S → Z be σ-additive where (S, S) is a measurable space with S as a δ-ring. An element N ∈ S is µ-null if µ | S(N ) ≡ 0 where S(N ) is the trace σ-ring (actually an algebra now) on N and the symbol is the restriction of µ to S(N ). [So if λ is controlling µ then the ring of λ-null sets being the same as kµk(·) null sets, is contained in S(N ).] An extension of the D–S integration of scalar measurable functions relative to a vector measure on a σ-algebra has been systematically explored and extended by Dinculeanu (1967), and the necessary modification for δ-rings has been the focus of Dobrakov in a series of works starting in 1970, for which some modifications are needed. Thus for instance f : S → Z is µ-measurable on (S, S) if there is a sequence of simple functions fn : S → Z such that, as n → ∞, fn (s) → f (s) for s ∈ S − N, kµk(N ) = 0 or equivalently outside of a λ-null set with λ as a controlling finite positive measure which exists. The following simple characterization will be useful later on. Proposition 8.2.4 Let (S, S) be a measurable space (S a σ-ring), and µ : S → Z be a vector measure where Z is a weakly sequentially complete Banach space. Then a measurable scalar function f : S → C is weakly µ-integrable if and only if it is Dunford-Schwartz integrable, and both have the same integral. Proof. The Dunford-Schwartz integral of a scalar function relative to a vector measure is defined in terms of the strong convergence of approximating sequences and hence it implies the weak convergence also. This means that f is weakly integrable for µ, and X can be any Banach space. It is the converse that uses the weak sequential completeness, and we include the detail in directionR by showing that for any weakly integrable f : S → C, one has Rthis w w f dµ → A f dµ. It is to be shown that for A ∈ S, z ⋆ εZ⋆ , the following A n holds:

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8 Vector Measures and Integrals

(i)f ∈ L1 (z ⋆ ◦ µ), (ii)z ⋆ and (iii)

Z

w

f dµ =

A

Z

w

A

Z

f dµ

=

w

f χA dµ =

S

Z

A

Z

f (z ⋆ ◦ µ),

f dµ

(11)

A Rw A f

the last being the Dunford-Schwartz integral. Here dµ is the weak integral in the sense that for each simple function f → f pointwise a.e. outside of an n R N, kµk(N ) = 0, A fn dµ → xA ∈ X, A ∈ S, for a unique element xA of X Rw which is denoted A f dµ. We now establish (i) – (iii) to complete the argument and then explain Pnits consequences. Let fn = i=1 ain χAin , Ain ∈ S, ain ∈ C be a simple function (Ain disjoint) such that fn → f pointwise on S − N, kµk(N ) = 0 so that R R z ⋆ ( A fn dµ) = A fRn d(z ⋆ ◦ µ) where z ⋆ ◦ µ is a signed measure on S, and by hypothesis {z ⋆ ( A fn dµ), n ≥ 1} is a numerical convergent sequence for each z ⋆ ∈ Z⋆ and A ∈ S. Since L1 (z ⋆ ◦ µ) is a weakly complete (scalar) Lebesgue space, fn → f˜ weakly in this space for each z ⋆ it follows that f˜ remains the same for each z ⋆ so that f˜ ∈ ∩{L1 (z ⋆ ◦ µ) : z ⋆ ∈ Z⋆ − {0}. If f∞ = f˜, then {fn , 1 ≤ n ≤ ∞} ⊂ L1 (z ⋆ ◦ µ) is a weakly compact set and so is {|fn |, 1 ≤ n ≤ ∞} by the classical results (cf., Dunford-Schwartz R (1958), Also it converges weakly so that one has A |f˜|d(z ⋆ ◦ µ) = R IV.8.10). limn A |fn |d(z ⋆ ◦ µ), A ∈ S. Consequently by the Vitali-Hahn-Saks theorem {fn , n ≥ 1} is uniformly integrable relative to z ⋆ ◦ µ, z ⋆ ∈ Z⋆ . It then follows by the Vitali convergence theorem that fn → f˜ in norm of L1 (z ⋆ ◦ µ) so that fn → f˜ in |z ⋆ ◦ µ|(·) measure, implying that f˜ = f a.e.[z ⋆ ◦ µ], and hence f ∈ L1 (z ⋆ ◦ µ) for all z ⋆ ∈ Z⋆ , proving (i). This also establishes (ii) above since Z Z Z lim z ⋆ ( fn dµ) = lim fn d(z ⋆ ◦ µ) = f˜d(z ⋆ ◦ µ) n n A A A Z w Z f d(z ⋆ ◦ µ) = z ⋆ = f dµ , by hypothesis. A

A

So far the weak completeness of Z is not used. This is needed at this point to invoke a result of D.R.Lewis (1970, Theorem 2.4) which is recalled below. It implies that under present conditions, f Ris Dunford-Schwartz integrable for the R w vector measure µ and by (ii) A f dµ = A f dµ for A ∈ S. This gives (iii) and (ii) hence the proposition. 2 We now state for reference the following form of: Theorem 8.2.5 (Lewis) Let Z be a sequentially complete Banach space (or a locally convex linear [separated or Hausdorff ] topological space). If f : S → C is a complex measurable (for S) function and µ : S → Z is a vector measure, then f is µ-integrable (as defined above) if and only if either (i) there is a sequence of simple functions fn → f pointwise on S − N, kµk(N ) = 0, for which

8.2 Vector Integration with Measures of Finite

Semivariation, Part I

383

R

{ A fn dµ, n ≥ 1} is Cauchy in Z for each A ∈ S, or (ii) there is a sequence of bounded measurable fn eachR of which is µ-integrable with fn → f pointwise on S − N, kµk(N ) = 0, and { A fn dµ, n ≥ 1} is Cauchy uniformly relative to A ∈ S. Remark 8.1 In our application, we used the result of part (i) with simple functions to deduce the f integrability. The proof of (i) was not detailed in the Lewis’s paper, and that is obtainable through the Vitali convergence of Theorem 1, using the uniformity hypothesis, or the dominated convergence of Corollary 2 when the {fn , n ≥ 1} is shown to be uniformly integrable essentially as in Theorem 1. In any case the result is convenient for applications, although some detail in the original paper has to be filled in, for completeness and helping some others here. Also S the fact that S is a σ-ring can be replaced by the δ-ring hypothesis if S = ∞ n=1 Sn , and Sn ∈ S, n ≥ 1, a small but useful improvement. If Z = B(X), the algebra of bounded linear operators on the separable Banach space X, then µ : S → Z has a controlling measure, and the integral R f dµ is defined for all simple S-measurable X-valued functions. The mapA R ping A 7→ A f dµ is again a vector measure with values in X. Also note that if fn is integrable relative to µ in the above R sense and fn → f , pointwise on S − N, kµk(N ) = 0, and that ν : A 7→ A fn dµ, A ∈ S, n ≥ 1, is uniformly σ-additive with respect to µ, then as in the preceding analysis f is also µ-integrable and (5) holds in this context. If X is not necessarily separable, and B(X) is replaced by Z = B(X, Y), the Banach space of continuous linear mappings from X into Y, then new conditions need be assumed to extend the Z-valued measures µ which may not have controlling measures under the previous hypothesis. This problem was considered by Dinculeanu (2000), and the additional new conditions are somewhat more difficult to verify in specific applications discussed below than those of B(X). They are therefore not fully detailed. An application of the above work is discussed here to show that the integral of a vector valued function relative to a vector measure cannot be understood in the standard Lebesgue-Stieltjes sense. In fact, we consider an example. Let µ : S → L2 (Ω, Σ, P ) = L2 (P ) where (S, S) is a measurable space (S a δring). Suppose that for each A ∈ S, µ(A) is a (real) Gaussian random variable with mean zero and variance λ(A), i.e. E(µ(A)) = 0 and E(µ(A)2 ) = λ(A) where λ(·) is in addition a nonatomic measure on S. For instance (S, S) can be taken as S = Rk , k ≥ 1, and S as the δ-ring of bounded Borel sets, λ as the Lebesgue measure and µ(·) will be the Brownian measure. Such a choice is possible since by the classical Kolmogorov existence theorem we can construct a BM measure, as discussed in earlier chapters. However µ(·)(ω) is additive but not σ-additive, and hence does not define a (signed) measure for each ω except for those ω(∈ S) of a set of λ-measure zero. There is no selection (or a “lifting”) theorem to have µ(·)(ω) and a σ-additive function. But µ(·) : S → L2 (P ) is defined and σ-additive in norm, having finite

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8 Vector Measures and Integrals

R semivariation. Consequently one can define the integral A f dµ, A ∈ S, for all simple and later µ-integrable functions f. This point is elaborated as follows. Recall that a mapping µ : S → B(X, Y)(= Z), the Banach space of continuous linear operators from the Banach space X (or is only a normed linear space, i.e., non-complete space) to a Banach space Y (completeness is essential here) with the uniform norm. Now µ is additive on disjoint sets. It is said to have finite semivariation if kµk(A) < ∞ where n o n X µ(Ai )xi kY : kxi kX ≤ 1, Ai ∈ S(A), disjoint . kµk(A) = sup

(12)

i=1

P Since for simple functions f = ni=1 xi χAi , the integral of f relative to µ is just the corresponding sum, this is equal to expressing it as: n Z o

kµk(A) = sup f dµ : kf k ≤ 1, f is S(A)-simple . (13) A

The following representation for Gaussian mappings can be obtained for which the existence of a controlling measure λ is assumed. This is automatically true if X = Y, and separable:

Proposition 8.2.6 Let (S, S) be a measurable space, S as a δ-ring. Suppose µ : S → B(X, Y) is a vector measure with a controlling σ-finite measure λ on S where X, Y are Banach spaces. If LpX (S, λ) is the Lebesgue space of X-valued (measurable and) pth -power λ-integrable functions (in Bochner’s sense) p ≥ 1, let V : LpX (λ) → Y be a bounded linear mapping, which may (but need not) be defined as: Z V (f ) = f dµ, f ∈ LpX (λ), (14) S

and in the general case there exits a σ-additive measure µ ˜ : S → B(X, Y) of finite semi-variation again having λ as its controlling measure such that (14) holds with µ replaced by µ ˜. In particular if X = R, Y = L2 (Ω, Σ, P )(= L2 (P )) and V : L2 (λ) → L2 (P ) is a continuous linear mapping whose range is a Gaussian subspace so that it is of centered Gaussian random variables all of whose finite linear combinations are Gaussian denoted G. Then there exists a Gaussian measure µ ˜ so that {˜ µ(A) : A ∈ S, λ(A) < ∞} spans G ⊂ L2 (P ), and the representation (14) holds with p = 2, X = R, Y = L2 (P ).

Proof. A quick sketch will be given, starting with the fact that in the last part (the application) µ ˜ (A) : Ω → R cannot be considered as a pointwise σ-additive measure, i.e., µ ˜(·)(ω) is not a scalar measure for all ω ∈ Ω, already emphasized by Neveu ( 1968, pp.64–65); such were originally the reasons (among others) for Wiener to introduce the new [Wiener] integral. By hypothesis on µ, with a control measure λ, the integral (14) is defined since µ, is assumed to

8.2 Vector Integration with Measures of Finite

Semivariation, Part I

385

1 p

have finite variation, and then with kµk(A) ≤ kV k(λ(A)) for p ≥ 1, where A ∈ S, λ(A) < ∞. It is the converse that is of interest. By hypothesis again for A ∈ S, λ(A) < ∞, and x ∈ X, χA x ∈ LpX (λ), let µ ˜ x (A) = V (χA x). Next note that µ ˜x (·) is additive and linear in x. Hence µ ˜(A)x = µ ˜x (A) satisfies (as in the Riesz representation): k˜ µ(A)xkY = k˜ µx (A)kY = kV (χA x)kY

1

≤ kV kkχA xkp ≤ kV k · kxkX λ(A) p , 1

(15)

˜ has λ as a controlling measure and is σso that k˜ µ(A)k ≤ kV kλ(A) p . Also µ additive. It has finite semi-variation as in (13). The representation (14) obtains with µ ˜ for all simple functions. But by definition of LpX (λ) every f can be approximated by such simple functions since f is (Bochner) pth power integrable and (5) applies. Hence (14) with µ ˜ there (determined by V ) holds and further details can be omitted. In the Gaussian case where Y = L2 (P ), and X = L2 (λ), the range of V required to be a Gaussian subspace, is a nontrivial restriction. In particular if the mapping V has the property that with S = R (or Rn ), V (χA x) and V (χA+t x) − V (χA x) are independent then the measure µ ˜ representing V will have: (i) λ nonatomic and (ii) µ ˜(A), µ ˜(B) to be independent when A ∩ B = ∅, for all bounded Borel sets A, B. Thus the vector stochastic integral must be, and is, subsumed here. 2 The preceding result asks for the relation between µ and µ ˜. If V is defined by µ as in (14), then the last part implies that µ − µ ˜ represents a zero integral and by (13), it follows that kµ − µ ˜k(A) = 0 for all A ∈ S of bounded λ-measure. This shows that µ = µ ˜. But the point of the above representation is that if V is given as a linear bounded operator, it determines a µ ˜ of finite semi-variation, and it is unique. This is a form of the classical Riesz representation in this context. An immediate question raised in this application is to obtain an integral representation of one vector measure µ1 relative to another such measure µ2 . This problem also arises in Whitney’s (1957) study of Geometric Integration Theory to be considered later. But here the first casualty is the uniqueness of the “density” when it exists. In the infinite dimensional case both the existence, uniqueness and even (Lebesgue type) decomposition present new problems. The question is necessary to settle here since a pair of conditional (probability) measures are typically vector valued and the preceding analysis applies. We therefore devote the entire next section elucidating some of these questions. Later on a few related matters are included in the complements section. Motivated by the above work, it is possible to define a class of µ-integrable functions f : S → R (or a Banach space Z) relative to such a µ of finite semivariation. In fact since µ is σ-additive implies that kµk(·) is σ-subadditive (an “outer measure”) on S, one can employ the classical Carath´eodory procedure and also consider kµk(·) measurable sets of S which are known to form a

386

8 Vector Measures and Integrals

σ-algebra S1 containing S. But if f is S1 -measurable in the sense that it is a pointwise limit of simple functions, or if f is such a scalar function but µ a vector measure, define a norm kf kµ , by the equality: n Z o

kf kµ = sup g dµ : |g| ≤ |f |, a.e. [kµk], g simple S n Z o = sup g d(z ⋆ ◦ µ) : g ≤ |f | a.e., kz ⋆ k ≤ 1 (16) S

R 1 where z ⋆ ◦ µ is a signed measure. Replacing f by |f |p and taking k S g dµk p R 1 in (16) or | S |g|p d|z ⋆ ◦ µ|| p for p ≥ 1 one gets the LpZ (µ) spaces by standard computations. These spaces are also complete and hence are Banach spaces. This needs a detailed (nontrivial) but familiar computations which we omit and nevertheless use the result on occasion. (The details may be found in Dinculeanu ( (2000) Sections 5–7), or in Panchapagesan (2008, Chapter 3). The completeness arguments typically need weak sequential completeness of Z, as in Proposition 4 above which is automatic when Z is reflexive, but has to be added on otherwise.) We now turn to some related aspects of the subject.

8.3 Vector Integration with Measures of Finite Semivariation, Part II The above section is devoted to integration of scalar as well as vector functions relative to a vector measure so that the indefinite integral defines another vector measure. We now consider comparing a pair of vector measures and discuss conditions under which one is an integral of the other for some possibly vector or operator valued “density”, and present a few consequences and applications. [Two vectors can be multiplied using tensor or duality pairing.] If µi : S → X, i = 1, 2 are a pair of vector measures each of which has a controlling measure λi , we can take λ = λ1 + λ2 and have a common measure dominating both the λi and µi . But for a serious study and comparison, a further restriction called s- (or strong) boundedness introduced by Rickart (1943) is needed for comparing vector measures and so it is given as follows: Definition 8.3.1 A vector measure µ : S → X, a Banach [or only a normed linear] space, is s-bounded if for each disjoint sequence An ∈ S, limn→∞ µ(An ) = 0 in X. The following alternative, but equivalent statements of s-boundedness introduced above, even when µ is merely additive and S is a ring of S, were discussed by Traynor (1973): (1) For each An ⊂ An+1 , An ∈ S ⇒ {µ(An ), n ≥ 1} ⊂ X is Cauchy.

8.3 Vector Integration with Measures of Finite

Semivariation, Part II

387

(2) For each An as above and A ∈ S, {µ(An ∩ A), n ≥ 1} is again Cauchy in X, uniformly in A. P∞ (3) For any disjoint sequence {Bn , n ≥ 1} ⊂ S, the series n≥1 µ(Bn ), P∞ ( n=1 µ(Bn ∩ A)) is unconditionally (and uniformly for A ∈ S) convergent in X which is assumed complete. We shall not include the equivalence proofs which are left as exercises. Also it should be noted that for scalar (or signed) measures the s-boundedness is automatic, and it plays a key role in cases of infinite dimensional range spaces X. In this connection Traynor also observes that the range X can also be any commutative (not necessarily locally compact) topological group, and the proofs go through with simple modifications. Two other concepts of interest in this analysis are as follows.

Definition 8.3.2 Let (S, S) be a measurable space where S is either a σ-ring, or a δ-ring with the restriction that S = ∪∞ n=1 Sn , Sn ∈ S. If X, Y are a pair of normed linear spaces and µ : S → X, ν : S → Y are σ-additive, then (i) ν is µ-continuous, denoted as ν ≺ µ if for each A ∈ S with kµk(A) = 0, one has ν(A) = 0 in Y where kµk(·) is the semi-variation of µ, which is subadditive, and (ii) ν is singular relative to µ, denoted ν ⊥ µ, if there is a set B ∈ S, kµk(B) = 0, but ν(A) = ν(A ∩ B), A ∈ S so that ν(·) is nontrivial only on the trace algebra S(B). A useful connection of s-boundedness and the Lebesgue type decomposition of a pair of vector measures for our applications is as follows: Proposition 8.3.3 Let (S, S) be a measurable space with S as a σ-ring, and µ : S → X, ν : S → Y be vector measures where X, Y are Banach spaces. Then an s-bounded ν can be uniquely decomposed relative to µ as ν = ν1 + ν2 where νi : S → Y, i = 1, 2 are vector measures such that ν1 ≺ µ and ν2 ⊥ µ in the sense of Definition 2 above. Proof. Since µ is a vector measure, the semi-variation kµk(·) is a σ-subadditive positive function on S, and so can be identified as an outer measure on S. Let N be the class of A(∈ S) with kµk(A) = 0. The subadditivity of kµk(·) implies for for A ∈ N, B ∈ S one has kµk(A ∩ B) ≤ kµk(A) = 0 so that N is (algebraically) an ideal in S since kµk(·) is σ-subadditive. On the other hand ν : S → Y is s-bounded. Hence by a result of Rickart ( (1943) Theorem 3.3) there exist νi : S → Y, i = 1, 2, ν = ν1 + ν2 , uniquely such that ν1 ≺ µ and ν2 ⊥ µ, and the ν1 , ν2 are σ-additive since ν is, implying the proposition. 2 The s-boundedness condition is powerful enough that the argument given by Gould, employed in the proof of Theorem 1.3 of this chapter extends with simple modifications. These direct changes have been detailed by Traynor (1973), and were also given by Brooks (1971) using a different argument [in fact reducing to the Dunford-Schwartz (1958) case through the Stone isomorphism mapping]. The desired result in Traynor’s form can be stated as follows:

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8 Vector Measures and Integrals

Theorem 8.3.4 Let (S, S) be a measurable space, S a σ-ring. Let M be a set of X-valued uniformly σ-additive and s-bounded vector measures where X is a Banach space. Then there exists a finite measure λ : S → R+ such that each µ ∈ M is λ-continuous and λ(A) ≤ sup sup kµ(B)k, A ∈ S.

(1)

µ∈M B⊂A B∈S

Moreover, if M is countable so {µk , k ≥ 1} = M, and limk µk (B) exists for each B ∈ S, then the family is uniformly λ-continuous and that the limit (= µ(·) say) is an s-bounded vector measure. The result holds if X is replaced by a (complete) topological group with simple modifications on the analogous operations instead of sums. The details will not be repeated here. (These are given in Traynor (1973).) We now analyze the part ν1 of the above Proposition 8.3.3, since ν1 ≺ µ and taking this as our new hypothesis. We first present a simple Radon-Nikod´ ym type result and then an example showing that one can expect no uniqueness property of density in general. Proposition 8.3.5 Let (S, S, λ) be a finite measure space f : S → X and g : S → Y be a pair of Bochner integrable (for λ) funcR R strongly measurable tions so that µ : A 7→ A f dλ, ν : A 7→ A g dλ, A ∈ S define vector measures such that ν ≺ µ. Then (i) there exist simple (vector) functions fn : S → X, gn : S → Y with fn → f a.e., (λ) and gn → g a.e.,(λ) as Rwell as in mean such R that if fn (ω) = 0 implies gn (ω) = 0 with µn (A) = A fn dλ, νn (A) = A gn dλ so that the variations satisfy |µ − µn |(Ω) → 0, |ν − νn |(Ω) → 0 as n → ∞, and (ii) there exists a (finite valued or) simple operator Tn (ω) : X → Y such that for some n0 , n ≥ n0 it implies gn (ω) = Tn (ω)fn (ω),a.e., where the Tn (·), n ≥ 1, is a sequence of “densities” of νn relative to µn . Proof. Since f and g are Bochner integrable, the variations |µ|, |ν| are finite measures and ν ≺ µ further implies that A = {ω : f (ω) = 0} has the folR lowing property. If |µ|(A) = A kf (ω)kdλ = 0 then ν(A) = 0. Also for all B ∈ S(A), |µ|(B) = 0 implies ν(B) = 0 as well. Since f, g are Bochner integrable, f = 0 on A(ν ≺ µ) gives g = 0 on A and there exist simple fn′ , gn′ such that fn′ → f a.e., gn′ → g a.e.(λ) as well as in mean. It is no restriction to assume that fn′ (ω) = gn′ (ω) = 0 for ω ∈ A. Given ε > 0, consider the following sets: Bn,ε = {ω : |f − fn′ |(ω) ≥ ε}, Cε = {ω : 0 < kf (ω)k < ε}

Dn = {ωfn′ (ω) = 0, f (ω) 6= 0}.

(2)

Then Bn,ε ∩ Dn and Cε ∩ Dn are disjoint with kf (ω)k ≥ ε on the first set and kf (ω)k < ε on the second set so that

8.3 Vector Integration with Measures of Finite

Semivariation, Part II

0 ≤ lim sup λ(Dn ) ≤ lim sup[λ(Bn,ε ) + λ(Cn )] ≤ ε · λ(Ω). n→∞

389

(3)

n→∞

Since fn′ → f a.e., it follows that limn→∞ λ(Dn ) = 0. Now let fn = fn′ and gn = gn′ (1 − χDn ). Then fn (ω) = 0 ⇒ gn (ω) = 0 and fn → f a.e. and in L1 (λ)-mean. The same holds for gn also. To see the L1 (λ)-mean property consider Z Z Z ′ kgn − gk(ω)dλ ≤ kgn − gn k(ω)dλ + kgn′ − gk(ω)dλ S S S Z Z kgn′ − gk(ω)dλ kgn′ k(ω)dλ + = S D Z Z n kg(ω)kdλ + 2 kgn′ − gk(ω)dλ → 0 ≤ S

Dn

fn′ ,

gn′ ,

as n → ∞ by the choice of (2)and (3). So (i) follows. For (ii) consider representations of fn , gn thus obtained: fn =

m X

xni χEni , gn =

m X

yni χEni ,

(4)

i=1

i=1

where Eni are measurable and taken to be disjoint as i varies and 0 6= xni ∈ X, yni ∈ Y. It is no restriction to take the same m and Eni in (4) and by (i) xni are non vanishing but yni = 0 can happen for some i. Let k0 = mini≤n kxni k and k1 = maxi≤m kyni k so that kfn (ω)k ≥ k0 > 0 on ∪m i=1 Eni , and kgn (ω)k ≤ k1 on Ω (also kfn (ω)k ≤ k2 < ∞ on Ω). Here the ki can depend on n. We now construct an operator Tn that sends xni to yni , i ≤ m. Define Bi , Ci as mappings from L1 (λ) to the one dimensional space generated by h = χEni satisfying the relations: Bi h = xni λ(Eni ), Ci h = yni λ(Eni );

(5)

and more generally for h ∈ L1 (λ), G ⊂ supp (h), let F = G ∩ Eni ∈ S, and Z Z Bi h = xni hdλ; Ci h = yni hdλ, (6) F

F

so that kBi k = kxni k ≤ k0 and kCi k = kyni k ≤ k1 . From (5), (6) and the definition of fn , gn , we have Bi h = 0 in X entails Ci h = 0 in Y, for h ∈ L1 (λ). It then follows from a result of Sard ((1948), Theorem 1) that there exists a bounded operator Ui such that (Ui = Ci Bi−1 can be taken) Ci h = Ui Bi h, h ∈ L1 (λ), i = 1, . . . , m, (7)

and kUi k ≤ kCi kkBi−1 k ≤ k1 k0−1 (= k3 ) < ∞. Now the domain of Ui is a one dimensional subspace of X and hence there exists a bounded projection on X onto that one-dimensional subspaces having

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8 Vector Measures and Integrals

X with norm 1, which is shown below. Hence Ui can be extended from X onto this space with the same norm (cf., e.g., discussion in Dunford-Schwartz (1958, p.554)). Thus for A ∈ S, Z

gn dλ =

A

=

m X

i=1 m X i=1

yni λ(A ∩ Eni ) =

m X

Ci h

i=1

Ui xni λ(A ∩ Eni ) =

Z

Tn (ω)fn (ω) dλ

(8)

A

where Tn = Un on Eni , i = 1, . . . , m and = 0 otherwise. Then kTn (ω)k ≤ k3 < ∞ and since A ∈ S is arbitrary, gn = Tn fn , a.e., n ≥ 1 establishing (ii). 2 Remark 8.2 Regarding the projection on the one-dimensional subspace used, let f ∈ L1X (λ), λ(S) < ∞, and let {f } be the one dimensional span of f (so f 6≡ 0). To normalize it let kf kL1 = 1. By the Hahn-Banach theorem, there exists a linear functional ℓ on L1 of norm 1, onto {f }, such that ℓ(f ) = kf k, and define P : L1X (λ) → {f } as P g = ℓ(g)f , for g ∈ L1X (λ). Then P 2 = P, P g ∈ {f } and kP k = kf kL1 = 1. But P g is strongly measurable and hence P (·) is a strongly measurable operator function as desired in the above proof. The result (ii) of the preceding proposition may be expressed as: Z Z Z νn (A) = gn dλ = Tn fn dλ = Tn dµn . A

A

(9)

A

using the change of variable in the Bochner integral. In this form the result of (9) implies a simple and formal Radon-Nikod´ ym property (existence) of “derivative” for a (particular) pair of vector measures. Even in this special case the “density” constructed above, namely Tn , is not unique, as we show by a pair of examples below. If X and Y are finite-dimensional (Banach) spaces, then a result corresponding to part (ii) of the above proposition has been presented by Whitney (1957, Chapter XI, Sec.5) with the remark that the R-N density is no longer unique. The derivation of his result is essentially noncomponentwise which does not easily extend to infinite dimensional spaces. We present two examples showing that even if X = Y(= R2 ) the density T (= T2 ) of (9) need not be a scalar function and that in general it is not unique so that later on the existence of a suitable R–N density will be presented and also conditions for a unique scalar density when X = Y are certain types of (function spaces). Several possibilities, applications and related analysis are seen to open up in the ensuing analysis. Example 8.3.6 Let X = Y = R2 and S = {s1 , s2 }, S = power set of the two point space S, and x1 , x2 ∈ X be a pair of linearly independent vectors

8.3 Vector Integration with Measures of Finite

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391

in the space R2 . Let µ({si }) = xi = ν({si−1 }), i = 1, 2. Then ν ≺ µ (and also µ ≺ ν). So T2 (= T say) of (8) is Ra 2 × 2 matrix function, that cannot be scalar. For otherwise, ν({s1 }) = {s1 } T dµ reduces to x2 = T (s1 )x1 , so that the vectors x1 , x2 will not be linearly independent contradicting the original choice. Thus T cannot reduce to a scalar density here. In fact it also dν does not exist in this case and such measures have to be follows that dµ excluded! Example 8.3.7 In this example it is shown that the existing “density” T (·) is not unique. Indeed let S be the set of positive integers with S as the power set and X = Y be a Banach space with the closed subspace given by X0 = sp{x ¯ n : sup kxn k < ∞, n ∈ S} (10) n

onto which there exists no bounded projection of X. Then there is a bounded linear operator T with domain X0 in X that can not be extended from X0 to X (cf., Dunford-Schwartz (1958), p.554). Consider the vector measures µ, ν for A ∈ S and xn ∈ X0 defined by; µ(A) =

X T xn X xn ; ν(A) = . n! n!

n∈A

(11)

n∈A

It is clear that ν ≺ µ and λ(·) as the counting measure. Also Z T (n) dµ, T (n) = T, λ(A) < ∞. ν(A) =

(12)

A

Now if there is a T1 (·), bounded and satisfies (12) with T1 in place of T , then by the assumed uniqueness, T1 (n) = T (n), n ∈ S and T1 : X → X. But this contradicts the choice of X0 implying that T is not extendible from X0 to X. Hence T of (10) is not a unique density. Thus agreeing for a not necessarily unique ‘density’ we now establish the dν existence of a derivative dµ if ν ≺ µ for a pair of vector measures into X and Y, possibly distinct (range) Banach spaces. Later we look for special spaces where there may be uniqueness. The general strategy is to reduce the problem to the case of measures, µ, ν to be of the form given in Proposition 5 above and then find a “density” T as in the second part of the result. Since we already know that a vector measure has a finite positive controlling measure thus taking the sum of these two for both µ, ν, it is convenient to assume that both have a common finite positive controlling measure λ. So the next question is to find dν conditions on µ, ν such that dµ dλ , dλ exist in some suitable sense, and then dν construct “T = dµ ” as in that proposition. This is the planned procedure here. Since in the infinite dimensional spaces X, the finite dimensional condition dµ ,several conditions enabling that µ ≺ λ need not imply the existence of dλ

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such a representation have been devised, the spaces for which such a representation holds are termed “spaces with the R − N property”, i.e., each µ of bounded variation |µ| has the R − N derivative for λ = |µ|. Numerous results in this direction have been given in the book by Uhl and Diestel (1977). Here some simple conditions will be recorded for use in the case under study with pairs of vector measures µ, ν satisfying ν ≺ µ. Since µ : S → X is σadditive and “controlled” by λ (same for ν : S → Y), the infinite dimensionality of the range (Banach) space and x⋆ ◦ µ is a scalar measure for each x⋆ ∈ X⋆ (the adjoint) and also x∗ ◦ µ ≪ λ. But from this one cannot conclude that x⋆ ◦µ should somehow give dµ dλ dλ , valued in X. Several special conditions additionally are needed. A reasonably general result due to Phillips (1943), is based on weak compactness of a “locally average range of µ”, introduced as follows. Definition 8.3.8 Let µ : → X be a vector measure with λ as a controlling measure λ : S → R+ . The average range Aλ of µ for λ on A is defined as µ(B) ˜ is a σ- finite posi: B ⊂ A, B ∈ S, λ(B) > 0} ⊂ X. If λ Aλ (µ) = { λ(B) ˜ be the set of vector measures µ : S → Z such tive measure on S, let V(Z, λ) ˜ that kµ(B)k ≤ cλ(B), B ∈ S, and the average range set Aλ˜ (µ), is (relatively) weakly compact in the Banach space Z, (i.e., the closures in the weak topology of Z are compact). In some from this type of compactness is always assumed in the class of vector integrals and in many cases this is also the best. With this concept Phillips ((1943), Section 5) established the following result which is of interest for our work: Theorem 8.3.9 A vector measure µ : S → Z, a Banach space, with λ as a controlling measure such that µ ∈ V(Z, λ), admits an integral representation relative to λ as: Z f dλ, B ∈ S, (13) µ(B) = B

where f is strongly measurable and weakly (or Pettis) integrable function on S into Z. If moreover |µ|(B) < ∞, so that µ is of bounded variation, then the integral is in the sense of Bochner (all these concepts are restated below).

Recall that a function f : S → X is strongly measurable if its range is sep¯+ arable and the inverse image of each Borel set of X is in S. If λ : S → R is a measure, then the above can be weakened to say that fn : S → X, fn , simple, and fn → f a.e.. Now an X-valued R f is Bochner integrable relative to λ, if it is strongly measurable and S kf (ω)kX dλ(ω) < ∞. The mapping f is weakly measurable if for each x⋆ ∈ X⋆ (adjoint Banach space of X), the scalar (or numerical) function x⋆ ◦ f is (λ)-measurable. It is known that f is Bochner integrable if and only if there is Ra sequence of simple functions fn → f pointwise a.e. (λ) and that limn S kf − fn kX dλ = 0.

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393

For weak measurability the situation is more complicated. If f : S → X is weakly measurable, and x⋆ ◦ f is λ-integrable for each x⋆ εX⋆ , then the mapping Z xfA : x⋆ 7→

A

(x⋆ ◦ f ) dλ,

A∈S

(14)

is a closed operator on X⋆ for each A, and defines an element in X⋆ by the closed graph theorem, but can fail to be in X itself since X need not be a reflexive space. If xfA ∈ X, A ∈ S, then the weakly measurable f is called Pettis integrable. It can be shown without much difficulty that the mapping xf(·) : S → X is then σ-additive so it defines a vector measure. Unlike the Bochner integral, there is no such simple characterization of Pettis integrable functions, even after the deep analysis on the subject by Talagrand (1984), although many sufficient conditions are known (cf. Uhl and Diestel (1977), Chapter II and Dinculeanu (1967) for details). Thus every Bochner integrable f : S → X is Pettis integrable and the latter is a much larger collection, as noted in connection with (13) above. We shall not include the details (they are given in the above references streamlining the original Phillips’s theorem). See also the Bibliographical Notes on the domination problem. With this preparation, it is now possible to present a general form of Proposition 5 above, for measures µ, ν of the form (13) satisfying ν ≺ µ, with values in X and Y respectively as before. In view of Proposition 8.2.3, it suffices to consider the case that ν ≺ µ where ν, µ are dominated (or controlled) by a common λ, a σ-finite (or even a localizable) measure each of which admits a vector integral representation as in (13) above. Although an extension of Proposition 5 can be given in a series of steps, it is somewhat longer than an alternative method found later. This uses different arguments, based on certain geometric properties of Banach spaces of independent interest. It is presented here which is motivated by Whitney’s (1957) finite dimensional case. This is shorter than the other older method, although it is still not elementary, and this gives our R-N density T (·): Theorem 8.3.10 Let (S, S, λ) be a σ-finite measure space, X, Y be Banach spaces, with µ ∈ V(X, λ), ν ∈ V(Y, λ) as in Definition 8.3.8, and let ν ≺ µ. Then there exists a strongly measurable T : S → B(X, Y), an operator valued mapping, such that the following representation holds: Z T (ω) dµ(ω), A ∈ S, λ(A) < ∞, (15) ν(A) = A

this being Bartle’s bilinear vector integral discussed earlier. Proof. The membership of µ, ν in the spaces V(Z, λ) where Z = X or Y, implies that the ranges of µ, ν are separable and with (the Phillips type) integral representation in Theorem 8.3.9 above, one has: Z Z µ(A) = f dλ, and ν(A) = g dλ, A ∈ S, λ(A) < ∞, (16) A

A

394

8 Vector Measures and Integrals

where f : S → X, g : S → Y are strongly λ-measurable. Replacing X, Y by the essential ranges of f, g they can and will be assumed separable. Since |µ|(·) ≤ cλ(·), we may take |µ|(·) in place of λ in (13) so that µ ∈ ν(X, |µ|), ν ∈ V(Y, |µ|) and use the classical Radon-Nikod´ ym theorem to have h = d|µ| dλ , and similarly for ν of (16), as: Z Z g0 d|µ|, A ∈ S, |µ|(A) < ∞, (17) f0 d|µ|, and ν(A) = µ(A) = A

A

so that g = g0 h a.e. in (13). We proceed with (17) for the rest of the argument, using an extension of Whitney’s ideas in the following three steps using some key results of Functional Analysis with references. I. We assert that there is an equivalent norm, denoted k · k1 , for X with the adjoint norm k · k⋆1 , for X⋆ and a lower semi-continuous mapping ϕ : X → X⋆ where X⋆ is endowed with X-topology, also called the weak*-topology, such that kϕ(x)k⋆1 = 1 and ϕ(x)(x) = kxk1 . Indeed, as we can and do take for this proof that X is separable, by a well-known theorem of M. M. Day (1955, p.519), X is linearly homeomorphic to a Banach space which has a weakly (or Gˆateaux) differentiable or “smooth” norm that is strictly convex. Consequently X can be given an equivalent norm, denoted k · k1 , under which (X, k · k1 ) has both the above strict convexity and smoothness properties. Let S and S ⋆ be the unit spheres of (X, k · k1 ) and (X⋆ , k · k⋆1 ), and let ψ : S → S ⋆ be the spherical image (set valued) map defined as follows. Recall that by the geometric form of the Hahn-Banach theorem for a convex set K(= the sphere S here) there is a closed hyperplane that lies strictly on one side of K and the plane is determined by a continuous linear functional x⋆ , |x⋆ (x0 )| > |x⋆ (K)|, x0 ∈ X − K, and if K is strictly convex there is a unique such x⋆ . This implies, for (X, k · k1 ), that for each x ∈ S there is a unique x⋆ ∈ S ⋆ , the unit sphere of X⋆ such that ψ : x 7→ x⋆ is well-defined and gives the desired spherical map. The smoothness of X further allows us to conclude that ψ is lower semi-continuous. Now the desired ϕ : X → X⋆ can be taken as x ϕ(x) = kxk1 ψ( kxk ), if x 6= 0, and = 0 if x = 0, for all x ∈ X. Then 1 ⋆ kϕ(x)k1 = 1, ϕ(x)(x) = kxk⋆1 and this ϕ is the desired mapping. (See Cudia (1964),Corollary 4.5, on a related use of the Hahn-Banach theorem in a similar context.) II. There is a strongly (or in norm topology) measurable mapping f0⋆ : Ω → ⋆ X , satisfying kf0⋆ (ω)k1 = 1, ω ∈ Ω, and for µ Z |µ|(A) = f0⋆ dµ, A ∈ S, |µ|(A) < ∞. (18) A

For, using (16), for the representation of µ, and the standard substitution results of measure theory, Z µ(A) = f0 d|µ|, kf0 (ω)k1 = 1 a.e. (|µ|), A ∈ S, |µ|(A) < ∞. (19) A

8.3 Vector Integration with Measures of Finite

Semivariation, Part II

395

By the preceding step there is a lower semi-continuous ϕ : X → X⋆ such that if f0⋆ = ϕ ◦ f0 , then f0⋆ : S → X⋆ and f0⋆ (ω)(f0 (ω)) = kf0 (ω)k = 1 a.e. (|µ|), and kf0⋆ (ω)k⋆1 = kf0 (ω)k1 = 1. We assert that f0⋆ is strongly measurable which can be used to present a simpler formula for |µ|(·) as well. First observe that if k : S → X, is a step or countably valued function, then ϕ ◦ k : S → X⋆ is also step or countably valued. The lower semi-continuity of ϕ noted in the preceding step and the fact that f0 is strongly measurable so that there exist fn → f0 in X uniformly on S, imply that fn⋆ (ω) = (ϕ ◦ fn )(ω) → (ϕ ◦ f0 )(ω) = f0⋆ (ω) for ω ∈ S the limit in the X topology of X⋆ (or weak*-topology) so that f0⋆ is weak*-measurable. To see that f0⋆ is strongly measurable, note that fn⋆ (S) ⋆ ⋆ is a (finite or ) countable set and so sp(∪ ¯ ∞ n=1 fn (S)) = X1 (say) is a sepa⋆ ⋆ rable closed linear subspace of X in k · k1 which is a strictly convex norm. Now fn (ω) → f0 (ω) in norm k · k1 uniformly in ω, implies by the norm-toweak*continuity of ϕ, f0⋆ (ω) ∈ X⋆1 because X is separable by the initial reduction. Also kfn (ω)k1 = kϕ ◦ fn k⋆1 (ω) → kf0 (ω)k1 = kϕ ◦ f0 (ω)k⋆1 . But by Dunford-Schwartz ( (1958), IV.8.8) there is a closed subspace Y of X such that Y⋆ and X⋆1 are equivalent. Hence X⋆1 may be assumed an adjoint separable strictly convex space for the present argument. Thus fn⋆ (ω) → f0⋆ (ω) in the Y-topology of X⋆1 and kfn⋆ (ω)k⋆1 → kf0⋆ (ω)k⋆1 , ω ∈ S. From this and a well-known theorem of Kade˘c and Klee (1960, p.27) we can conclude that kfn⋆ (ω) − f0⋆ (ω)k⋆1 → 0 for all ω ∈ S so that f0⋆ is strongly measurable as desired. It now follows from Bartle’s (1955) bilinear integral formulation (cf. also the remark in Dunford-Schwartz (1958), p.317 which holds here) we conclude that Z Z ⋆ f0⋆ (f0 ) d|µ| = |µ|(A), A ∈ S, |µ|(A) < ∞. (20) f0 dµ = A

A

This also implies (and simplifies (20)) that (for |µ|(A) < ∞) Z |µ|(A) = sup f ⋆ dµ : kf ⋆ (ω)k⋆1 ≤ 1, A

f ⋆ : S → X⋆ is a step function relative to S ,

(21)

and this formula is also useful for the work of Whitney’s (1957) among others. III. From (19) and (21), if T = g0 ⊗ f0⋆ : S → B(X, Y), defined as T (ω) = g0 (ω) ⊗ f0⋆ (ω), where this is the usual tensor product defined as (y ⊗ x⋆ )(x) = y · x⋆ (x) ∈ Y, x ∈ X, y ∈ Y, x⋆ ∈ X⋆ , then T satisfies (15) to conclude the theorem. For, observe that T = g0 ⊗ f0⋆ : S → B(X, Y) is strongly measurable and T (ω) is in B(X, Y) for almost all ω in |µ|(·) measure, which also holds here even with weak*-measurability. Moreover,

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8 Vector Measures and Integrals

Z

T (ω) dµ(ω) =

A

= =

Z

A

(g0 ⊗ f0⋆ )(ω)f0 d|µ|(ω)

A

g0 (ω) · f0⋆ (ω)(f0 (ω)) d|µ|(ω)

A

g0 (ω) · 1 · d|µ|(ω) = ν(A),

Z

Z

A ∈ S,

for all |µ|(A) < ∞. Since the change of equivalent norms in a Banach space implies no restriction on measurabilities, the representation (15) is established. 2 An alternative but a much longer proof of the result without using the theorems of Day, as well as of Kade˘c-Klee can be given. We give an indication with sketches of this procedure as a graded exercise later. A more general formulation of the above type result can be stated for certain integrals of vector functions f : S → B(X, Y) relative to vector measures µ : SP → B(X, Y) of finite n semi-variation that are strongly σ-additive. Indeed if fn = i=1 xi χAi , Ai ∈ S (disjoint), is a simple function and we set ν

fn

(A) =

Z

A

fn dµ =

n X i=1

µ(A ∩ Ai )xi ,

(22)

which is well-defined for all such simple functions. Suppose that fn → f pointwise, outside of a set of zero semi-variation of µ. Now νnf : S → Y is a vector measure for each n and if ν fn (A) → ν f (A) as n → ∞ (for fn → f as above) exists, then by the Vitali-Hahn-Saks theorem with Nikod´ ym’sRextension, ν f (·) is σ-additive and the limit is uniform on S, so that ν f : A 7→ A f dµ can be regarded as an integral. This point of view was taken up by Dobrakov (1970) who reworked and extended many of the standard measure and integration results through the dominated and Vitali convergences. The ideas were then applied to get an extension of the Radon-Nikod´ ym theorem by Maynard (1972) which we present here for comparison with Theorem 8.3.9 above. Recall that if µ is vector valued and σ-additive (weakly or strongly both being equivalent by Pettis’s classical result) it is of finite semi-variation, and in case it is operator valued then the same is true in the strong operator topology when the analysis of integrations are taken on the δ-ring S of S on which these (semi-) variations are finite. In this context Definition 8.3.8 above takes the following somewhat more restrictive form: Definition 8.3.11 Let (S, S) be a measurable space, S being a δ-ring, and µ : S → B(X, Y), ν : S → Y be σ-additive functions of finite semi-variation for ν and in the strong (not uniform) topology for µ. If for any ε > 0 and A ∈ S, consider the trace S(A) for kνk(A) > 0, and the collection A(A, ε) ⊂ X defined between the vector measures, ν and y0⋆ ◦ µ, by: A(A, ε) = {x ∈ X : F ∈ S(A), y ⋆ ∈ Y⋆ , |y ⋆ (ν(F ) − µ(F )x)| ≤ εky0⋆ ◦ µk(F )}, (23)

8.3 Vector Integration with Measures of Finite

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397

then the set A is termed localized in a non-empty set K ⊂ X if for each F ∈ S(A), we have A(A, ε) ∩ K 6= ∅. This technical condition, motivated by some geometric properties of the spaces X and Y, incorporates the weak compactness of the ranges. That property was used in Theorem 10 so that a Radon-Nikod´ ym “differentiation” can be obtained. Such a result can be given as: Theorem 8.3.12 Let µ, ν be vector measures of finite semi-variation where µ : S → B(X, Y) and ν : S → Y, ν ≺ µ. Then Z ν(A) = f (ω) dµ(ω), A ∈ S, kµk(A) < ∞, (24) A

if and only if there is a set A0 ∈ σ(S) on which ν is supported,and for each E ∈ S, there exists an F ∈ S(E) and a compact set K ⊂ X such that F is localized for both µ, ν in the sense of Definition 11 above. Now the integral of f in (24) is understood as the strong limit of simple fn → f pointwise outside of a set of kµk(·)− outer measure zero. The point of this result, due to Maynard (1973), is a complete characterization of the “R–N derivative” f above which is of interest. This contrasts with Theorem 10 which gives a sufficient (and relatively easier to verify) condition for an analogous representation. This existence result has the localization condition, together with the compactness restriction embedded in it, but which has as yet no recipe to verify the condition. Thus its application is somewhat restricted. We omit the proof referring to the original paper. Our purpose here is a comparison with the earlier result and in both cases to indicate the new area for analysis, complementing the available work on the subject wherein the dominating measure was always a scalar. The stochastic integration of earlier chapters also demands that we study the subject when the work is for vector measures of just finite semi-variation, motivated by random measures starting with the Brownian motion. As an application of the above analysis we include a pair of simple consequences. First we discuss an Lp (µ), p ≥ 1, with µ : S → X, and its norm topology. Then we present a Riesz type representation of a continuous linear operator on such a space which in fact has motivated the preceding generalization of the vector measure integration. The following property of the semivariation is a consequence of a result of Lewis’s ( (1970), p. 158) and it will be used below. Proposition 8.3.13 Let µ : S → X be a vector measure (X a Banach space and S a δ-ring of S). If An ∈ S and An → A, A ∈ S, then kµk(An ) → kµk(A). (Here An → A means as usual the symmetric difference An ∆A → ∅ for the outer measure kµk(·).)

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8 Vector Measures and Integrals

Proof. First let An ∈ S, An ↓ ∅. If the result is false, then there is an ε > 0 and for some sequence of the above type kµk(An ) > ε, n ≥ 1. Choose a subsequence n1 < n2 < . . . such that for some x⋆ ∈ X⋆ , kx⋆ k ≤ 1, satisfying |x⋆ ◦ µ|(An1 ) > ε, where x⋆ ◦ µ is a signed measure and |x⋆ ◦ µ|(An2 ) < 2ε , which is clearly possible. From the definition of semi-variation we also will have ε (25) 4 sup{kµk(B) : B ⊂ An1 − An2 } ≥ |x⋆ ◦ µ|(An1 − An2 ) > . 2 Hence dividing by 4 and using the definition of kµk(·) we have for some measurable B1 ⊂ An1 − An2 , kµ(B1 )k > 8ε . Now continuing the procedure we can obtain a sequence Bk ⊂ Ank − Ank+1 such that kµ(Bk )k > 8ε for k > 1. Since the Bk ’s are disjoint, this contradicts the σ-additivity of µ, and thus the result holds in this case. For the general case that An → A in S, (so An , A ∈ S), [ [ |kµk(A) − kµk(An )| ≤ kµk[ (A − Ak )] + kµk( (Ak − A)), (26) k≥n

k≥n

and that by the special case considered above the right side terms both tend to zero as n → ∞, and so does the left side. Hence the result holds as stated. 2 Remark 8.3 Since kµk(·) is an outer measure, one can consider the class of kµk(·)-measurable sets using the classical Carath´eodory procedure, and the above proposition would have followed if kµk(·) is Carath´eodory regular in the sense that kµk(E) = inf{kµk(B) : B ⊃ A, B is kµk(·) -measurable}, (cf., e.g., Rao (2004), Theorem 2.2.9). But kµk(·) need not be regular in the above sense, and hence the proof using the special property of the semi-variation of a vector measure is utilized. The σ-ring σ(S) will also contain the limits of its sequences. We now define function spaces Lp (µ) on (S, S) for a vector measure µ, which have pth power µ-integrable elements and introduce a norm to consider their topological structure. Since kµk(·) can be understood as an outer measure, f : S → R or C can be defined as kµk(·)-measurable (or µ-measurable for short), if there is a sequence of S-simple functions fn : S → R (or C) such that fn → f a.e. (kµk). The vector space of the µ-measurable f (scalar) is denoted by L0 (µ) (or L0 (S, S, µ) in full). [The definition of measurability relative to measures is usually given in books on the subject, but the concept with outer measures is often not studied generally. Keeping the above type of application in view, the definition with outer measures is given in the author’s book (Rao (2004) Section 3.2) and hence it will be assumed known.] If f ∈ L0 (µ) where µ : S → X is a vector measure, then the functional R k · kµ : f 7→ sup{ S |f |(s)dv(x⋆ ◦ µ) : kx⋆ k ≤ 1, x⋆ ∈ X⋆ } where, as usual, v(x⋆ ◦ µ) is variation measure of the signed measure x⋆ ◦ µ may be shown

8.3 Vector Integration with Measures of Finite

Semivariation, Part II

399

to be a semi-norm of f : S → R or C. More generally if |f |p ∈ L0 (µ), then k · kpp,µ : f 7→ kf kpp,µ defined as above with |f |p , in place of |f |, one finds that k · kp,µ is a norm functional for 1 ≤ p < ∞, and the subspace Lp (µ) of L0 (µ) for which k · kp,µ is finite can be shown to be a Banach space if kf − gkp,µ = 0 is taken as f = g a.e. (µ). One can follow the usual methods. (The particular space L1 (µ) was treated in Rao (1967), and the Lp (µ), p ≥ 1case is detailed in Panchapagesan (2008). As remarked in the early paper, even Orlicz types can be defined, and one will be indicated in the complements section.) As an application of the preceding analysis, the following result is presented as a typical example of a representation of operators on Lp (µ)spaces. Theorem 8.3.14 Let (S, S, λ) be a finite (or even σ-finite) space and µ : S → X be a vector measure satisfying µ ∈ V(X, λ) in the sense of Definition 8 above. If LpX (µ) is the Banach space of scalar functions f : S → R (or C) which are kµk(·)- (hence λ-) measurable, let W : L1X (µ) → Y be a weakly compact linear operator, a condition that is automatically valid when Y is reflexive or more generally weakly sequentially complete. Then W can be represented as Z h(s)T (s) dµ(s), h ∈ L1X (µ), (h-scalar!) (27) W (h) = S

relative to a strongly measurable essentially bounded operator function T : S → B(X, Y), such that R k E T (s)dµ(s)kX . (28) kW k = sup{kW hkY : khk1,µ ≤ 1} = kT (·)k = sup kµ(E)kX λ(E)<∞

On the other hand, an operator W defined by the integral (27) with a strongly measurable operator function T satisfying the (norm) condition (28), defines a weakly compact operator on L1X (µ) into Y, and one has kW k = kT (·)k. Sketch of Proof.R The hypothesis on µ implies that it admits the representation µ : A 7→ A f dλ, A ∈ S, λ(A) < ∞ where f is measurable and essentially bounded (on λ-finite sets). It is seen from this that the following holds: Z Z ⋆ kχA k1,µ = sup χA dv(x ◦ µ) = sup |x⋆ (f )| dλ kx⋆ k≤1

S

≤ kf k∞ · λ(A) < ∞.

kx⋆ k≤1

A

(29)

Now if ν(A) = W χA , then ν : S → Y is a vector measure and ν ≺ µ. Since W is weakly compact by hypothesis, (29) shows ν(A) χA : 0 < λ(A) < ∞ (30) : 0 < λ(A) < ∞, A ∈ S ⊂ W λ(A) λ(A)

400

8 Vector Measures and Integrals

is weakly compact and this easily implies that ν ∈ V(Y, λ) in the earlier notation. Now Theorem 10 above implies the existence of a strongly measurable T : S → B(X, Y) such that Z χA T dµ, A ∈ S, λ(A) < ∞. (31) W (χA ) = ν(A) = S

The earlier discussion shows that (31) holds for simple functions h (in lieu of χA ) in L1X (λ), and then for all h ∈ L1X (λ) by Proposition 13. Then (31) easily implies the norm equation (28). The last part is established using Theorem 10 with some results from Phillips’s paper ((1943), Theorem 3.6), and the fact that the hypothesis on µ also implies that W is weakly compact. 2 It may be noted that in the preceding work the uniqueness of the density is not mentioned because for it additional restrictions are necessary either on the measures µ, ν or on the spaces X, Y or both. Also the discussion on the adjoint spaces of the LpX (µ) is not touched. This depends on restricting X or Y to have the “Radon-Nikod´ ym” property (reflexivity or weak sequential completeness etc.) and so this specialization will not be considered here. To indicate a feeling for the subject we include below a special class of (infinite dimensional) Banach spaces for which a familiar type of Radon-Nikod´ ym theorem for vector measures µ, ν(ν ≺ µ) with even, a scalar derivative will hold. The LpX case is discussed further in Exercise 4(c). The space to be considered here is a vector lattice which also has the following least upper bound property for that space. If S is a compact Hausdorff space and C(S) is the set of all real continuous functions with the usual order (f ≤ g if f (x) ≤ g(x), x ∈ S), then according to the wellknown result due to M. H. Stone (1949), the space C(S) has the least upper bound property if and only if S is a “Stone space”, i.e., each open set O ⊂ S has the property that its closure is open. Since C(S) is also a Banach algebra, so that X = C(S) has this special structure and µ : T → X is a measure on a general measurable pair (T, T) which is nonnegative and S for En ∈ T, disjoint, n En ∈ T, it is only (order) σ-additive, in the sense that n [ X µ( En ) = sup µ(Ek ). n

n

(32)

k=1

However even in formulation thePright side of (32) can not always be rePthis ∞ n placed by limn k=1 µ(Ek ) (i.e., by n=1 µ(En )), because of the existence of examples showing that the monotone sequence on the right need not converge to the limit defined by the infinite sum. This is implied by the following surprizing result due to E.E. Floyd (1955):

8.3 Vector Integration with Measures of Finite

Semivariation, Part II

401

Proposition 8.3.15 Let (I, B, λ) be the Lebesgue unit interval I = [0, 1],and B the class of its Borel sets. If B ∞ (I) is the space of bounded real Borel functions on I and M is the ideal contained in B ∞ (I), whose elements vanish outside of a meager (or first Baire category) set, let L = B ∞ (I)/M , the quotient space with the usual ‘sup’ norm. Then the function (a, b) 7→ a − b, of L × L → L is not jointly continuous in both variables for any T1 topology for L which is σ-compatible with the partial ordering in the sense that fn ∈ L, fn ↓ f (i.e., inf n fn = f ) should imply fn → f in the topology of the space L. Because of this pathology special care is needed in many cases when the statements and even the arguments for the most part appear quite familiar but the result does not follow from the known cases. Thus (S, S, µ, A) is an A-valued (A- a Stone algebra) measure space if µ : S → A ∪ {+∞},and µ is additive, nonnegative and σ-additive in the order topology as defined above in (32) where one adjoins ‘∞’ by requiring f ∈ A to satisfy f < ∞ and supn fn = ∞ if fn ’s are unbounded. One verifies that for An ∈ P S, An ↓ ∅ and µ(A1 ) < ∞ then inf n µ(An ) = 0 (or ∧n µ(An ) =P0). If fn = ni=1 ai χAi , Ai ∈ S, a simn ple function, the mapping ℓ : fn 7→ i=1 ai µ(Ai ) ∈ A where A is as above, a Stone algebra, then ℓ(fn ) is well-defined and positive, linear and if fn ↓ 0, a measurable simple sequence, then ℓ(fn ) ↓ 0 (i.e., ∧n ℓ(fn ) R= 0); hence if fn ↑ f and gn ↑ f , then limn ℓ(fn ) = limn ℓ(gn ), and we set S f dµ = supn ℓ(fn ) for f ≥ 0 and S- measurable. Thus the integral is defined for all such f ≥ 0 R and if µ{s : f (s) > k} = +∞ one sets S f dµ = +∞. Hence given measur+ able f : S → R, one can define the integral of fR for µ provided f− R + f = Rf − − ± and f are integrable in which case one sets S f dµ = S f dµ − S f dµ. From this we may establish the monotone and dominated convergence theorems for this integral using the order structure of A and the real space of measurable functions. [Here the Daniell construction of the integral, as in the familiar scalar case, need not extend!] One can define Lp (S, S, µ, A) spaces using the standard procedure, but these are algebra valued spaces (A being the Stone algebra) and each result needs a separate proof. These assertions have been carefully described by Wright (1969a,b). Here we present a Radon-Nikod´ ym theorem for a pair of vector (Stone algebra valued) measures as a useful companion to Theorem 10 above. The norm of a functional in Lp (S, S, µ, A) is then defined as follows: kf k∞,µ = inf{k ≥ 0 : µ{x : |f (x)| > k} = 0}, and for R 1 p ≥ 1, kf kp,µ = k( S |f |p dµ) p k∞,µ . It is to be shown that (Lp (µ, A), k · kp,µ ) is a Banach space. To establish the H¨ older and Minkowski analogs need a detailed and nontrivial analysis. The unique Radon-Nikod´ ym derivative is obtained only for a subclass of A-valued measures which we now isolate to fully appreciate the intricacies of the vector measure integrals of which the Brownian and martingale measures form important individual classes, motivating the general study as well as its applications. As a consequence of one of the examples preceding Definition 8 above, we need to exclude vector measures that “twist” the coordinates. We present a large class of (vector) measures

402

8 Vector Measures and Integrals

for which this pathology can be eliminated. The desired condition is given by: Definition 8.3.16 Let (T, T) be a measurable space and C(S), the real continuous function algebra on a Stone space S. If µ : T → C(S) is a σ-additive nonnegative measure (= a Stone algebra valued measure) let L∞ (T, T, µ) be the A(= C(S))-valued space defined above with norm k · k∞,µ . If for an algebra homomorphism π : A → L∞ (T, T, µ) it is true that Z Z f dµ, a ∈ A, f ∈ L1 (T, T, µ), (33) π(a)f dµ = a T

T

holds, then µ is called a modular measure relative to π. The existence of modular measures is a consequence of the following: Proposition 8.3.17 Let T be a compact Hausdorff space, T as its Baire σalgebra and C(T ) the space of continuous functions. If π : C(S) → C(T ) is any algebra homomorphism and τ : C(T ) → C(S) is a positive linear mapping such that τ (π(a)f ) = aτ (f ) for all a ∈ C(S), which is a Stone algebra, and f ∈ C(T ), then there exists a unique outer regular (positive) measure µ : T → C(S), such that one has the Riesz type representation Z f dµ, f ∈ C(T ). (34) τ (f ) = T

Moreover if N : C(T ) → L∞ (T, µ) is the embedding map, then µ is modular relative to N ◦ π. [Recall that the positive µ is outer regular means that for each closed set A ⊂ T , we can approximate it by open sets from above in the sense that µ(A) = inf{µ(B) : B ⊃ A, B open}. (35) Thus modular measures form a large subclass of representing ones as in (34).] We omit the proof of this interesting representation which is presented here to isolate a large family of modular measures. The ensuing class of modular measures will exclude the “twisters”. If a Stone algebra C(S) is isomorphic to some L∞ (T, T, λ) where λ(T ) < ∞, then it will satisfy our requirements. More generally C(S) can be the Stone algebra isomorphic to a commutative von Neumann algebra based on a separable Hilbert space. A comprehensive function space subsuming both the above cases is that the algebra C(S) should satisfy the countable chain condition. This means that each non empty subset C which is bounded above and below has a countable set C0 ⊂ C such that C0 and C have the same set of upper and lower bounds. It is an interesting and nontrivial fact that a (vector) measure µ : T → C(S), on (T, T), into a Stone algebra satisfying the countable chain condition which represents a positive linear mapping τ : C(T ) → C(S) that is both modular and non-twisting, is termed

8.4 Some Applications of Vector Measure Integration,

Part I

403

an ample measure by Wright (1969b ), and for that class we can extend our analysis. With this preparation we can present the desired Radon-Nikod´ ym theorem for ample (vector) measures to be a desirable companion of Theorem 10 above, as follows: Theorem 8.3.18 (Wright 1969b ) Let (T, T) be a measurable space and C(S) a Stone algebra of continuous functions on a compact Stone space. Let µ and ν be a pair of ample measures such that ν ≺ µ. [The ample condition is automatic if C(S) is a Stone algebra satisfying the countable chain condition.] Then there exists a unique scalar T-measurable function f : T → R, such that Z ν(A) = f (s) dµ(s), A ∈ T. (36) A

It is clear that to strengthen the “density” of the vector measures considered in Theorem 14, the hypothesis on µ, ν and the spaces X, Y allowed should be strengthened (or spaces restricted) even to get uniqueness. For applications of such a result the additional conditions are not easy to verify in many cases. The proofs of the above results for Stone algebras and related interesting applications have been presented in several publications by Wright (1969) in great detail to which we refer the interested reader.

8.4 Some Applications of Vector Measure Integration, Part I The analysis of the preceding section shows that we have to choose between specialized spaces on which unique kernel or Radon-Nikod´ ym density function exists or going for larger classes of spaces with diverse applications but forgoing the uniqueness property. Here we consider some different but important applications from both points of view and moving to admit more spaces. Now it becomes necessary to extend certain aspects of Whitney’s (1957) treatment of geometric integration, again with a relaxation of uniqueness. A substantial part of this section is devoted to these aspects generalizing some key finitedimensional results. It should also be remembered that in real applications one has to employ special structural properties to formulate and solve such problems. So far we considered vector integration on a measurable space (S, S) into a vector space X (a Banach space) relative to a vector measure µ : S → X of finite semi-variation, for functions f : S → Y (also a Banach space). Here S is any point set with or without a geometric structure. Suppose now that S has some geometric properties itself. Then it is meaningful to demand that we utilize such a property and extend integration for some geometric figures, instead of just point functions f but objects (e.g., curves) with range again in a vector space. This is geometric integration and we

404

8 Vector Measures and Integrals

present a glimpse of it by extending the previous theory. Thus the point functions Pn will be replaced by geometric curves in which simple functions f = i=1 ai χAi go into elementary correspondence with such elements as simplexes initially. The following special case of Theorem 3.10 above, using its notation if X is a Banach space and Y = R (reals) will motivate the ensuing discussion. This result was given in Whitney (1957, Section IV.5) for finite dimensional X which is shown to be extendible. Theorem 8.4.1 Let (S, S) be a measurable space, µ : S → X a vector measure, µ ∈ V(X, λ) and ν : S → R be a scalar measure such that ν ≺ µ, satisfying also ν ≪ µ. Then there exists a (strongly) measurable mapping f : S → X⋆ , the adjoint space of X (non uniquely in general) such that one has the representation: Z f (ω) dµ(ω), A ∈ S, kf (ω)kX = 1, a.e., (1) ν(A) = A

and moreover, for the variation measure |µ|(·) of µ, one has: Z |µ|(A) = sup g(ω) dµ(ω) : kgkX⋆ ≤ 1 , |µ|(A) < ∞.

(2)

A

The existence and non-uniqueness of the integrand f in (1) is significant even if X is of two dimensions. On the other hand if |µ| : S → R+ with µ ∈ V(X, |µ|), then one can use the vector Radon-Nikod´ ym theorem and obtain the following result for an a.e. unique g with kgkX = 1 as: Z g(ω) d|µ|(ω), A ∈ S, (3) µ(A) = A

where g(·) is strongly measurable, a.e. unique, and Bochner integrable. We now present some aspects of the above theory applied to the geometric objects defined in linear spaces with values in scalar or vector (= Banach) spaces. Let us recall the desired concepts of “simplexes” and “polyhedral chains” and present their integral representations as consequences and also as extensions of the work of the preceding section. Many potentially new problems are suggested. Let us start with the ‘new’ concepts. These results will be needed below in a study of isotropic random currents. Definition 8.4.2 Let X be a vector space and {xi , 0 ≤ i ≤ n} ⊂ X be such that {yi = xi − xi−1 , i = 1, . . . , n} is a linearly independent set. Then its convex hull, co(x0 , x1 , . . . , xn ) is called an n-simplex set (of dimension n). An oriented n-simplex has two directions or orientations obtained by multiplying with +1 or −1. A pair of n-simplexes is called disjoint if their intersection has a dimension ≤ n − 1. A multiple of an (oriented) n-simplex by a P real number is unambiguously defined, and the formal sum A = ki=1 ai Sin of k disjoint oriented n-simplexes is given as an oriented n-simplex, so that for

8.4 Some Applications of Vector Measure Integration,

Part I

405

x ∈ (Sik )o , an interior point, A(xi ) = ai or = 0 if no such points exists. Such an A is termed a polyhedral n-chain and Pn (X) denotes the set of all these chains of order n in X which is a finite dimensional space. Two elements A, B of Pn (X) are said to be equivalent, written A ∼ B, if the dim{x : A(x) 6= B(x)} < n. This gives the basic format corresponding to the classical operations of point functions. The vocabulary is extended in order to define a vector (and a Banach) space of these objects and to introduce a suitable integration relative to vector valued measures as well as integral representations using our vectorial integration developed earlier. This extension also has important stochastic applications exposed later. The set Pn (X) of all polyhedral n-chains is thus an n-dimensional subspace of X spanned by {vi = xi − xi−1 , i = 1, . . . , n}. If τu : X → X defined by τu (x) = x + u, the translation operator, then τu (Pn (X)) = Pn (X) whence it is closed under translations (and being finite dimensional) and that it becomes a locally compact group under addition. Hence there exists a unique (except for a multiplicative constant) translation invariant measure (a Haar measure) denoted by m(·) on Pn (X). Considering the oriented n-chain as an element of X, the mapping k · k : x(n) 7→ m(x(n) ) on n-simplexes xn (treated as a [finite] Pn n subset of X) defines a norm, and for A(x) = the n-chains Pn i=1 ai xi , (since n xi are disjoint by definition) we have kAk = i=1 kai km(xni ). The norm functional is well-defined. It is easily seen that k·k is a translation invariant norm on the space of polyhedral n-chains, Pn (X), the (finite dimensional) vector space formed of elements of X. The preceding layout clearly shows that the geometric integration is very much dependent on the finite dimensionality of the subspaces of the space X, and the deeper analysis is moreover based on introducing useful norm functionals on Pn (X). There are two types of these functionals that make Pn (X) into a kind of abstract (M ) and (L) spaces of the Kakutani (-Yosida) type and yet are quite distinct from the latter. In the present context these spaces have to distinguish the geometry of the problems concerned. They are called flat and sharp norm functionals. After introducing these, generalizing Whitney’s definitions which are essentially tied to the Euclidean (and hence the inner product) functionals, to a general Banach space context, we intend to extend them to infinite-dimensional spaces. This will be accomplished using direct limits, the counterpart of projective limit concepts, appearing in our earlier chapters in the (extended) form of Kolmogorov existence theory for stochastic processes. The geometric integration in Banach spaces extending Whitney’s theory has been obtained by Nolite (1975,1986) and we shall sketch it here partly as application of the preceding sections and mainly because it opens up many areas of new research problems both for random analysis (called random currents which are extensions of random Schwartz distributions) and also classical counterparts such as the Stokes theory considered in Chapter 7. All this needs some (new) terminology and concepts which are

406

8 Vector Measures and Integrals

introduced here. Then the desired norm functionals are given starting with the following: Definition 8.4.3 Let X, Y be a pair of (real) infinite dimensional Banach spaces and i : X → Y be an embedding which preserves the linear structure, and let i : Pn (X) → Pn (Y) be the induced mapping on the spaces of polyhedral n-chains of the two spaces as given. [This can be the general set up, but for simplicity we take now and hereafter X = Y.] Then a pair of norms will be given as follows: (a) the flat norm on Pn (X) is defined and denoted by kAkb = inf{kA − ∂Dk + kDk : D ∈ Pn (X)}, A ∈ Pn (X),

(4)

P P where ∂ is the boundary given as: if D = ki=1 ai xi , then ∂D = ki=1 ai ∂xni , P P (n−1) , and ∂ is a linear operator. ˜i ˜ni ) = i bi x with ∂xn = ∂( i bi x (b) the sharp norm on Pn (X) is again defined and denoted as: ♯

kAk = inf

X n i=1

A=

n X |ai |kxni kkvi k ai τvi (xni )kb : +k n+1 i=1

n X i=1

ai xni , simplex, xni ∈ Pn (X), vi ∈ X , A ∈ Pn (X), (5)

where τv is the translation operator acting on Pn (X), the linear space of n1 simplexes of X. [Here in (5) the factor n+1 is used in the first sum for a certain normalization in computations occurring in later work.] The concepts of both these norms are unmotivated and targeted to the specific problems appearing in the theory of geometric integration. That they are semi-norms is easily verified, and that they vanish only at zero, ‘0’, is nontrivial but true and needs more work. Taking vi = 0, one sees that the set on which infimum is taken is larger for the sharp case than for the flat set, so that from (4) and (5) one obtains kAk♯ ≤ kAk♭ , A ∈ Pn (X). It is clear that {Pn (X), k · k♭ } and {Pn (X), k · k♯ } are normed linear spaces, and their elements are termed flat and sharp chains. The members of their adjoint spaces are termed flat and sharp cochains respectively. Thus a cochain is a continuous linear functional on the normed space of (oriented) polyhedral chains. Similarly if ∂ : Pn (X) → Pn (X) so that ∂(xn (p0 , . . . , pn )) = P n j n−1 (p0 , . . . , pˆj , . . . , pn ) where the ‘hat’ notation on pˆj means that j=0 (−1) x th the j point is omitted and one verifies that ∂(∂A) = 0 for A ∈ Pn (X), (see Whitney (1957),p.153 for details). The adjoint of “∂” is denoted by “d”, called the coboundary of xn ∈ Pn (X), which gives (dx)(A) = x(∂A), and dx ∈ Pn+1 (X). Their (adjoint) norms are termed (co) mass which are given as follows. If AP∈ Pn (X), its mass (i.e. norm) P is given after Definition 2 above n n n n |a |m(x ) where A(x) = as kAk = i i=1 i i=1 ai xi . If X(·) is a cochain,

8.4 Some Applications of Vector Measure Integration,

Part I

407

or an n-cochain, so that X(A) is defined, we have the comass (i.e.,the adjointn norm) of X as okXk = sup {|X(A)| : kAk = 1} which is the same as : kAk 6= 0 . The Lipschitz comass of a cochain X is defined simisup |X(A)| kAk larly as: |X(τu A − A)| L(X) = sup : A 6= 0, u 6= 0, u ∈ X, A ∈ Pn (X) . (6) kAk · kuk This concept gives a connection between flat and sharp norms of cochains, and comasses. We give these relations here leaving the easy proofs to the reader (or consult Whitney (1957)): Proposition 8.4.4 Let A and X be a polyhedral chain and a cochain based on Pn (X). Then, using the Hahn-Banach theorem and kAk ≥ kAk♭ one has:(∂=boundary, d=coboundary) (i) |X(A)| ≤ min{kXk♭, kXk♭kAk♭ , kXkkAk}, A ∈ Pn (X); (ii) k∂Ak♭ ≤ kAk♭ , and if X is a flat cochain (i.e., (Pn (X), k · k♭ )⋆ , the adjoint space), so that d ◦ X is also one, and kXk♭ ≤ max(kXk, kdXk). (iii) A linear functional X : Pn (X) → R, with max(kXk, kd ◦ Xk) < ∞ is necessarily a flat cochain, and we have the adjoint norms (=comasses) as: kXk = sup{|X(σ)| : kσk = 1, σ − simplex}, (7) kd ◦ Xk = sup{|X(∂σ)| : kσk = 1, σ a simplex} and the sharp norm is given for a sharp X, by using (6) when A is an nsimplex, kXk♯ = max{kXk♭, (n + 1)L(X), L(X)} < ∞. (8) A proof of this proposition is deduced from the definitions of various terms, and is also detailed in Noltie (1986). Our aim is to present the basic results on these objects and obtain integral representations relative to suitable (vector) measures of sharp and flat chains in (infinite dimensional) Banach spaces under appropriate conditions using the finite dimensional results given above, through a procedure based on direct systems and their limits. This shows that there are many possibilities with different generalizations of these results to Banach spaces and later applied to “random currents” based on the works of G. de Rham and L. Schwartz. Hereafter we consider real vector spaces unless stated otherwise. In the present work if X is a vector space for x, y ∈ X, we should have an operation (a binary one) such that their composition (or “product”) is again a vector just as in a group algebra L1 (G) if (f1i , . . . , fni ), i = 1, 2, is a pair we can form (f11 ⋆ f22 , . . . , fn1 ⋆ fn2 ) as a similar vector of elements where fi1 ⋆ fi2 is a convolution. In the general space of n-dimensional vectors we similarly want to find an operation (binary at first) to obtain a new vector which preserves linearity and anti-commutativity, i.e., f 1 ⋆ f 2 = −f 2 ⋆ f 1 and so f 1 ⋆ f 1 = 0 (f 1 6= 0). It

408

8 Vector Measures and Integrals

is fortunate that such an operation exists, which will be denoted V by ∧ (wedge product) and consider theVspace of all such vectors, denoted (X) satisfying (i) the (real) element 1 ∈ (X), (ii) spanned by 1 and the elements of X with the same property, (iii) for x ∈ X (x 6= 1), x ∧ x = 0 and if the dimension of V P m m X is m(< ∞), then the vector space (X) has i=0 i = (1 + 1)m = 2m V linearly independent vectors. It results that if xi ∈ (X), i = 1, . . . , m, then from xi ∧ xi = 0, xi ∧ xj = −xj ∧ xi , i, j = 1, . . . , m, and has vectors of the of course vanishes if form xi1 ∧ · · · ∧ xik , i1 < i2 < · · · < ik , k ≤ m, which V xij = xi′j , for some 1 ≤ j = j ′ ≤ m. Such a space (X) is called an exterior algebra (or a Grassman algebra) and the operation ‘∧’ exterior product. The elements xi1 ∧ · · · ∧ xim , are if not zero linearly independent. This is true if x1 , . . . , xm are linearly independent in X, and are often called simple or deV composable m-vectors and (X) is spanned by 1 and all such vectors in case X has m linearly independent xi ’s. This algebra has numerous applications and here we indicate just a few. [There is also an “interior product”, between an r-covector and an s-vector, to be an |s − r|-vector so that if s = r it becomes a scalar product. But we shall defer this until “currents” are considered. See Whitney ((1957), Section I.7, and we will discuss it only in the next secV V tion.] Note that from (X) we can consider its k th exterior power k (X), and when Vk X is a normed vector space, we will also introduce a norm functional on (X). Vk V Indeed a norm for (X) is defined as follows. If k = 1, (X) = X and the X-norm is given to it and if k > 1, we can consider the tensor V Vk (X), and if t = ⊗ki=1 xi for linearly independent product ⊗ki=1 (X) = Vi xi ∈ X, then t ∈ ⊗ki=1 (X) and then let ′ ∼′ be the equivalence class Vk of these vectors so that (X) may be identified with the quotient space V ⊗ki=1 i (X)/ ∼ and a “cross-norm” can be defined on it. [ There exist several such cross norms as discussed in Schatten (1950).] Earlier in the section we assigned another type (a translation invariant one ) that was given Vk to (X) = Pk (X), the space of polyhedral chains, through the Haar meaVk sure. In all these cases, we find (and thus may assume) that (X) can be Vk ′ given a norm. Thus ( (X), k · k ) becomes a normed algebra and we set Pm Pm kxk0 = inf{ i=1 kxi k′ : x = i=1 xi , a decomposible chain}. This norm will Vk also be called a mass on (X) and its adjoint, termed comass on the adVk Vk joint space ( (X))⋆ , or for short, (X)⋆ , denoted k · ko and is given by o kyk = sup{|y(x)| : x decomposable, kxk′ = 1}. Just as in the general theory of linear analysis, we first show that linear Vkfunctionals on spaces of simplexes determine their behavior on the spaces (X), and show that the mass and its comass are conjugate norms. Then we proceed with the general integration. The following is basic: Vk (X) be the k th -power exterior algebra and y : Proposition 8.4.5 Let Vk (X) → R be a linear functional on simple vectors x. If x˜ =

8.4 Some Applications of Vector Measure Integration,

Pk+1

409

0 for any (k + 1) simplex V σ = σ(p0 , . . . , pk+1 ), then there exists uniquely a y˜ ∈ k (X)⋆ , a covector, which extends y so that y˜(x) = y(x) for simple x. Moreover the functionals ‘mass’ and Vk ‘comass’ are conjugate norms on (X). j=

(−1)σ(p0 , p1 , . . . pˆj , . . . pk+1 ), y(˜ x)

Part I

=

Proof. The result is established for the familiar Rn in place of X (of dimension n) in Whitney ((1957), Theorem 9A on p.165), and we extend it to the abstract space as follows. Let {ei , i = 1, . . . , n} be the standard basis of unit vectors in Rn and p0 , p1 , . . . , pk for X with vi = pi − pi−1 , i = 1, . . . , k. Let Φ : Rk → X with Φ(ei ) = vi and e1 ∧ · · · ∧ ek 7→ vi ∧ · · · ∧ vk which is isomorphic, and Vk n Vk V⋆ Vk n Φ: (R ) → (X). If Φ⋆ : (X)k → (R ) is the adjoint of Φ, then for a Vk Vk given y : (X) → R, we associate y˜ : (X) → R by setting y˜ ◦ Φ−1 = y. Since Φ preserves simple multivectors, if σ = σ(Φ−1 (p0 ), . . . , Φ−1 (pk+1 )), σ being a k + 1 simplex, we have k+1 X j=0

[ −1 )(p ), . . . , Φ−1 (p (−1j )˜ y {σ(Φ−1 (p0 ), . . . , Φ j k+1 )}

=

k+1 X

(−1)j y˜(Φ−1 σ(p0 , . . . , pˆj , . . . .pk+1 )})

j=0

=

k+1 X

(−1)j y{σ(p0 , . . . , pˆi , . . . pk+1 )} = 0.

j=0

Thus y˜ satisfies the result if X = Rn , and by Whitney’s theorem, there is a y˜ of the kind in the theorem such that y˜ ◦ Φ−1 = y. Since Vkasserted ⋆ y˜ is defined on (X) , agreeing with y on simplexes, the first part is established. Regarding the second part, if x 6= 0, then kxk0 ≥ kxk′ > 0 since k · k′ is a norm. As the subadditivity and positive homogeneity of k · k0 are clear, it follows that the mass functional k · k0 is a norm. Also if y 6= 0 is a covPm ector, there is a chain x = i=1 xi , xi -simple, such that y(xi ) 6= 0 for some i and so kxk0 > 0 implying that the comass is also a norm. To see Pmthat these are conjugate norms, let x ∈ V k (X), P y ∈ V k (X⋆ ) and x P = i=1 xi m m 0 ′ kx where xi are simple chains. Then |y(x)| ≤ |y(x )| ≤ kyk ik , i i=1 i=1 0 which implies that |y(x)| ≤ kyk kxk0 on taking the infima over such x. Hence the supremum over the set kxk0 = 1, gives kyk0 = sup{|y(x)| : kyk0 = 1}. Since kxk′ = kxk0 for simple chains, by the Hahn-Banach theorem kxk0 = sup{|y(x)| : kyk0 = 1}, so that k · k0 and k · k0 are conjugate norms. 2 Just as for cochains, we can define the Lipschitz constant for differential forms, and a comass for y as: L(y) = sup{ky(p)−y(q)k0 /|p−q| : p 6= q ∈ X}; kyk0 = sup{ky(p)k0 : kpk ≤ 1}.

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A differential k-form on a subset of Rn is a mapping with values in (R ) , the adjoint space of a subset of dimension k,i.e., the values are kcovectors, and a k-vector is a (finite) linear combination of k-exterior multiples of the basis vectors vi of Rn so that k- is the degree (or dimension) of the k-vector. If σ is a k-simplex, let ασ be the unit k-vector that corresponds to the oriented k-dimensional subspace of X determined by σ, and ασ is termed the k-direction of σ which is uniquely determined. [These concepts and their properties are discussed in Whitney (1957), Chapter I.] Since we have V already seen that there is a normalized Haar measure on the subspace of k (X), for each k-simplex σ k say µ ˜σk and setting µσ (A) = µσk (A ∩ σ) for each Borel set A, one can define the integral of the differential k-form y as R R y. Now if we let d˜ µ = ασ dµσ , a vector meay · α dµ abbreviating it as σ σ σ X R R µ, is a bilinear vector integral. Let us also denote by sure, then σ y = X y · d˜ R ˜ = sup{| y|/kσk : σ ∈ Pk (X)} in the earlier notation. All these special ∆y ∂σ concepts and symbols are needed in the geometric theory! It is desired to obtain integral representations of differential forms for both sharp and flat types. First consider the former: Vk (X) is called sharp if kyk0 , L(y) Definition 8.4.6 A differential form y on ˜ and ∆y defined above are all finite. Then the sharp norm of y is defined as ˜ (k + 1)L(y)}. kyk♯ = max{kyk0, ∆y, (9) n ⋆

For X a finite dimensional Banach space (not necessarily Euclidean) sharp k- cochain can also be given an integral representation as follows: Theorem 8.4.7 Let y be Ra sharp kR cochain. Then there exists a unique kform Dy such that y(σ) = σ Dy (= X Dy · ασ dµσ ) for all k- simplexes σ in X. Moreover, an isometric equivalence is given as: kDy k0 = kyk, kDy k♯ = kyk♯ , and L(Dy ) = L(y).

(10)

The point of this result is to extend Whitney’s work from Rn to a finite dimensional normed space X without using the Euclidean inner product. Thus if α is a given direction, Dy and Dy′ are two sharp k-forms then letting {σm , m ≥ 1} be a sequence of simplexes with p ∈ σn , which converge to zero in the sense that kσn k → 0 as their diameters in X tend to zero when all σn have αy as their k-direction, then for any p, q ∈ X, Z Z 1 Dy (q) · αdµσi → 0 Dy (p).αdµσi − kσi k X X R as i → ∞, by the continuity of Dy (·). Thus for fixed p, Dy · α = kσ1i k σi Dy . R R ¯ = Dy′ · α ¯ for all α ¯ . Hence Dy = Dy′ so that the Also σi Dy = σi Dy′ or Dy · α sharp k-form does not depend on the {σi , i ≥ 1} used implying uniqueness. y·σi The actual construction of Dy is defined by Dy (p, α) = limi→∞ kσ , and it ik

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411

should be shown to satisfy the three norm conditions. It involves, a systematic extension of Whitney’s analysis given for Rn . This was also done by Noltie ((1986) pp.219-222) in detail, and we refer to it, since it is readily available. Note that αµσ is the vector measure of Theorem 1 above. Our aim is to extend the result to general Banach spaces. We note, in passing, the fact that the sharp and flat norms given in Proposition 5, shown there as only semi-norms, are actually genuine norm functionals since k · k♯ ≤ k · k♭ . Then it is enough to verify this for the sharp case, and this involves some nontrivial work. This was also given in Noltie (1986) extending Whitney’s Euclidean case. The procedure employed here is to apply the methodology of direct limits, from Bourbaki (cf., also Rao (1981), Chapter III ), and postulating an additional condition which will be seen to be satisfied for Hilbert spaces as well as, more generally, for Banach spaces that are continuous images of Hilbert spaces. We now sketch the necessary details. A family {Xα , fαβ , α < β, α, β ∈ I} of indexed spaces is a directed system if {I, <} is a directed index set and for α < β, the mappings fαβ : Xα → Xβ are defined subject to the conditions: (i)fαα = identity, and (ii)α < β < γ ⇒ fγα = fγβ ◦ fβα . We denote it as {Xα , fαβ , α < β α, β ∈ I}. The direct limit of measure spaces can be given as follows. Let X be the free union or set sum ˜ = ∪α∈I (Xα × α), and declare that xα ∈ Xα , xβ ∈ Xβ in the sense that X are in the same equivalent class, written xα ∼ xβ if and only if for some γ ∈ I, γ > α, γ > β, and fγα (xα ) = fγβ (xβ ). This “ ∼′′ defines an equivalence ˜ and the quotient space denoted X(= X/ ˜ ∼), also written X = relation on X lim→ (Xα , fαβ ), is called the direct limit of the family of spaces (Xα , fαβ ). The ˜ is a lattice homomorphism and fα = f |Xα canonical mapping f : X → X (the restriction with Xα where Xα × {α} is identified with the former) takes Xα into X and for α < β < γ, we have fγβ ◦ fβα = fγα as before. Moreover fαβ ◦ fβ = fα . These mappings are also homomorphisms and for x, y ∈ X there exist xα ∈ Xα , yβ ∈ Xβ satisfying fα (xα ) = x, fβ (xβ ) = y. In our present application, the Xα are vector spaces and so the fαβ are linear mappings into X; also the fα are linear. Moreover if {Xα , fαβ , α < β, α, β ∈ I} and {Yα , fαβ , α < β, α, β ∈ I} are two such systems and if there exists a mapping hα : Xα → Yα connecting the two systems by the relation gαβ ◦ hα = hβ ◦ fαβ for α < β in I, then it can be shown that there exists a unique h : X → Y such that gα ◦ hα = h ◦ fα : Xα → Y. Then the hα -family is termed a direct system of mappings. It is desired to define, in an infinite dimensional Banach space X, sharp and flat differential k-forms with sharp and flat norms extending the finite dimensional case considered earlier, using the direct limit theory described above. Special care is needed because several new problems arise. We treat the sharp case here and the flat chains in the following section. All this is needed in order to apply the general vector integration of the earlier sections to important problems of analysis. They show the need for further extension of the preceding

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effort, emphasizing all this work for general analysis (here these are infinite dimensional Banach spaces) which is needed to solve several interesting “applied” problems including random currents, to be discussed later. Let {Xα , α ∈ I} be the collection of finite dimensional subspaces (= closed linear manifolds) of X directed by containment. If iαβ is the inclusion mapping of Xα ֒→ Xβ (for α < β) so that {Xα , iαβ , α < β in I} k forms a directed system, let SX (Xα ) be the space of k-chains in Xα , of dik k mension at least k ≥ 1. The adjoint maps i⋆αβ : SX (Xα ) → SX (Xβ ), in⋆ ⋆ ⋆ duced by iαβ , form a direct system (iγβ ◦ iβα = iγα for α < β < γ) and k let S0k (X) = lim→ {SX (Xα ), i⋆αβ } be the direct limit of the system of k-chains k so constructed. It is a normed linear space. P∞ If {Ai , i ≥ 1} ⊂ S0 (X) such that P ∞ ♯ i=1 Ai is termed a sharp k-chain of X. i=1 kAi k < ∞, then the sum A = We set ∞ ∞ X X kAk♯ < ∞}, (11) Ai , Ai ∈ S0k (X) : S k (X) = {A = i=1

i=1

¯ + by (+)kAk♯ = limm→∞ k Pm Ai k♯ . and define the map k · k : S (X) → R i=1 ♯

k

Proposition 8.4.8 The functional k · k♯ : S k (X) → R+ given by (+) just above is well-defined, independent of the sequence {Ai , i ≥ 1}, and is a norm, which satisfies: ) (∞ ∞ X X k ♯ ♯ Ai ∈ S (X) . (12) kAi k : A = kAk = inf i=1

i=1

P∞

Pj Proof. Let A ∈ S k (X), A = i=1 Ai and Bj = i=1 Ai so that since A ∈ S k (X) by (11), {Bj , j ≥ 1} is Cauchy in k · k♯ . Hence k · k♯ being a norm, |kBj k♯ − kBℓ k♯ | ≤ kBj − Bℓ k♯ → 0, and the expression (12) is defined unambiguously. To seeP that it does not P depend on the particu∞ ∞ ♯ lar representation let kAk′ = inf kA k : A = i i=1 i=1 Ai . The standard ′ and familiar arguments show that k · k is a positively homogeneous subadk ♯ ditive functional If for ε > 0, we chose P∞ ′on S0 (X) since P∞k · k′ ♯is a norm. ′ ′ Ai , A = A , satisfying kA k < kAk + ε, let mε be an integer i i i=1 i=1 such that mε ∞ X X ε ε kA′i k♯ < . (13) A′i k♯ < kAk♯ + , and k 2 2 i=mε +1 i=1 P mε ′ This is possible definition of k·k′ . If B1 = i=1 Ai , Bj = Amε +j−1, j≥2 P∞ from theP ♯ ♯ then A = i=1 Bi and ∞ kB k < kAk + ε. Taking the ‘inf’ on the left, we i i=1 get kAk′ ≤ kAk♯ . Since we opposite inequality is always true kAk′ = kAk♯ . But S0k (X) is dense in S k (X). So it follows that {S k (X), k · k♯ } is a normed linear space, and k · k♯ is unique as well. 2 V V As before we consider the exterior algebra k (X), k ≥ 1, ( 0 (X) = R) generated by {xi1 ∧ · · · ∧ xik , i1 < · · · < ik } where {xi , i ∈ I} is a baVk Vk ⋆ Vk ⋆ (X ) ⊂ ( (X))⋆ with a gener(X ) is defined and sis of X. Similarly Vk ally strict inclusion when X is infinite dimensional and k > 1. Also (X)

8.4 Some Applications of Vector Measure Integration,

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413

Vk

(X)⋆ is defined as a difmay be incomplete. The adjoint mapping y : X → ferential k-form (same way as in finite dimensional case of X), and denote the norm (or mass) and adjoint norm (or comass) written again as k · k0 and k · k0 . The Lipschitz constant L(y) is also defined for a k-simplex σ and the k-direction ασ . Then the integral of the differential form is given as follows: Z ˜ ∆y = sup y /kσk . (14) ∂σ

In this infinite dimensional case we will also introduce: Vk (X)⋆ is sharp if its norm Definition 8.4.9 A differential k-form in satisfies ˜ (k + 1)L(y)} < ∞. kyk⋆ = max{kyk0 , ∆y, (15) and the elements of S k (X)⋆ [ or(S k (X), k · k♯ )⋆ ] are termed sharp k-cochains in X.

Our aim is to get an integral representation of sharp k-chains, corresponding to Theorem 7 above, in the infinite dimensional case of X. For this we intend to define sharp k-forms as a direct limit of these from finite dimensions, and with them obtain the desired integral for all sharp k-chains in X relative to a vector measure, valued in X, having a suitable variation property. Because of the way the sharp norm defined on (S k (X))⋆ , the space is not generally complete (a new complication in infinite dimensions), and X being a Banach (not Hilbert) space, some additional restrictions are needed. We now present the necessary outline to better understand applications of the general theory for which some new and nontrivial problems arise and they must be solved. In the finite dimensional case, Theorem 7 provides the desired integral representation of a k-cochain y relative to a unique k-form defined on ksimplexes with isometric equivalence. Now to generalize this, observe that for a directed family {xα , iαβ , α < β, α, β ∈ I} where I is a directed index and Xα ⊂ X are finite dimensional let iαβ : Xα → Xβ (α < β) be inclusion mapVk (Xα )⋆ , y ∈ S k (X)⋆ (with pings. We may define sharp k-forms Dyα : xα → ♯ k the sharp norm k · k for S (X) of Definition 9 above). Now we want to deVk (X)⋆ by the direct limit process because a direct system of fine Dy : X → Vk (Xα )⋆ nonempty spaces always has a direct limit. Although Dyα : Xα → is well-defined and iαβ : Xα → Xβ is available, there need be no mapping V V eαβ : k (Xα )⋆ → k (Xβ )⋆ so that for α < β, Dyβ ◦ iαβ and eαβ ◦ Dyα should agree. Also the eαβ need not exist. We give an example later to illustrate this difficulty. Consequently a little more work is needed to circumvent this problem. To understand the issue better, we first state the following result from Bourbaki (1966, pp.204 to 207) and comment on it briefly.

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i Proposition 8.4.10 Let {Xiαβ , fαβ , α < β, α, β ∈ I}, i = 1, 2 be a pair of direct systems of spaces and let Xi = lim→ Xiα be their direct limits, which always exist. If fαi : Xiα → Xi , i = 1, 2 are the canonical mappings into the limit spaces, define uα : X1α → X2α as a mapping such that for 2 1 each fβα ◦ uα = uβ ◦ fαβ , α ≤ β. Then there exists a map u : X1 → X2 such that for each α ∈ I, one has u ◦ fα1 = fα2 ◦ uα and that the mappings commute. Hence the collection {uα , α ∈ I} which is the direct system of i mappings of {Xiα , fαβ , α < β, α, β ∈ I} has u as the direct limit, written u = lim→ uα . In case the spaces X1α and X2α are topological for α ∈ I and all the connecti ing maps fαβ , uα , i = 1, 2 are continuous (and injective, subjective or homeomorphisms) for each α, β, then the same property holds for the limit map u. The result survives if I is replaced by a sub directed set I0 (⊂ I) which is cofinal in the sense that for each α ∈ I there is α0 ∈ I0 , α < α0 and for β, β ′ ∈ I0 , there is α ∈ I0 with β, β ′ < α, so that again there is α ˜ 0 ∈ I0 satisfying α < α ˜0 .

The last part with topological conditions (to be used below) is established (along with the main part) in Dugundji ((1966), Appendix Two) and the general case in Bourbaki (1966). The reader is referred to these standard sources for details, and here we limit our diversion. Our immediate use of this result is to consider the sharp cochains Dyα : Xα → Vk (Xα )⋆ and uα with iαβ : Xα → Xβ as inclusion maps, where Dyα is D(y |S k (Xα )) in the above proposition. Although Dyβ (pi ) |Vk (Xα ) = Dyα (pi ), i = 1, 2, in general Dyβ (p1 ) 6= Dyβ (p2 ) for p1 6= p2 , can happen but with equality of the same for α. Hence the direct limit of the uα = Dyα (·) need not exist. Fortunately a weakening gives a “weak direct limit ” of the “uα ’s”, sufficient for us, and the desired concept will now be introduced. The idea here is to utilize the (special) result that {Dyα (·), α ∈ I} are functions on Xα , and so we need to consider the special case of this function families obeying the direct limit conditions. This is accomplished by recalling the related concept of projective limit and to connect both these concepts as follows. If {Xα , fαβ , α < β} is a direct system and Z is a set, consider the function space Yα = Z Xα , the class of all mappings or functions on Xα valued in Z. If f˜αβ : Yα → Yβ is defined as f˜αβ (h) = h ◦ fαβ , h ∈ Yα , then for α < β < γ we get f˜αβ ◦ f˜βγ = f˜αγ , f˜αα = id = identity, so that {f˜αβ , α < β, α, β ∈ I} forms a projective system of mappings. If Z is a topological space (e.g., Z = R) then one can give the Tychonov or product topology for Yα from Z for any set Xα called the pointwise convergence topology denoted Jα p with {(x, V ) | x ∈ Xα , V ⊂ Z open} as a sub base of this topology, and Q the f˜αβ are continuous in this case. The points y = {yα , α ∈ I} ∈ α Yα may satisfy πα (y) = gαβ ◦ πβ (y) for a system {gαβ , α < β, gαα = id}, and gαβ ◦ gβγ = gαγ for α < β < γ. If this holds then {Y Qα , gαβ , α < β in I} is called the projective system, and the subspace Y ⊂ α Yα of the points y is

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415

called the projective limit, denoted Y = lim← (Yα , gαβ ), when Y 6= ∅. Unlike the direct system (and limit) the projective system even when all gαβ are onto, can be empty. The {yα , α ∈ I} ∈ Y are called threads the mapping gα : Y → Yα , i.e., gα (y) = yα is the canonical or a coordinate projection and thus many y’s can have the same imageQyα . [Note that gα is the restriction to Y of the Q coordinate projection from α Yα into Yα i.e., if projα : α Yα → Yα then gα = (projα | Y ) : Y → Yα .] The following connecting property of the projective and direct limits is of great interest here and it will be related to our own {Dyα (·), α ∈ I} system defined above. Proposition 8.4.11 Let {Xα , fαβ , α < β in I} be a direct system of (nonempty) spaces and Z be a nonempty set. Let Yα = Z Xα with the induced mappings f˜αβ . Then {Yα , f˜αβ , α, β in I} is a projective system of spaces and the projective limit lim← {Yα , f˜αβ , αβ in I} exists and it is isomorphic to Z lim→ {Xα ,fαβ , α,β in I} when all these product spaces (regarded as function spaces) are given the topology Jp of pointwise convergence as defined above. A proof of this useful result and its consequences are detailed in the wellknown text book by Dugundji (1966) and need not be repeated here. Its application in our context is given by the following: Theorem 8.4.12 If y ∈ S k (X)⋆ , a sharp k-cochain, then there is a unique Vk (X)) one has the represharp k-form Dy Rsuch that for any k-simplex σ(∈ sentation y(σ) = σ Dy , in the notation of Theorem 7, and the corresponding norm (or isometric) relations given in (10) all hold. Vk V (Xα ) can be identified with (X) the norm kDy k0 = Note. Although lim→ kyk which is defined as kyk = sup{|y(A)| : A ∈ lim Pk (Xα ) ⊂ S0k (X), kAk = 1}

(16)

→

is not related in general to the direct limit topologies employed in getting the k-form Dy (·). Proof. For an appreciation of the difference between this result and the classical linear analysis representations, we include Vthe details. Here take Z = R k in Proposition 11 above and Vconsider Rlim→ (Xα ) Vas the vector space of k k (Xα ))⋆ , which is thus (Xα ), i.e., on (lim→ real linear mappings on lim→ identifiable as a system of linear functions, denoted as Dyα (·), on Xα and they satisfy a consistency relation in α (as part of a ‘thread’ defined on Q ˜ = {xα , α ∈ I} and f˜αβ (xβ ) = xα , α < β, α Xα ), so that letting x α we have idR (Dy (x)) = Dyβ (x) ◦ f˜αβ (= xαβ (Dyβ ) in the standard composiVk

tion). This means we have a function Dy : lim→ (Xα , fαβ ) → Rlim→ (Xα ) = Vk (Xα ))⋆ . The identifications imply that the element Dy (·) is a lin(lim→ Vk (X). We now show that it is continuous, or equivalently ear functional on

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bounded. The problem is to establish the properties on component spaces Xα and lift them to lim→ Xα . This is called a “weak” direct limit, since it works only for such special function spaces! The details are given in steps for convenience: Vk Vk (Xα ))⋆ as above. (Xα ) and Dy : lim→ Xα → (lim→ 1. Let x1 , x2 ∈ lim→ If α1 , α2 ∈ I, such that xi = fαi (xαi ), i = 1, 2 where fα : Xα → lim→ Xα is the canonical inclusion family of mappings giving the direct limit, then by directedness of I, there exists α > α1 , α2 such that Xαi ⊂ Xα and xi = fαi (xα ). If p ∈ lim→ Xα , then we have if h·, ·i denotes the bilinear pairing on the vector space and its adjoint [for hx⋆ , xi or x⋆ (x)] : (use fγ : Xγ → X, the direct limit space) hDy (p), ax1 + bx2 i = hDy (p), (fγ (axα1 + bxα2 )i

= hDy (p), axα1 + bxα2 i = ahDyα1 (p), xα1 i + bhDyα2 (p), xα2 i

= ahDy (p), xα1 i + bhDy (p), xα2 i,

Vk so that Dy (p) is linear on lim→ ( (Xα ), fαβ ). For boundedness, let x ∈ Vk Vk (Xα ) and take β > α, xβ ∈ (Xβ ) and fβ (x) = xβ . Then we lim→ have |hDy (p), xi| = |hDy (p), fβ (xβ )i| = |hDyβ (p), xβ i|

≤ kDyβ k0 kxβ k0 = kyβ k kxβ k0 ≤ kyk kfα (xβ )k0

= kyk kxk0 .

Vk Taking the supremum as p-varies on lim→ Xα , we get Dy : lim→ Xα → (X)⋆ , to be continuous and linear. We now show that Dy represents y on simplexes as an integral kernel. 2. If σ ˜ ∈ lim→ Pk (Xα ) ⊂ S0k (X), select β ∈ I with f˜β (σ) = σ ˜ where k σ ∈ P (Xβ ). Then by Theorem 12, we can evaluate hy, σ ˜ i as: Z Z ˜ σ idµσ Dy β = hDyβ , α hy, σ ˜ i = hyβ , σi = σ Z Zσ Dy . Dy · α ˜ f˜β (σ) dµiβ (σ) = = f˜β (σ)

Hence y(σ) = Also

R

σ

σ ˜

Dy for all k-simplexes σ in S0k (X) (replacing σ ˜ by σ above).

kDy k0 = sup{kDyβ (p)k0 : p ∈ Xβ , β ∈ I}

= sup{ky α k : α ∈ I} = sup{|hy, Ai| : kAk = 1, A ∈ S0k (X)} = kyk.

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One can similarly show that the Lipschitz norm of y is the same as the norm of y. If σ ˜ is a k + 1 simplex of S0k (X), then for the sharp norm we need to evaluate the value of y for the boundary operators too. Thus σ ˜ = f˜β (σ) with σ ∈ Xβ and ∂ σ ˜ = f˜β (∂σ) so that |y(∂ σ ˜ )| = |y β (∂σ)| = |(dy β )(σ)|

≤ kdy β k kσk ≤ ky β k♯ kσk ≤ kyk♯ k˜ σ k,

˜ y ) ≤ kyk♯ < ∞ implying that y is a sharp k-form. We also need to whence ∆(D calculate the sharp norm of Dy . 3. It is now asserted that the correspondence Dy → y is isometric and onto, to conclude the theorem. However, using the result of Theorem 7, and the preceding analysis, one has ˜ y ), (k + 1)L(Dy )} kDy k♯ = max{kDy k0 , ∆(D β 0 ˜ yβ )(k + 1)L(Dyβ )} = sup max{kDy k , ∆(D β

= sup kDyβ k♯ = sup ky β k♯ , by Theorem7, β∈I

β

= sup ky |S k (Xβ ) k♯ = kyk♯ . β

Since the correspondence between Dyβ and y β is one-to-one, and then using the uniqueness of the relation in Proposition 11, one concludes that the same property holds here. Now regarding the surjectivity property of Dy , for any V given sharp k-form y˜ : lim→ Xα = X → k (X)⋆ , consider y˜α (= y˜α (p, x), p ∈ Vk Vk (Xα )⋆ is sharp, and (using the fi(Xα )). Then y˜α : Xα → Xα , x ∈ Vk α α nite dimensional case) y˜ and y : Xα → (Xα )⋆ correspond to each other Vk α ⋆ uniquely. Then {y : Xα → (Xα ) , α ∈ I} is a direct system of functionals and {y |S k (Xα ) = y α , α ∈ I} will be compatible and Dy = y˜ holds. MoreV over the system {˜ y α (p), k (Xα ), f˜αβ , α < β ∈ I} forms a direct system of real continuous linear functionals having a limit that is identifiable with Dy . Then the surjectivity may be inferred from this as desired in the statement. 2 Most of the details are included here to emphasize that a simple direct application of the abstract analysis is not possible and significant additional argument is needed. Because (three) different functionals appear in defining the sharp norm, the normed linear space of k-chains S k (X) is only normed but will not be complete, as seen from the following computation when X is infinite dimensional. It is also referred to later and so the example is given a serial number. Example 8.4.13 Let H(= X) be an infinite dimensional Hilbert space with {un , n ≥ 1} as an orthonormal set. Consider starting with a 1-chain in H given as: if {pn ∈ H1 , n ≥ 1} is chosen such that, defining A1 = σ(0, p1 ) satisfying

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the distance condition dist(p1 , 0) = 1. For convenience write the resulting chain as A1 = [0, p1 ], and let A2 = [0, p21 ] ∪ [p21 , p1 ], where dist (p21 , A1 ) = 12 , and then A3 = [0, p31 ]∪[p31 , p21 ]∪[p32 , p1 ], where dist (p3i , A2 ) = 212 , for i = 1, 2. By induction for k ≥ 2 we set Ak = [0, pk1 ] ∪ [pk1 , pk−1,1 ] ∪ · · · ∪ [pk,2k−2 , p1 ], with dist (pkj , Ak−1 ) = 21k , j = 1, 2, . . . , 2k−2 . Thus the Ak satisfy kAj − Ak k♯ ≤ P∞ 1 k kAj − Ak k♭ ≤ ℓ=min(i,k) 2ℓ . Hence the finite chains {Ak , k ≥ 1} ⊂ S (H) form a Cauchy sequence in the sharp norm. Let A = sharp norm limit of Ak . But A 6∈ S k (H), and for a finite dimensional subspace X0 ⊂ H1 , X0 ∩ supp(A) is a closed nowhere dense set and has H 1 - measure zero where H k denotes the Hausdorff k-dimensional metric measure. [If X0 is Rn and we consider the Lebesgue outer measure, then as is well-known, Leb. meas. and H 1 -are equivalent, but the general metric spaces demand H k (even more general H ϕ for ϕ(t) > 0) and increasing metric measures - - for a brief account of this class, see (Rao (2004), p.97).] It may be verified that the support of A is − ∞ contained in the boundary set ((∪∞ i=1 support(Ai )) − ∪i=1 support (Ai )), 1 and has H -metric outer measure zero. But these chains are important in geometric integration, and recalling that the Cantor set C ⊂ [0, 1], has Lebesgue outer measure zero, but has positive Hausdorff α-dimensional metric outer measure 1 > α > 0, so there are important sharp (hence flat) chains living on such sets. It follows that our proposed representation of such chains should use properties of Hausdorff (and other) metric measures which are not σ-finite. This non-trivial difference will be explained further in our representation. We have seen in earlier discussion that both the flat and sharp norms are defined using the class Pk (X), of polyhedral k-chains. Consequently the limit elements as well as those of S k (X) are determined by the class Pk (X), in the sense of the following observation. Here the elements of (S k (X), k · k♯ )− , the sharp norm closure of S k (X), will be denoted by GS k (X). These may be called general sharp (respectively flat) chains. One has the following: Proposition 8.4.14 If A ∈ GS k (X), then there exist Ai ∈ f˜α(i) (Pk (Xα(i) )) for some index α(i) ∈ I, the latter index depending on i. The same type result also holds for the generalized flat chains. Proof. By definition of A ∈ GS k (X), for each i there exists Ai ∈ f˜α(i) (S k (Xαi )) satisfying kA− Ai k♯ < 21i . But Pk (Xα ) ⊂ S k (Xα ) and is dense in it. Hence there exists a Bi ∈ f˜α(i) (Pk (Xαi )), with kAi − Bi k♯ < 21i . So it results by the triangle inequality that kA − Bi k♯ ≤ kA − Ai k♯ + kAi − Bi k♯ <

1 . 2i−1

Since i is arbitrary this implies the assertion. Using the flat norms the corresponding assertion obtains for the generalized flat chains as well . 2 Since Pk (X) is identifiable with lim→ Pk (Xα ) ⊂ S k (X), the Cauchy sequences and their limits in respective flat and sharp norms, the polyhedral chains (and

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cochains) play a key role in the computations on these spaces of chains. A generalized chain A of GS k (X) has finite mass (or norm) if there is a polyhedral sequence An ∈ Pk (X), satisfying limn inf kAn k < ∞, and then we set kAk to be the infimum of the norms sequences {An → A} in sharp norm. This means: kAk = inf{lim kAn k : An ∈ Pk (X), kA − An k♯ → 0}. (17) n

Pk Let A ∈ P (X) be a polyhedral chain. For a k-vector of A(= i=1 ai σi where σi is the simplex determined by the vectors σi (pi0 , . . . , pik )), and if vji = pij − pij−1 , j = 1, . . . , k, then {σi } = v1i ∧ · · · ∧ vki is the k-vector of Pk σi and then {A} = i=1 ai {σi } for scalars ai . Thus a k-vector of the chain A which is a linear combination of k simplexes is then the vector {A} as a k-vector polyhedral chain. This vector has a special relation with the k-differential form for members of GS k (X) which will be used in the main integral representation that is presented below. The bracket operation {·} is a technical one, and its properties are detailed in Whitney ((1957), pp. 81-82), which are utilized here. It was shown there that for each A ∈ Pk (X), {A} is well-defined and if A ∈ Pk+1 (X), then the boundary ∂A satisfies {∂A} = 0. Moreover, for any x ∈ X, and a k-vector α there exists a sharp chain A with support x, and {A} = α. Our representation is a non-trivial extension of the finite dimensional case Xα through the direct limit technique. Since in finite dimensional cases the space is isomorphic to a Euclidean space, we want to present a condition that abstracts and generalizes the latter. Recall that a k-form can be multiplied by a scalar function. If L(X) denotes the vector space of bounded scalar functions ϕ satisfying a Lipschitz condition Lβ (ϕ) of order β(0 < β ≤ 1), then ϕ ∈ L(X) is said to have a finite sharp norm, kϕk♯ = max{Lβ (ϕ), |ϕ|u } where |ϕ|u is the uniform norm. We then introduce the following technical: k

Condition M: S k (X)⋆ is a left module over L(X), regarded as a space of sharp k-forms on X for k ≥ 1, and satisfies for y ∈ S k (X)⋆ , ϕ ∈ L(X), i.e., for sharp y and sharp ϕ one has: ˜ ∆(ϕ) = sup{kϕyk♯ : kyk♯ ≤ 1, kϕk♯ ≤ 1} < ∞,

(18)

[Thus L(X) acts as the scalar field in the case of the vector spaces such as X.] This condition seems to be complicated and unmotivated, but it is valid for all X = H, Hilbert spaces and all those Banach spaces that are continuous linear images of H. This useful result is established by Noltie ((1986), p. 246) If (S, S) is a measurable space and Z is a Banach space, let µ : S → Z be a (vector or σ-additive) measure, of P finite variation ν, so that ν(·) = |µ|(·) and µ ≪ ν (i.e., ν(B) = sup { kµ(Ai )kZ : Ai ∈ S(B), disjoint.}) which is σ-additive but ν(B) = +∞ for all B 6= ∅ is possible. Here we take Vk (X) with sharp and flat S = X and S = Borel σ-algebra of X and Z =

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norms as the case may be. Let Mk (X) ⊂ GS k (X) be the set of generalized sharp k-chains in X of finite mass as in (18) above. The corresponding general representation of Mk (X) is given by the main result of this section as follows: Theorem 8.4.15 Let X be a Banach space satisfying Condition (M). Then with each element A ∈ Mk (X)(⊂ GS k (X)) there exists a unique vector measure Vk µA : B(X) → (X), of finite (Vitali) variation such that for all sharp cochains y ∈ S k (X)⋆ , we have Z (19) y(A) = hDy , dµA i, kAk = |µA |(X) < ∞, kAk♯ = kµA k♯ . X

Moreover {A} = µA (X) and Dy is a sharp k-form determined by y (cf. Theorem 12).

As may be expected, the proof needs several properties of k-chains and k-cochains as well as other facts that we have discussed above. Since the complete details are given in Noltie ((1986), Sec.2.4), which is available easily, we omit the proof here and invite the reader to look up this article. Also an extension of this result for all elements of GS k (X), beyond those of Mk (X) may need vector measures of finite semi-variation, and perhaps sacrificing uniqueness but utilizing the ideas of our earlier work of this chapter. Because of the geometric properties of the chains and the intricate nature of their structure and supports one may need to make a serious effort on this as a research problem. In the following section we discuss briefly the second important class of chains called the flat chains, contained in the sharp class, since k · k♯ ≤ k · k♭ . Consequently they will have integral representations, similar to (19). Their supports are more intricate than those of sharp chains, corresponding to the classical abstract (M )-spaces (termed (AM)-class) which will now be discussed below.

8.5 Some Applications of Vector Measure Integration, Part II In his detailed analysis of “Chains and Additive Set functions” Whitney ((1957), Chapter XI, p. 327) presents an open problem as:“Let A be a flat chain of fiR nite mass. Then it is sharp and hence y(A) = Rn Dy (p)dµA (p), for all sharp cochains y. [In our case it is extended as Theorem 4.15 above.] Does this representation also hold for flat y?]” Since the supports of these chains can be of the first category (or meager) sets, one may anticipate a negative solution in general. In fact F.J. Almgren gave a counterexample to such an expectation (he told this author in 1984 [at the IAS in Princeton] of this fact). The

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question as interpreted by the author in 1973 was to find general conditions on the flat chains for which some suitable refined integral representation may hold. Since flat chains spaces correspond to Kakutani’s analysis of (AM)-type class, a result somewhat on the lines of representing L∞ (µ)-spaces (relative to finitely additive measures), can be expected and one has to work with Hausdorff measures to incorporate the special fact that metric spaces are supporting these chains. We describe briefly what may be expected, based on an unpublished thesis by Noltie (1975) which should be of interest for further study, as there seems to be very little available on the representations of the type studied here, on such an important problem. Another consequence of this work is an application to some (isotopic) random currents, a topic which was introduced by Itˆ o (1956) and analyzed further by Wong and Zakai (1989). These two aspects are the next and last key applications to be considered in this chapter. We first explain the problem and a strategy of attack. As seen already, a flat chain is always in the sharp class. So the vector integral for a k-cochain y acting on a flat chain A has the representation on Rn as: Z y(A) = hDy (p), dµA (p)i (1) Rn

and since µA (and consequently Dy (·)) may be supported on arbitrarily small sets of α−Hausdorff k-dimensional measure, denoted Hkα (·), 0 < α ≤ 1 and of zero Lebesgue or Hk1 measures, it will be desirable to convert (1) into some Hkα (·)-integral. But one should note immediately that Hkα (·) is a metric outer measure and by a fundamental theorem due to Carath´eodory, it is outer regular relative to the class of open Gδ - sets. However it is also known that each Borel set (of Rn ) is Hkα (·)-measurable (cf., e.g., the text book by the author, Rao (2004), pp.94-96). Moreover integrals of scalar (or vector) functions relative to such outer measures are well-defined and are subadditive. But are additive on (Borel) measurable functions. (See, again Rao (2004), Sections 4.1 and 4.2.) These facts will be used to extend (1) since µA is Hkα (·)˜ of Rn , continuous. Now restricting Hkα to the δ-ring of bounded Borel sets B

dµA α using a (hopefully legitimate) one may want to replace dµA by dH α · dHk k Radon-Nikod´ ym derivative. This may be possible since Rn , being finite dimensional, has the well-known Radon-Nikod´ ym property. But unfortunately Hkα (·) does not satisfy the σ-finiteness condition, and so a further finer analysis is needed. A basic representation given in Whitney ((1957), p.261) for the ksimplex σ and k-cochain y, determined in the subspace generated by σ, is of the form Z y(σ) = Dy (p, ασ ) · ασ χσ dHk1 (p) (2) Rn

where ασ is the k-vector corresponding to the oriented k-dimensional subspace of X(= Rn here ) and χσ is the indicator function of σ, Hk1 (·) being the Hausdorff k-measure (for “α = 1”). It is a Carath´eodory regular (metric)

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outer measure for all α > 0, as noted above, with Dy (·, ασ ) as Borel measurable so that the integral is well-defined. This special case of σ should be generalized for A, a flat k-chain. Now a flat k-chain y, still in the finite dimensional case, should be extended if Rn is replaced by a Banach space X. This may be done as in the case of sharp chains of the last section, with a suitable identification of the integrand of (2). Here one turns to a certain (welldefined) Radon-Nikod´ ym derivative of a (general) vector measure µA relative to the Hausdorff measure Hkα on B(X). Now we may use the (nontrivial) facts that the Hausdorff α-dimensional measure H α in Rn , has the property (for α = n) that bounded Borel sets in Rn have finite Hausdorff measure H n as well as Lebesgue measure. [Note that λn (H n (·) = kn λn (·), kn > 0 is a constant depending only on n, λn being the Lebesgue measure in Rn (see Rogers ((1970),p.54) for a proof). But for 0 < α < n, H α (B) > 0 while λn (B) = 0 can happen!] Thus Hausdorff α-dimensional (0 < α < n) measure is particularly effective in the geometric integration which is considered here. In fact it is desirable to identify the quantity that is multiplying Dy in (2) with a (vector) measure µσ that is Hk1 (·)-continuous. Using the finite dimensionality of the spaces, µσ should be “differentiable” relative to Hk1 (·) which is not σ-finite, and may not even be a decomposable measure. Fortunately it is localizable and this is the best possible condition for the [Radon-Nikod´ ym] differentiability of µσ [or µA ] relative to Hkα . [The ‘localizability’ concept is recalled below.] We shall now outline the details of implementation of the procedure, after recalling the necessary concepts. Definition 8.5.1 (a) In Rn , if Pk (Rn ) stands for the class of polyhedral kchains, let Ek (Rn ) be the completion of its linear span under the mass norm. Elements of En (Rn ) are termed elementary chains so that if Fk (Rn ) is the set of flat k-chains, then one will have the inclusions Ek (Rn ) ⊂ Fk (Rn ) ⊂ S k (Rn ), the last one being sharp k-chains. (b) Let Nk (Rn ) be the subclass of S k (Rn ) that are absolutely continuous relative to the Hausdorff measure Hk1 (·) which thus will have Radon-Nikod´ ym derivations. These are called normal k-chains, (hence may be identified with the Lebesgue space L1 (Rn , B(Rn ), Hk1 ), to be shown). To appreciate the Radon-Nikod´ ym property of a (finite dimensional vector) measure relative to Hk1 (·), we also recall the concept of localizability. Definition 8.5.2 Let (S, S, β) be a measure space with the finite subset property, i.e., β(E) > 0, E ∈ S implies the existence of E0 ⊂ E, E0 ∈ S, 0 < β(E0 ) < ∞ and S is the completion for β of the ring of measurable sets of finite β-measure. Then (i) β is called localizable if every (not necessarily countable) collection S1 ⊂ S has a supremum in the sense that there exists E0 ∈ S such that (a) for each E ∈ S1 , β(E − E0 ) = 0 and (b) if E˜ ∈ S satisfies (a) then β(E0 − E) = 0. (ii) β is strictly localizable or has the direct sum property, if there is a collection {Ej , j ∈ I} ⊂ S with 0 < β(Ej ) < ∞, and I not necessarily countable, such that (c) the set S − ∪j∈I Ej ⊂ N, β(N ) = 0 and (d) if

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D ∈ S, 0 < β(D) < ∞, then the collection {j ∈ I : β(Ej ∩ D) > 0} is at most countable. The second part of the above definition is a generalization of sigmafiniteness, and in 1978, D. H. Fremlin constructed an example to show that there are measure spaces (S, S, β), localizable but do not have the direct sum property. Also the finite subset property in the above definition is not restrictive. (For a discussion of this concept and references, see the author’s text book [Rao (2004),p.84].) We discuss the usefulness of this concept in the context of Hausdorff measures in Rn which is of immediate interest. Some relevant facts for use here are that a general Hausdorff (outer) measure Hk on a metric space is outer regular relative to the class of open Gδ -sets and is a metric outer measure and hence every Borel set is Hk -measurable. [For details again see e.g., the author’s book (2004), pages 94-98).] The standard Carath´eodory process for an outer measure β shows that the class of β-measurable sets forms a σ-algebra (β being σ-subadditive) on which β is σ-additive. Consequently the Hausdorff (outer) measure Hkα restricted to the class of its (Carath´eodory) measurable sets, which is a σ-algebra, is a measure and so the integrals relative to Hkα -are well-defined, and moreover (although this restriction need not give a decomposable measure), it is (and it can be shown which is actually detailed in Federer (1969) differently using Daniell’s approach) localizable. All of this is needed and will be employed below. With the notation of Definition 1 above, we shall obtain a special representation, of flat k-cochains on Fk (Rn ), that shows how a different approach is needed compared to the sharp case. Consider the set L = En (Rn ), the space of n-chains and for 1 < k < n let L = Ek (Rn ) × Ek+1 (Rn ), the product space where Ek (Rn ) was given in Definition 1(a) above. Thus for k = n, the second factor drops out. Consider the null “ideal” N = {(B, C) ∈ L : B + ∂C = 0} of L (∂ just denoting the boundary operator), and if A = (A1 , A2 ) ∈ L, one defines a semi-norm on the quotient space L/N, with equivalence classes (mod N): kA/Nk = inf{kB1 k + kB2 k : (B1 , B2 ) ∈ A( , mod N)}. We now identify the space Fk (Rn ) with L/N isometrically, and then obtain the first integral representation of flat chains and extend it. Actually this is a consequence of an interesting result of Whitney’s (cf.Whitney (1957),Theorem VIII.5A on p.246) implying that these spaces can be identified if λ : Fk (Rn ) → L/N, λ(A) = λ(B + ∂c) = (B, C)/N with A = B + ∂C and kλ(A)k = inf{kB1 k + kB2 k : A = B1 + B2 } = kAk♭ . Also the boundary ∂A of an r˜ of an r- cochain X ˜ is an chain A is an (r − 1)-chain and the coboundary dX (r + 1)-cochain (dX)(A) = X(∂A) (so that ∂∂A = 0, ddX = 0). We may now obtain the first representation using the above notation: Theorem 8.5.3 Let A be a flat k-chain not necessarily of finite mass, and y a flat k-cochain. Then

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y(A) =

Z

Rn

hDy , dµA1 i +

Z

Rn

hdDy , dµA2 i

(3)

for the decomposition of A as λ(A) = (A1 , A2 ) and µAi are the vector measures determined by Ai , such that (with the above notation) kAk♭ = inf{kµA1 k + kµA2 k : (A1 , A2 ) ∈ λ(A)}. If y is a coboundary, then (3) reduces to Z hDy , dµA i. y(A) =

(4)

(5)

Rn

Proof. We sketch the detail to give an idea of the extension. Consider A1m ∈ Pk (Rn ), A2m ∈ Pk+1 (Rn ) such that kA1n − A1 k → 0 and kA2n − A2 k → 0 where A = A1 + ∂A2 , A1 ∈ Ek (Rn ), A2 ∈ Ek+1 (Rn ). Then A1m + ∂A2m ∈ Pk (Rn ), and since ∂ is a linear contraction in k · k♭ , kA − Am k♭ ≤ kA1 − A1m k♭ + k∂(A2 − A2m )k♭ ≤ kA1 − A1m k♭ + kA2 − A2m k♭ ≤ kA1 − A1m k + kA2 − A2m k → 0 as m → ∞. Next from Whitney’s work ((1957), Theorem RIX.5.A) it follows that if B ∈ Pk (Rn ), y ∈ Fk (Rn )⋆ , the representation y(B) = Rn hDy , dµB i is valid where µB is a vector measure. So Z Z hd′ Dy , dµA2 i|, hDy , dµA1 i − |y(A) − Rn

Rn

d′ being the adjoint of ∂ (to avoid confusion with derivative),

≤ |y(A − Am )| + Z + |y(∂A2m ) −

|y(A′m )

Rn

−

Z

Rn

hDy , dµA1 i|

hd′ Dy , dµA2 i|

≤ kyk♭ kA − Am k♭ Z Z Z +| hDy , dµA1m i − hDd′ y , dµA2m i hDy , dµA1 i| + | n Rn Rn Z R hDd′ y , dµA2 i|, − Rn

on substituting the integrals for y(Ajm ), j = 1, 2,

≤ kyk♭ kA − Am k♭ + kDy k0 kµA1m − µA1 k + kDd′y k0 kµA2m − µA2 k

≤ kyk♭[kA − Am k♭ + kA1m − A1 k + kA2m − A2 k].

(6)

Here we have used kyk♭ = max(kyk, kdyk), y ◦ ∂ = d′ y, Dd′ y = d′ Dy , as well as the theorem of the preceding section and the classical work of Whitney’s

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(1957, p.273). The right side of (6) goes to 0 as m → ∞. The assertions (4) and (5) are immediate now. 2 This covers only a portion of flat chains. There are the more general classes– normal, Lipschitz and other chains which act on k-cochains on Fk (Rn ), as the classical characterization of the adjoint of the abstract M -space, i.e., (AM )⋆ space (of functional analysis) for whose member characterizations give only finitely additive vector measures may be used and then decomposing them into σ-additive and purely finitely additive classes with vector Yosida-Hewitt decomposition. We now indicate these classes which lead to ‘currents’ and their important and useful random counterparts, thereby showing interesting and deep new applications of this analysis. If σ k is a k-simplex and f : σ k → Rn is a Lipschitz function (of order 1) then f (σ k ) defines a flat chain, called a simple Lipschitz chain. If Lk (Rn ) is the set of simple Lipschitz chains then it can be verified (nontrivially) that Lk (Rn ) ⊂ Nk (Rn ) properly so that each simple Lipschitz k-chain is a normal k-chain (but not conversely). This class of Lipschitz k-chains is sufficiently interesting and nontrivial to have the following representation. Namely if A is a simple Lipschitz chain in Rn so that A = f (σ) for a Lip(1) function f and a k-simplex σ, then the Jacobian Jf of f exists at almost all interior points σ 0 of σ, (relative to the Lebesgue measure), and then by a deep result of geometric integration theory, f maps sets of null Lebesgue measure in σ to sets of null Hausdorff measure Hk (cf. Federer (1969), Theorem 3.2.3 on p.243). But then there exists a vector measure µa of finite variation on dµA (the Radon-Nikod´ ym density) exwhose support Hk is localizable so that dH k ists as discussed above. [Since Hk in general need not have the direct some property, the general R-N theorem is needed here!] Then for A = f (σ), a simple Lipschitz chain and the vector measure µA with values in Rn , one J (p) dµA dµA can compute dH and evaluate it as dH (p) = kJff(p)k0 a.e., [Hk ], k · k0 bek k ing the mass (or norm) for multi-vectors so that each has unit mass a.e., and Jf (p) is the Jacobian of f at p. This can be verified by the R-N calculus (cf., e.g., Noltie (1975)). Because of this (differential) structure, Lipschitz chains are of special interest. Two such chains f (σ k ), g(σ k ) are equiv0 alent at p ∈ (σ k ) (interior of the k-simplex), if there is a neighborhood of 0 p in (σ k ) on which f, g agree. Then the class Lk (Rn ) has an equivalence “∼” defined (by the group property of Rn this equivalence does not depend on a particular ‘p’), and let GLk (Rn )(= Lk (Rn )/ ∼) be the resulting genVk k G eral class. Then one has the mapping y : Rn → [ (Rn ) ] , the indexing set being G = (GLk (Rn )), and is termed a Lipschitz differential vector field. If for any simple A ∈ Lk (Rn ), the associated differential field y is defined Vk n ⋆ for each direction α, A 7→ y(p, α, G(A)(P )) ∈ (R )) , is a Borel function, then A is a measurable simple k-chain. We thus can obtain the following representation: Theorem 8.5.4 Each A ∈ Lk (Rn ) can be represented as:

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y(A) =

Z

Rn

hWy , dµA i,

(7)

for any flat k-cochain y in Rn , for a Lip-differential vector field Wy = Wy (p, α, ˜ G ◦ σ) = Dy (p, α ˜ ) a.e. [Hk ] and any k-simplex σ, oriented in the α ˜ -direction. Outline of proof. Any flat cochain has finite comass and so y here can be extended to a cochain y˜ onto Nk (Rn ) without changing the (mass or) norm. Since Nk (Rn ) can be identified with L1Vk (Rn ) (Rn , B, Hn ) where B is the Borel σVk n ⋆ algebra of Rn , y˜ can be represented by a vector measure my˜ : B → (R ) which is σ-additive with finite variation from the standard results in functional analysis. [Nk (Rn )]⋆ can be represented by my˜ so that [since Nk (Rn ) itself may be identified with {dµA /dHk , A ∈ Lk (Rn )} and extended to Nk (Rn )] we have: Z dµA · dmy˜, A ∈ Nk (Rn ). (8) y˜(A) = dHk Rn dµA Here we use the fact that dH is well-defined since the Banach spaces in which k µA is valued, is finite dimensional and Hk is localizable. But my˜ also has finite variation |my˜|. Consequently (8) can be expressed as: Z dµA dmy˜ · · d|my˜|, ∀A ∈ Lk (Rn ). (9) y(A) = dH d|m | n k y ˜ R Pn0 ai fi (σ), and then since the |my˜|(·) is also This may be extended if A = i=1 absolutely continuous relative to Hk on suppfi (σ) = Bi , one has:

y(A) =

n0 X i=1

ai

Z

supp(fi (σ))

dµA dmy˜ d|my˜| · |Bi dµfi (σ) . · dHk d|my˜| dHk

(10)

If the integrand in (10) is set equal to Wy˜(p, α, G(fi (σ))(p)), the whole expression can be written as: Z Wy dµA . (11) y(A) = Rn

Here Wy is well-defined since if f (σ) = g(σ) on some neighborhood of f (p) in the d|m | d|m | set f (σi ) then dHky˜ |suppf (σ) = dHky˜ |suppg(σ) by our equivalence of Lipschitz chains noted earlier. Finally if we set Dy (p, α ˜ ) = Wy (p, α ˜ , G(σ0 (p)), a.e.[Hk ], where f (σ) is a simplex σ0 , then (11) gives Z Dy (p)˜ α σ0 dµσ0 . y(σ0 ) = σ0

under a suitable identification indicated above.

2

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Remark 8.4 If Hk is not localizable, then Dy (·) will be a “quasi-function” in the sense of E.J. McShane as detailed in (Rao (2004), p.234). The full discussion can be found in Noltie (1975). Using the preceding representation, Noltie (1975) was able to obtain a solution of Whitney’s problem stated at the start of this section for the flat k-chains in the following form with the analysis of Lipschitz chains, which is the completion of polyhedral k-chains in flat norm, denoted Fk (Rn )(⊂ S k (Rn )) as in our Definition 1 (a) above. Here we use the fact that a vector measure representing y ∈ Fk (Rn )⋆ for A ∈ Fk (Rn ) will only be finitely additive which can be decomposed in the Yosida-Hewitt (1952) form. The corresponding vector integration is available in standard treatments, since we are dealing only with finite dimensional vector valued additive set functions, (see, e.g., in Dinculeanu (1967) or Uhl and Diestel (1977)). It was noted for (8) that Nk (Rn ) may be isometrically identified with L1Vk (Rn ) (Rn , B, Hk ) where Hk (·) is the k-dimensional Hausdorff measure, and B is the Borel σ-algebra. The space Fk (Rn )⋆ denotes the flat k-cochains in Rn and then Hk (Rn )⋆ can be isometrically identified as a subspace of [L1Vk (Rn ) (Rn , B, Hk )]⋆ , the adjoint space, which Vk n k (Rn , B, Hk )] in the mass norm, because Hk is localizis equivalent to L∞ (R ) able here. If Hk is replaced by the Lebesgue measure in Rn , Whitney’s work gives this correspondence. In the general case considered now, one has to use the fact that L∞ and Fk (Rn )⋆ admit a strong lifting operation in the sense of Ionescu Tulcea and Ionescu Tulcea ((1969) Chapter VIII). This is essential since we need to select continuously elements from equivalence classes to work with k-cochains and such a lifting does not exist in Lp -spaces for 1 ≤ p < ∞ if the measure has a nonatomic part (cf. the Tulcea book (1969), Sec.IV.4). The localizability property of the underlying measure, such as Hk here, is also essential. Using these properties and the above work the general representation of the flat k-cochains in Rn , due to Noltie (1975), can be presented as follows: Theorem 8.5.5 For each y ∈ Fk (Rn )⋆ , a flat k-cochain and A ∈ Fk (Rn ) a flat k-chain of finite mass, there exists a vector measure µA and a purely finitely V additive (vector) set function µ ˜A , both valued in k (Rn ), such that µA has finite (Vitali) variation, where µ ˜A is bounded, and there is a sharp form Dy regarded Vk n ⋆ (Rn , B, Hk ), in terms of which we have a general as an element of L∞ (R ) representation Z Z hDy , d˜ µA i, A ∈ Hk (Rn ). (12) hDy , dµA i + y(A) = Rn

Rn

However, the correspondence of y and µA + µ ˜A for each A is not necessarily unique. Remark 8.5 Typically {DyA , A ∈ Hk (Rn )} is a quasi-function obtained for the measure Hk , but determines an Hk (·)-unique function Dy (a vector field) by the localizability of Hk (·) in this context and gives (12).

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Brief sketch of ideas of proof: First consider Hk (Rn ), identifying it with its Vk n ⋆ (Rn , B, Hk ), with the adjoint of the second adjoint, using (Fk (Rn )⋆ ) ֒→ L∞ (R ) Vk n ⋆ (Rn , B, Hk ))⋆ , each of whose elements is identified as latter, i.e., with (L∞ (R ) a finitely additive vector measure of finite variation in our context. To separate those that correspond with the k-chains Dy , consider a linear form on these latter elements as ℓA (Dy ) = y(A), A ∈ Hk (Rn ). Then ℓA (·) is continuous since |ℓA (Dy )| = |y(A)| ≤ kyk kAk = kDy k0 kAk. Hence ℓA can be extended (by the Hahn-Banach theorem) to all of Vk n ⋆ (Rn , B, Hk ) with the same bound. Using the fact that the adjoint L∞ (R ) of the latter is the space of bounded additive vector measures, each ℓA will be represented by µ ˜A , an additive vector measure, in terms of which we have Z ℓA (Dy ) = y(A) = Dy d˜ µA , y ∈ Hk (Rn )⋆ . (13) Rn

Vk n (R ) is finite dimensional, idenWe now analyze (decompose) µ ˜A . Since tifiable as Rm for some 1 ≤ m < ∞, V let πr be the coordinate projeck tion onto Rr , 1 ≤ r ≤ m from Rm (∼ (Rn )),and the scalar measures = πr ◦ µ ˜A can then be decomposed (Yosida-Hewitt) into πr ◦ µ ˜A = µ ˜cAr + µ ˜fAr , the completely additive and purely finitely additive parts respectively. Now V for the basis of k (Rn )(∼ ˜cA = = Rm ), denoted {er , r = 1, . . . , m} let µ Pm Pm f f c µAr , er ), µ µAr , er ), using the inner product notation. This ˜A = r=1 (˜ r=1 (˜ gives µ ˜A = µ ˜cA + µ ˜fA . [The decomposition is not unique but the latter property is not used here.] It is to be verified that µ ˜ cA = µA and µ ˜fA = µ ˜A of (12). Now one has to choose consistently, a member of Dy which is given by a sharp cochain y, using the earlier result on the representation of sharp cochains. Here the existence of a (strong) lifting is needed, since S k (Rn ) ⊃ Hk (Rn ) Vk n (Rn , B, Hk ) gives a measure µ so that S k (Rn )⋆ ⊂ Hk (Rn )⋆ ⊂ L∞ ¯A such (R ) that Z Dy (d¯ µA − dµA ) = 0, ∀ sharp y. (14) Rn

Using a detailed argument one shows that πr ◦ (µA − µ ¯A ) and then µ ˜ A − µA = P m µA − µA ) are purely finitely additive. Now with some shuffling the r=1 πr (˜ terms one produces µA and µ ˜A of the desired type that is asserted in (12). For the completion of the argument and to use Yosida-Hewitt decomposition, it is crucial that one needs the existence of a Borel linear lifting to assert that for each flat k-cochain y in Rn and the corresponding flat k-form Dy there n

is a bounded Borel function f : Rn → (ℓ∞ )m , where m = k (the binomial coefficient), such that if Lim (·) is the Banach limit on ℓ∞ , a linear functional (cf. Dunford-Schwartz (1958),p.73) satisfying Lim fy = Dy , for a bounded Borel

8.5 Some Applications of Vector Measure Integration,

Part II

429

k- form. This involves some nontrivial analysis, which then will give the desired relation: Z y(A) = Lim (fy ) · dµA (15) Rn

for all flat chains A of finite mass. In particular if y is a flat k-cochain for which Dy is continuous then (15) becomes Z y(A) = hDy , dµA i Rn

as a bilinear vector integral as a final result from which (12) follows. 2 We now specialize the flat k-cochains to the differential forms and then outline an important application with generalized (Schwartz) distributions of higher order that leads to the corresponding representation (and discussion) of random fields complementing the earlier analysis of Section 2.2. Note that the preceding development and representation of flat (co-)chains in (Rn ) has not used the differential structure of the ambient space. This can be considered by restricting ourselves to the special case of ‘differential forms’ and a remarkable specialized theory follows. Vk n (R ) Consider the natural basis e1 , . . . , en of Rn and the k-covector ak ∈ which can be expressed as: X X ak = ai1 ik ei1 ∧ · · · ∧ eik = ai ei (say) (16) 1≤i1 <···
i

the summation n over all ordered distinct k-sets out of the n basis vectors so that there are k terms here, the binomial coefficient. In addition to the exterior V product, denoted ‘∧’ (a wedge), used above for the exterior algebra k (Rn ) one needs an interior product also, denoted by ‘∨’and defined as follows. [This was already briefly noted in the last section.] If y r is a covector (or vector in the adjoint space of V = Rn ) and xs is a vector s (x ∈ Rs ), then y r ∧ xs is a vector z r−s in Rr−s space if r ≥ s, so that if a new multiplication is introduced symbolically as ‘∨’ to get another r − s vector z r−s to satisfy (xs ∨z r−s ) acting on y r we must have y r ·(xs ∨z r−s ) = (y r ∧xs )· z r−s for all z. (This is called the interior product.) Since y r ∧ xs is a vector in Rk−s , by the Riesz theorem for the ‘dot’ (or inner) product, there is a unique vector so that the right side i.e., (y r ∧ xs ) · V z r−s is uniquely Vr−s Vs isn well-defined, r n (R⋆ )n . Stated R for a y ∈ R and z ∈ defined for any given x ∈ r differently, a product ‘∨’ is well-defined satisfying for y (covector) and xs (vector) to get for all z using the dot products, the following relations (similarly for r ≤ s): z s−r · (y r ∧ xs ) = (z s−r ∨ y r ) · xs , if r ≤ s, (17) (y r ∧ xs ) · ur−s = y r · (xs ∨ ur−s ), if r ≥ s.

This and some properties of it are found in Whitney (1957, p.42).

(18)

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8 Vector Measures and Integrals

We may give the above notion a different point of view as follows leading quickly to the concept of currents of interest here, and this uses interior products. Let U be an open set in Rn (can be the whole space) and L(U ) be the space of infinitely differentiable f : U → Rn with compact supports. Next let (m) fk ∈ L(U ), k ≥ 1 and say fk → f if the sequence {fk , k ≥ 1} vanishes (m) outside a compact set and fk converges to f uniformly as k → ∞, for all Pn m (m) fk m ≥ 1 where fk = ∂xm∂1 ···∂x mn , m = i=1 mi , so that L(U ) is given the n 1 Schwartz topology (as in Chapter 2), to make it a locally convex linear topological space. Equivalently it can be given the topology defined by the family of semi-norms: kϕkK,m = sup{k(Dj ϕ)(x)k : 0 ≤ j ≤ m, x ∈ K ⊂ U, compact} where Dj ϕ is the j th order partial derivative, j = (j1 , . . . , jm ) and jℓ ≤ m, ℓ = 1, . . . , m, as the compact sets K ⊂ U and integers m ≥ 1 vary. It is known that the adjoint space L∗ (U ) is the vector space of continuous linear mappings called the space of distributions. These are special (sub)classes of flat (co-)chains, and the earlier work on these geometric (based) functionals is also essential for the important ‘forms’ connecting various aspects of analysis. These include L. Schwartz’s distributions. The analysis is somewhat involved, and we present a brief but useful sketch to illustrate the prospects and problems. n Vp If ne = (e1 , . . . , en ) is the standard orthonormal basis of R , and ap ∈ (R ), then it can be expressed, with index i = {i1 < i2 < · · · ip } as: X ap = (19) ai (ei1 ∧ · · · ∧ eip ) = ai · ei (say). i

Thus if x1 , . . . , xp ∈ Rn , and ap as above, then with x = (x1 , . . . , xp ), X ap · x = ai det(hej , xk i, j, k = 1, . . . p)

(20)

i

n where hej , xk i is the ninner product of R and ai depends on i which carries the order relations of p distinct elements giving the pth order determinant. Note Vp n Vk n (R ) isVa given element, (R ). If λ ∈ that (20) gives an inner product in n then the mapping ν 7→ λ ∧ ν is linear in ν ∈ ∧n−p (Rn ) into (Rn )(≃ Rn ) so that λ∧ν = fλ (ν)·e, where fλ (·)is real linear. Now by the Riesz representation, ˜ ∈ Vn−p (Rn ) such that λ ∧ ν = (λ, ˜ ν)σ. The mapping λ 7→ λ ˜ there is a unique λ ˜ is denoted by ‘⋆’called Hodge Vn−pstarn operator so λ = ⋆λ. The operator Vp the (R ), and has important properties. For (Rn ) into “⋆” is linear from instance, if σ H = σi1 ∧ · · · ∧ σik with the basis vectors of Rn , so that the σ H Vk n also form a basis of (R ) and (σ H0 , σ H1 ) = det(hσ ij , σ ij′ i, j, j ′ = 1, . . . , k). Now if H0 6= H1 , this determinant vanishes but not if H0 = H1 . In fact it is ±δ H0 ,H1 where δ H0 ,H1 is the Kronecker delta. Also one finds with H = {1, . . . , p},

8.5 Some Applications of Vector Measure Integration,

Part II

⋆λ = (σ k , σ k )σ k (= ⋆σ H ), k = {p + 1, . . . , n}.

431

(21)

Since the determinant changes sign as the rows or columns are interchanged one finds that ⋆σ k = (−1)p(n−p) (σ H , σ H )σ H , and moreover ⋆(⋆σ H ) = (−1)p(n−p) (σ, σ)σ H , σ is defined above, 1

= (−1)p(n−p)+ 2 (n−t) σ H , |t| < n,

(22)

and is called the signature. [The full details may be found in Flanders ((1989), p.15) or Bishop and Goldberg (1980, Sec 2.22).] The signature is the number of +1’s minus the number of −1’s (called index ) in the determinant (σ, σ). Using the Hodge star operator we can give an alternative definition of the interior product for a pair of p-covectors [cf. (17), (18)] as follows. Let yp , zp be a pair of covectors, then we set (yp ∨ zp ) = ⋆(yp ∧ (⋆zp ), 1 ≤ p ≤ n

(23)

and more generally for 1 ≤ p ≤ q ≤ n, (yp ∨ zq ) = (−1)(q−p)(n−q) ⋆ (yp ∧ (⋆zq )). Consider a p-covector ϕp (·) which is a differential form, given by X ϕp (t) = ϕi (t)ei , t ∈ Rn ,

(24)

(25)

i

where ϕi (·) = ϕi1 ,...,ip (·) for 1 ≤ i1 < . . . < ip ≤ n is a measurable scalar function. In the standard work the p-covector ϕp (·) is taken as a C ∞ function and then the exterior derivative of ϕp is given by the (p + 1)form X ∂ϕi (t) ej ∧ ei (26) dϕp (t) = ∂tj (j,i)

and the codifferential is the (p − 1)-form given by the ⋆-operation δϕp = (−1)np+n+1 ⋆ (d ⋆ ϕ).

(27)

Hereafter for all p-covectors ϕp of (25), the coefficients ϕi (·) are taken to be C ∞ -functions. If the coefficient functions ϕi (·) in (25) are taken as C r - (instead of C ∞ -) functions, then the form is termed a C r -class p-form. Let U ⊂ Rm be an open set and V be a finite dimensional (real) vector space, say of dimension m and E(U, V ) be the vector space of smooth (or C ∞ -) mappings f : U → V with the Schwartz topology (as in C ∞ (R)). Thus the topology J in E(U, V )is given by saying fk → f as k → ∞, if all {fk , k ≥ 1} vanish outside of a compact set and on it they and all their (partial) derivatives (Dj fk ) converge uniformly to those of f , which thus is in E(U, V ). This also makes the latter a complete locally convex topological space. Now consider E(U, V ) as the space of differential forms. Let D(U, V ) ⊂ E(U, V ) consisting of elements, each of which vanishes outside compact sets so that

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8 Vector Measures and Integrals

the inclusion is a continuous embedding. Then the adjoint (or dual) spaces satisfy E′ (U, V ) ⊂ D′ (U, V ). [These are endowed with the weak*-topology.] The elements of D′ (U, V ) are called m-dimensional currents on U , in the sense of de Rham which when m = 0 is the Schwartz space of distributions that were studied in Chapters 2 and 4. Now considering Schwartz distributions as scalar functionals, the many applications to (partial) differential equations and other applications prompted Itˆ o (1954) and independently Gel’fand (1955) to replace the range (scalar) space by the more inclusive space of (scalar) random variables and thus opening up the study of a new class called generalized random processes. These are linear mappings on Schwartz (type) spaces into the space of (scalar) random variables, or to subspaces of L0 (P ). Thus we have {F (ϕ), ϕ ∈ E(U, R)}, with E(U, R) as an indexing set so that this again becomes a new probability area since for each k ≥ 1, the set {(F (ϕ1 ), . . . , F (ϕk )), k ≥ 1} defines finite dimensional distributions which satisfy the Kolmogorov consistency conditions. Hence the class {F (ϕ), ϕ ∈ E(U, R)} is a well-defined random field which may be Gaussian, Poisson or of other classes. A substantial analysis appears if L0 (P ) is replaced by its subspace Lp (P ), particularly if p = 2 (the second order fields) to use the Hilbert space methods, (cf. Gel’fand-Vilenkin (1964)). It is thus natural to consider the analogous study if the (Schwartz) distributions are replaced by the more inclusive currents (of de Rham), and this has been done again by K. Itˆ o (1956) which was further studied and elaborated by E. Wong and M. Zakai (1987, 1989), motivated also by some interesting engineering applications. These are called random currents and they are discussed here as an application of the preceding vector and random measure theory which can be used to analyze many new applications. Definition 8.5.6 If U ⊂ Rn is an open set, let D(U, Rn ) be the space of smooth differential forms on U and F : D(U, Rn ) → H be an H (or Hilbert) valued m-dimensional current (in the sense defined above i.e., that of de Rham). When [H = L0 (P ) or] H = L20 (P ), the space of centered square integrable random variables, then F is termed a random current if ϕk → 0 in the topology of D(U, Rn ) implies F (ϕk ) → 0 in [probability or] the metric of H. If τh (ϕ)(x) = ϕ(x + h), when U = Rm , so that τh is a translation operator (for h ∈ Rm ) and the inner product of F (τh ϕ) is invariant under the action of τh i.e., (F (τh ϕ1 ), F (τh ϕ2 ))H = (F (ϕ1 ), F (ϕ2 )) = B(ϕ1 , ϕ2 ), is also termed covariance bilinear form, then the random current F is called homogeneous or stationary. It is possible to characterize the covariance bilinear form of a homogeneous random current. For this we need to connect it with appropriate probabilistic formulation using the structure of currents. Since B(·, ·) is positive definite and measurable, or equivalently in the present context, a continuous bilinear functional, it admits an integral representation as in the case of results on the stationary random Schwartz distributions (see Chapter 2). The Itˆ oGel’fand theory shows that the covariance bilinear form B(·, ·) of a station-

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433

ary random distribution obeys the hypothesis of the Bochner-Schwartz representation of a positive definite continuous bilinear form since the space of complex valued Schwartz distributions is “nuclear” and so can be expressed as: Z (ϕˆ1 , ϕ¯ˆ2 )(y)dµ(y), ϕi ∈ D(Rn ), (28) B(ϕ1 , ϕ2 ) = Rn

where ϕˆi is the Fourier transform of ϕi on Rn , (over“bar” denoting complex conjugation) and µ : B(Rn ) → R+ is a tempered measure in the sense R that A (1 + |v|2 )−k dµ(v) < ∞ for some integer k ≥ 0, and any Borel set A ⊂ Rn . Here the “nuclearity” condition of the infinite dimensional space D(Rn ), which is not locally compact, is essential for the desired extension of this representation to homogeneous random currents of order ‘p’ using de Rham’s theory (0 ≤ p ≤ n). With the notation of (16), if ϕ1 , ϕ2 ∈ Dn−k , then the covariance of the k th order random current Uk is given by τh (Uk (ϕi )) = Uk (τh ϕi ), and the covariance bilinear form is defined for homogeneous currents as : (τh Uk (ϕ1 ), τk Uk (ϕ2 )) = (Uk (ϕ1 ), Uk (ϕ2 )). (29)

Note that if Vi (ϕ) = {Uk (ϕei ) : ϕ ∈ D, |i| = k}, then one has the gener n alized k-vector random field in the old notation and thus Vi (ϕ) has k components, each component representing a (generalized) random Schwartz distribution. n But D0 is a Schwartz space and Dk can be isometrically identified as the k -tensor product of D0 , and it is again a nuclear space to which the

Bochner-Schwartz theorem is applicable. [The nuclearity condition is a critical property for this non-locally compact space in order to invoke the integral representation of the homogeneous covariance bilinear mapping (into C) relative to a tempered measure.] The preceding discussion may be summarized in the following representation, a form of which was originally obtained by Itˆ o (1956). Theorem 8.5.7 If Uk : Dn−k → L20 (P ) is a homogeneous random current with its covariance bilinear functional B(·, ·), then it is representable as : Z ϕˆ1 (x)ϕ¯ˆ2 (x)dµij (x) (30) B(ϕ1 ei , ϕ2 ej ) = Rn

where µij is a tempered measure on B(Rn ), the Borel σ-algebra. Moreover, the random current admits the following integral representation : Z ϕ(x)Z ˆ (31) Uk (ϕ · ei ) = i (dx), ϕ ∈ D Rn

where Zi : B(Rn ) → L20 (P ) is a random measure with orthogonal values (i.e., Zi (A) ⊥ Zi (B) if A ∩ B = ∅), ϕˆ being the Fourier transform, and moreover one has (Zi (A), Zj (C))L20 (P ) = µij (A ∩ C), µij (·) being a tempered measure on B(Rn ).

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Proof. Only the representation (31) needs to be established. For the direct part, namely that if Uk is given by (31), the condition on Zi (·) easily implies that the covariance bilinear functional of Ui (·) verifies (31). Thus the representation (31) is the new and nontrivial part which is obtained as follows. Let H0 = sp{F (f ) : f ∈ D} ⊂ L20 (P ) and denote its completion by H in 2 L0 (P ). If (·, ·)L20 (P ) denotes the inner product of L20 (P ), then (F (ϕ1 ), F (ϕ2 ))L20 (P ) = B(ϕ1 , ϕ2 ),

[(ϕ1 , ϕ2 )B ],

(32)

defines an inner product on the space D so that ϕ 7→ F (ϕ) is an isometry of H0 onto D. Now using (30) one sees that there is an extension of this correspondence between L20 (P ) and the completion of D, denoted as L2 (µij ) where µij is the tempered measure determined by F . This is the key observation. Then (30) and (32) imply, for a unique tempered µ12 the following: Z ¯ˆ (ϕ, ˆ ϕ)(x)dµ (33) (F (ϕ1 ), F (ϕ2 ))L20 (P ) = 12 (x). Rn

Letting sequences ϕ1n → χA and ϕ2n → χB uniformly in the topology of D, (with some additional approximation argument) that for elements of D, the sequence F (ϕ1n ) tends to a limit, denoted by Z(A), and is verified that Z(·) is σ-additive in L2 (P ), so that kZ(A) −

m X i=1

Z(Ai )kL20 (P ) = kχA −

m X i=1

k → 0, χAi kB = kχ∪m i=1 Ai B

as m → ∞ which implies that Z(·) is a random measure with orthogonal values. Here A = ∪m≥1 Am is a disjoint union of Borel sets of Rn . It can then be inferred with a familiar argument that Z F (ϕ) = ϕ(x)dZ(x), ˆ ϕ ∈ Dk . (34) Rn

This is equivalent to (31) and completes our sketch of the representation of the homogeneous random current. 2 It is now natural to specialize the homogeneous random currents and obtain representations from (30) when they are also isotropic. This means: if G is the orthogonal group of transformations on Rn , consider the mapping σg : ϕk → g −1 ◦ ϕk ◦ g, where g ∈ G, so that X g −1 ϕk = ϕi1 ,...,ik g −1 ei1 ∧ . . . ∧ g −1 eik (1 ≤ i1 < . . . < ik ≤ n)

and then let

(σg Uk )(ϕn−k )(x) = Uk (σg−1 ϕn−k )(x) = Uk [gϕn−k (g −1 x)].

(35)

Here Uk is said to be isotropic if the covariance bilinear mapping of σg Uk is invariant for all g ∈ G. A characterization and further decomposition into

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435

“irrotational” and “solenoidal” classes was discussed by Itˆ o (1955) as well as Wong and Zakai (1989) based on certain applications. These will not be included here. On the other hand it is also possible to go further in a general direction without assuming stationarity of the random currents. We indicate this as a last topic of this chapter. Consider the random current {Vi (ϕ), ϕ ∈ D} which forms a vector field belonging to L20 (P ). This may be treated as a second order random field indexed by Dk where i = {1 ≤ i1 < . . . < ik ≤ n}. Note that n

n

it is an k -vector, so the “index set” Dk is k -vector valued. The random field is separable in L20 (P ) since it is indexed by Dk a nuclear space which is separable (it is determined by a countably Hilbert space in the sense of Gel’frand and Vilenkin (1964),p.73) hence separable.) This gives the following representation: Theorem 8.5.8 If {Vi (ϕ), ϕ ∈ D, i = [1 ≤ i1 < . . . < in ≤ n]} is a random field with values in L20 (P ), then it is separable and is representable with orthogonally valued random measure Z and a kernel K(·, ·) as : Z Vi (ϕ) = K(ϕ, u)dZi (u), (36) Rn

and its covariance bilinear form admits a representation as : Z K(ϕ1 , u)K(ϕ2 , u)dµij (u), ϕi ∈ D, (Vi (ϕ1 ), Vj (ϕ2 ))L20 (P ) =

(37)

Rn

for some tempered measure µij on B(Rn ), and K(ϕ1 , ·) is µij -measurable. The proof of this theorem is a simple modification of that detailed in (Chang and Rao (1986), pp.57-58 in Theorem 5 of Sec.7). Following the general formulation it will be called a random current of Karhunen type, which includes the homogeneous class. There are also several (nonhomogeneous) subclasses of this family. The following is one such. Let {Vi (ϕ), ϕ ∈ D, i = [1 ≤ i1 < . . . < in ]} be a second order centered random current (i.e., range is L20 (P )) and let B(·, ·) be its covariance bilinear functional on D×D. It can be called the harmonizable random current, if B(·, ·) is continuous and representable as : Z Z B(ϕ1 , ϕ2 ) = ϕˆ1 (x)ϕ¯ˆ2 (y)d2 µ1,2 (x, y) (38) Rn

Rn

relative to a positive definite tempered bimeasure on the product Borel σalgebra, B(Rn ) × B(Rn ) where ϕˆj is the Fourier transform of ϕj , j = 1, 2 and the integral in (38) is in the sense of Morse-Transue as in Section 2.2. A representation of such random currents extending Theorem 7 above and sharpening (36)-(37) is possible. In fact almost all the results of multidimensional generalized random fields given in (Rao (1969), Secs.3–4) are extendable for the

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8 Vector Measures and Integrals

corresponding currents. The details of this extension are not trivial, but they will be left for future works. Finally it is of interest to observe that on recalling that the differential forms are special cases of flat cochains, the preceding application is an important particular analysis which motivates the corresponding study for the representation theory of not only sharp, but flat (co-) chains with values in Banach spaces that are (at least) linear images of Hilbert spaces. In this effort, the classical (non random) applications presented here and in the preceding sections should be of particular interest. The analysis given in this chapter based on Noltie’s work (1975, 1986) will be important and can clearly be extended, motivated by the applications of probabilistic analysis discussed above. First one has to generalize the finite dimensional analysis of Noltie’s on flat (co-) chains to infinite dimensions as in the sharp chain case. The recent extensions by Ambrosio-Kirchheim (2000) and Adams (2000) do not include our case, but can be of interest in such an extension. Then the theory, based on de Rham’s currents to Hilbert case of the Itˆ o-Gel’fand type should be possible for a similar analysis. We thus leave this aspect for future studies, and add some complements with proof sketches, and related remarks as exercises to conclude this chapter.

8.6 Complements and Exercises 1. Let (S, S) be a measurable space with S as a σ-ring and µ : S → X, a mapping where locally convex (Hausdorff) vector space. If P∞(X, J) is a ∞ µ(∪∞ A ) = µ(A ), ∪ n n=1 n n=1 An , An ∈ S, An disjoint, and the series n=1 converges in the weak topology, induced by the adjoint space of X. Verify that (with the uniform boundedness principle) the range of µ is bounded in the weak (or X′ ) topology of X, and that the σ-additivity hypothesis is not dependent on the topology J making (X, J) locally convex. [A number of properties of such vector measures, as µ above, are discussed by I. Tweddle (1970) where it is observed that the essential condition is the property of its dual X′ separating points of X, and this may be of interest to the reader.] 2. This problem gives conditions under which it is possible to strengthen Theorem 2.8 and obtain essentially unique densities for a class of vector measures. Thus let (Ω, Σ, λi ), i = 1, 2, be a pair of finite measure spaces where Σ is a σalgebra of Ω and let B ⊂ Σ be a fixed σ-subalgebra. Set Xi = L∞ (Ω, Σ, λi ) and Yi = L∞ (Ω, B, λi ) so that Yi ⊂ Xi , i = 1, 2 are closed subspaces of the Banach spaces Xi . By the classical Radon-Nikod´ ym theorem, there exist vector measures νi : Σ → Yi , σ-additive, called conditional measures satisfying Z νi (A)(ω)dλi (ω) = λi (A ∩ B), B ∈ B, A ∈ Σ, i = 1, 2. B

8.6 Complements and Exercises

437

Show by an easy computation that νi : Σ → Yi are vector measures and if λ1 ≪ λ2 (absolute continuity) then ν1 ≺ ν2 holds.RMoreover there is a unique Σ-measurable g : Ω → R+ , such that ν2 (A) = A gdν1 , A ∈ Σ. [The νi have finite variations and the vector integral (of the scalar ‘density’ g) is easily obtained first for simple functions and then generally.] Further, the class of bounded B-measurable scalar functions B(Ω, B), which is a module in the sense that for each ν2 -integrable f and g ∈ B(Ω, B), show that Z Z f dν1 . f gdν1 = g Ω

Ω

[These are easy consequence of the definitions. See, for more details of this type of special study, the author (Rao (1973), Sec.3).] 3. Let (S, S) be a couple where S is a ring of subsets of a set S. An additive function ν : S → X, a Banach space, is s-bounded (or strongly bounded in Rickart’s sense) if for any disjoint sequence {An , n ≥ 1} in S, ν(An ) → 0 as n → ∞. Verify that this concept is equivalent to saying that for any increasing sequence {Bn , n ≥ 1} ⊂ S, the collection {ν(An ), n ≥ 1} is Cauchy in X. Here the word “sequence” can be replaced by “net” and the same conclusion holds. This extends Theorem 1.3.4 and an indirect proof is effective. If ν is a vector measure (i.e., it is σ-additive), then it is always s-bounded. [See Traynor (1973) for extensions and details.] 4. (a) If (S, S) is a measurable space with S as a σ-algebra, let µ : S → X and ν : S → Y be σ-additive into (real or complex) Banach spaces X and Y, so that each is (by Excercise 3) s-bounded. Then there is a (unique) decomposition ν = ν1 + ν2 into σ-additive Y-valued vector measures such that ν1 ≺ µ and ν2 ⊥ µ in that ν1 (A) = 0 for any A ∈ Σ satisfying kµk(A) = 0, called a Lebesgue type decomposition of ν relative to µ. [Here one employs the result of Exercise 3 above that a vector measure is always s-bounded. The proof uses some results of Rickart (1943), where the key concept of s-boundedness was first introduced.] (b) Let (S, S) and µ, ν be as in (a) above. Suppose that X is reflexive and ν ∈ V(Y, λ) where λ is (finite) controlling both measures ν and µ, with V(Y, λ) as in Definition 3.6. Also let the variation |µ|(S) < ∞. Then there is an essentially bounded strongly measurable operator function T (ω) : X → Y such that for all E ∈ S it follows that |µ|(E0 ) = 0 and Z T (ω)d µ + ν2 (E), (ν2 (E) = ν(E ∩ E0 ), E ∈ S), ν(E) = E

where the integral is bilinear Bochner type as in Theorem 3.10. [The difference is that we use (a) for the decomposition of ν and then the fact that reflexive Banach spaces have the Radon-Nikod´ ym property. Then the same argument as in the above noted theorem applies, and it is not much simpler.]

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(c) On a measurable space (S, S) and a Banach space X, let µ : S → X be a vector measure with finite semi-variation and λ as its (finite) controlling measure. If kµk(·) treated as an outer measure, let J be the set of Carath´eodory generated kµk(·)-measurable sets so that it is a complete σ˜ If f : S → R (or C) is measurable for S, so that (by the algebra and S ⊂ S. structure theorem) there is a sequence of simple functions fn (for S) satisfying |fn | ↑ |f | (and fn → f ) pointwise, define (v(x∗ ◦ µ)(·) for variation measure of the scalar (x∗ ◦ µ)) : Z kf kp,µ = sup |f |p dv(x⋆ ◦ µ) : kx⋆ k ≤ 1, x⋆ εX⋆ . S

Then k · kp,µ is a (semi-) norm and LpX (S, S, µ), p ≥ 1 is a Banach space when equivalence classes are identified. [This result was sketched for p = 1 ˜ the space is in Rao (1967,p.122).] If S is replaced by the larger σ-algebra S, ˜ µ) is weakly normed linear and completeness also holds whenever LpX (S, S, sequentially closed or using a classical theorem of Bessaga and Pelczy´ nski ˜ µ). This holds if (1958) if the null convergent sequence space c0 6⊂ LpX (S, S, X is reflexive or has the R-N property. For an extended discussion of these spaces see Panchpagesan (2008).] (d) Extend the work of (c) above to Lϕ X (S, S, µ) for a Young function ϕ by a modification of the L2X (µ), using the work on the corresponding Orlicz space analysis (see e.g., Rao and Ren (1991) on these spaces). 5. We present another adjunct both to the above exercise and to Theorem 3.10. If (S, S) is a measurable space and µi : S → R+ , i = 1, 2 are a pair of (finite) positive measures, then µi ≪ νt where νt = tµ1 +(1−t)µ2 for all but a countable set of t ∈ R, and this fact has been extended by Rybakov (1970) to obtain a weaker version of the controlling measure problem for a vector measure F : S → X, a Banach space, than the original Bartle-DunfordSchwartz theorem given as Theorem 1.3.4 in Chapter 1. The desired result then is as follows which will be used to discuss a (weaker) companion of Theorem 3.10 above. Thus if X is a Banach space and µ : S → X is a vector measure, then there exists some x⋆ ∈ X⋆ such that the scalar measure x⋆ ◦ µ dominates µ, i.e., µ ≪ x⋆ ◦ µ. Then B. J. Walsh (1971) has noted that the set of x⋆ for which this domination holds forms a dense Gδ subset of X⋆ , although there seems to be no method of construction of any x⋆ that works; the controlling measure is a stronger statement (which also does not have a constructive method). Now suppose that Y is another Banach space with the R-N property and µ : S → X and ν : S → Y be σ-additive where ν has finite variation (|ν|(S) < ∞). Since Y has the R-N property, and by Rybakov’s result µ ≪ xR⋆ ◦ µ, suppose that ν ≪ x⋆ ◦ µ. Then by the R-N property of Y, ν(A) = A f d(x⋆ ◦ µ). Also x⋆ ◦ µ vanishes outside of a set A0 ∈ S, so that (x⋆ ◦ µ)(A) = x⋆ ◦ µ(A ∩ A0 ), A ∈ S. If we now take gx⋆ = x⋆ (χA0 − χAc0 ), andR define hx⋆ = f gx⋆ , then by the change of variable rule, we get ν(A) = A hx⋆ dµ, A ∈ S. This is a weaker form of Theorem 3.10, where in both cases the “density” hx⋆ is not unique. Here

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439

the x⋆ ∈ X⋆ is not constructed, and exhibits how only a class of “densities” is obtained even in this very special case. This example is from Uhl and Diestel ((1977),p.96) where the details of Rybakov and Walsh results are also given. 6. The work in this chapter is concerned with vector measures and integrals. Vector integrals also arise if one integrates vector functions relative to scalar measures (prominent among these are the Bochner and Pettis integrals), or integrating scalar functions relative to vector measures (significant among these are the Wiener and martingale integrals), and can have a combination of both (such as those of Theorem 3.10 and Problem 5 above). Here we include a characterization of the first type by restricting the basic measure triple since this clarifies the underlying structure better and throws light on the Carath´eodory process of generation in obtaining the largest measurable class. First recall the concept of a perfect measure, introduced by Gnedenko and Kolmogorov (1954). A measure space (Ω, Σ, µ) is perfect if for each measurable f : Ω → R, the image measure νf = µ ◦ f −1 is a Lebesgue-Stieltjes measure, i.e., for A ⊂ R such that f −1 (A) ∈ Σ, there is a Borel set B ⊂ R such that νf (A) = νf (B). Another condition is quasi-compactness of µ which means that for each{An , n ≥ 1} ⊂ Σ and ε > 0, there is a set Aε ∈ Σ with µ(Aε ) > µ(Ω) − ε such that the sequence {An ∩ Aε , n ≥ 1} has the “compactness” property meaning that if ∩∞ n=1 An ∩ Aε = ∅, there is an 0 n0 > 1 such that ∩nk=1 Ak ∩ Aε = ∅. The equivalence of these non-obvious conditions was established by C.Ryll-Nardazewskii. This property plays a key role in the projective limit theory of measure spaces (the Kolmogorov fundamental existence theorem). Several other related results were established by Sazonov (1965).[See Chapter III of the author’s book, Rao (1981) where some of these applications as well as the related “compact” and “pure” measures are detailed.] Thus if (Ω, Σ, µ) is a perfect finite measure space, X a Banach space, and f : Ω → X, then f is Pettis integrable if and only if there is a sequence fn : Ω → X of simple (measurable) functions such that (i) x⋆ ◦ fn → x⋆ ◦ f a.e. as n → ∞, for each x⋆ ∈ X⋆ (so f is weakly measurable), and (ii) there is a symmetric convex increas↑ ∞ as x ↑ ∞, ϕ(0) = 0, ing (or Young) function ϕ : R → R+ with ϕ(x) x and the set {ϕ(x⋆ R◦ fn ) : kx⋆ k ≤ 1, n ≥ 1k} ⊂ L1 (µ) is (norm) bounded (so that supkx⋆ k≤1 Ω |x⋆ ◦ f |dµ exists finitely). (This is equivalent to the uniform µ-integrability of {x⋆ ◦ fn , kx⋆ k ≤ 1, n ≥ 1}.) If X is separable (or only the range of f is separable) so (i) implies that f is (strongly) measurable, i.e., f −1 (B) R⊂ Σ where B is the BorelR algebra of R, then f is Bochner integrable if Ω kf (ω)kdµ(ω) < ∞, or Ω ϕ(x⋆ ◦ f )dµ is uniformly bounded as all kx⋆ k ≤ 1. This is called the strong or Bochner integrability of f for any space (Ω, Σ, µ), µ(Ω) < ∞. [This result shows the great difference between weak and strong integrabilities. An extended study of Pettis integrability is given by Talagrand (1984) and the nice characterization stated above is due to Geitz (1981). Thus if the measure space (Ω, Σ, µ) has no pathological sets when one goes from finite to

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infinite dimensions, only then the classical finite dimensional result may be extended.] 7. This problem deals with a restricted form of Theorem 3.10 but is also of interest, obtaining a ‘Radon-Nikod´ ym derivative’ of a random measure relative to another such measure. Thus let Zk : B0 (Rn ) → Lp (P ), p ≥ 1, k ≥ 1, be a sequence of random measures such that Zk (A) → Z(A), a.e., as k → ∞, ∀A ∈ B0 (Rn ), the δ-ring of bounded Borel sets of Rn . Verify that Z(·) is a random measure (i.e., σ-additive in Lp (P )). If V : B0 (Rn ) → Lp (P ) is of finite Vitali variation, and is bounded so that supA∈B0 (Rn ) |V (A)|(ω) < ∞ for all ω ∈ Ω, and if kZk (A)kp ≤ |V |(A)(ω), for all ω ∈ Ω, 1 < p < ∞, then Z(·) satisfies the same domination inequality and there is some operator random function T (·), (not necessarily unique) such that Z Z(A)(ω) = Ts (ω)ds Vs (ω), A ∈ B0 (Rn ). A

This can be extended if Z and V are also vector (random) measures. [Such a result was discussed by Klotz (1993) when X, Y are finite dimensional, and an extension with some related results are included for infinite dimensions in the author’s recent paper, Rao (2009), Sec. 6.] 8. (a) This problem describes a general form of the Karhunen random field which may be given a vector (even a B ⋆ -algebra valued) integral representation of considerable interest. Our basic Hilbert space is H = L20 (P ), the class of centered random variables with finite variances. Let T be an index set and {Nt , t ∈ T } be an abelian collection of (bounded) normal operators indexed by a separable topological space T with a distinguished t0 ∈ T so that Nt0 = identity. Let Xt0 ∈ H be a fixed element and {Xt = Nt Xt0 , t ∈ T } ⊂ H, referred to as a random field. If r(s, t) = (Xs , Xt )H which is considered as a “covariance functional” of the random field, then the following (unique) vector integral representation of Xt and the scalar function of the positive definite measurable r(·, ·) both hold. Namely there is an orthogonally valued σ-additive (vector) measure Z : B(I) → H, I = [0, 1] the unit interval, such that for a continuous function b : T × I → scalars, with (Z(·) = ZXt0 (·)) having the following properties: R R (i) Xt = I b(t, λ)dZ(λ), and (ii) r(s, t) = I b(s, λ)b(t, λ)dα(λ) where α(A) = (Z(A), Z(A))H for all Borel sets A ⊂ I, α(·), a finite measure. The result also holds if the inducing set T is an abstract (not necessarily topological) space. In this case the above representation becomes: The unit interval is replaced by a compact set M and the measure Z(·) is defined on the Borel σ-algebra of M . Here the class of functions b(t, ·) : M → C, t ∈ T , is a continuous family and α(·) is again defined through Z(·), as before. [These representations depend on a well-known spectral theorem due to von Neumann for commuting normal operators. This representation and some related extensions of this result were given in a paper by the author (cf., Rao (2006), Sec.3).]

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(b) We next indicate an extension of these ideas if the above Hilbert space H is replaced by a (Hilbert) module, but using the corresponding inner product as follows. Let the scalar field C of H be replaced by a module K, based on the complex Stone algebra C(S) where S is the Stone space, a compact extremally disconnected (or 0-dimensional) space and that there is one inner product (·, ·) on K valued in C(S) with the usual bilinearity properties, where C(S) plays the role of C of (a), i.e., a positive definite conjugate bilinear functional. Thus means for + each f ∈ K p and a ∈ C(S), af ∈ H and k · k : f → R is defined as kf k = sup (f, f ), and hence k(f, g)k ≤ kf k kgk from which the triangle inequality kf +gk ≤ kf k+kgk may be inferred. Then {K, k·k}when complete is called a Kaplansky-Hilbert (or K-H) module based on the Stone algebra C(S), in lieu of the complex scalar field, if the following two conditions hold: (i) there is a family {ej , j ∈ J} of orthogonal projections in the Banach (Stone) algebra C(S) with e = ∨j ej ; (in the order relation) such that for x ∈ K, ej x = 0, j ∈ J ⇒ ex = 0; and (ii) if the family in (i) satisfies ∨j ej = 1 (identity), then for any {xj , j ∈ J} in a ball of the algebra C(S), there exists an x ∈ K such that ej x = ej xj , j ∈ J. Since K = C(S) satisfies the above conditions, the set of K − H modules is substantial. Moreover, Kaplansky showed the following analog of the classical Riesz representation, namely if f ∈ B(K, C(S)), a bounded linear operator between the (Banach) spaces K and C(S), then it is representable as f (x) = (x, y) for a unique y ∈ K, and hence if T : K → K is a module homomorphism, then (T x, y) = (x, T ⋆ y), x, y ∈ K for a unique bounded linear operator T ⋆ , which can be called the adjoint of T and is a module homomorphism when (and only when) T is bounded. So the usual properties of adjoint are valid (with of course nontrivial proofs). A modulehomomorphism (linear) operator T : K → K is normal if T and its adjoint T ⋆ commute. Using the (by now classical) theory of Gel’fand-Na˘imark on commutative Banach algebras, Wright (1969) observed that an abelian B ⋆ -subalgebra A of B(K) containing the identity is isometrically isomorphic to the continuous function space C(M ) where M is the maximal ideal space of A. With such a structure, a spectral theorem for normal operators T ∈ B(K) was established by Wright, and so with his work one can find an extension similar to that of (a) to the K − H module space, based on the following sketch. Let {Bt , t ∈ I} ⊂ B(K) be an abelian collection of normal operators B containing the identity. Then there exists a compact set M and an operator valued function f (t, ·) : M → C(S) and a vector measure E on the Borel σ-algebra of M into A, the B ⋆ -algebra generatedRby {Bt , t ∈ I}, such that Bt is representable as (a spectral integral) Bt = M f (t, λ)dE(λ), t ∈ I, where E(·) is a projection operator function (or the spectral resolution). More details of the mapping can be obtained by using the general arguments of Wright (1969) (see also Theorem 3.18

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above) and the work of (a) with the references given there. The problem is an indication of the mutual relations between the ideas of abstract functional analysis and probability theory. Further details, which are interesting and deeper will be omitted. The interested readers can extend this study using the methods of (a) above. 9. We saw in Theorem 5.5 that an integral representation of a flat k-cochain using a flat k-chain consists of a kind of Yosida-Hewitt decomposition of a σ-additive and a purely finitely additive (vector) measure. Here we present a condition when there is only the σ-additive part of the function. Thus if X(·) is a flat k-cochain having its (flat) k-form DX (·, α) ¯ in R R γA i where γ˜A the direction α ¯ , then X · A = Rn hDX , dγA i + Rn hDX , d˜ is purely finitely additive. Verify however that γ˜A = 0 if DX (·, ·) is continuous for the flat k-cochain and the flat chain A has finite mass. [The conclusion is obtained by refining the construction of the proof of TheoremV5.5, and using the properties of a “strong lifting” available in the k L∞ ( (Rn ), dλ)-spaces and is somewhat technical. This was also studied by Noltie (1975).] 10. A smooth form F : D(U, Rn ) → L20 (P ), where D(U, Rn ) is the space of differential forms, on the open set U ⊂ Rn , is a random current which is continuous if ϕk → 0 in D(U, Rn ) implies F (ϕk ) → 0 in L20 (P )-mean. Recall that a random current F (·) is homogeneous if for ϕ, ψ ∈ S n−r (0 < r < n), the covariance distribution of F (ϕ) and F (ψ) of r-currents is translation invariant,i.e., E(τh F (ϕ) · τh F (ψ)) = (τh F (ϕ), τh F (γ)) = (F (ϕ), F (ψ)), where τh F (ϕ) = F (τh ϕ), P by definition, is given forPr-currents F. Similarly if σg is defined as σg ( i di (t) · ei )(x1 , . . . , xr ) = i ϕi (g · t)(g ′ ei1 ∧ · · · ∧ g ′ eir )(x1 , . . . , xr ), where g ′ is the adjoint of g and when it is a nonsingular transformation g ′ = g −1 . Hence σg Fr (ϕn−r ) = Fr (σg ϕn−r ). A random r-current is isotopic if for any ϕ, ψ ∈ S n−r and g ∈ G, the group of orthogonal transformations on Rn is (σg Fr (ϕ), σg Fr (ψ)) = (Fr (ϕ), Fr (ψ)). Then show that the following relations on the random current F (= Fr ) obtain: (i) If the random current Fr is isotropic, then so is dFr , ⋆Fr , and also the same holds for ∂Fr where ⋆Fr is the Hodge star mapping; d and ∂ are as before (i.e., boundary and coboundaries), (ii) If the random current Fr is isotopic, so are dFr , ⋆Fr and ∂Fr . (iii) If Fr = dFr−1 , Fr being homogeneous or isotopic, then Fr−1 is not necessarily homogeneous or isotropic respectively. [For (iii) if r = u = 2, x1 = at2 dt1 where a is a centered normal random variable then ax1 is homogeneous and isotropic, but x has neither of these properties. Regarding this problem and the new questions related to the extended analysis, see Wong and Zakai (1989) and also the original results due to Itˆ o (1956).]

Bibliographical Notes

443

Bibliographical Notes This chapter is devoted to the counterparts of functional analytic results when the measures are not necessarily taken to be random but are valued in one of the classical Banach spaces. The results are refined when the spaces are assumed to have more smoothness structure. Section 1 concentrates on integration relative to vector measures with only finite semi-variations but typically having infinite total (or even nonfinite local) variation, Wiener’s measure (generated by the BM) being the prime example. On this topic much effort was expended by I. Dobrakov whose work was somewhat simplified and detailed in T. V. Panchapagesan (2008) and in this connection we also follow Gould (1965) and Dunford-Schwartz (1958). Dinculeanu (2000) has treated the subject and immediately applied it to stochastic integration. Our treatment is also motivated to a large extent by the random measure theory that was studied in the preceding chapter. In this work D. R. Lewis’s (1970) analysis has special role and exemplifies the importance of sequential completeness restriction of the underlying Banach spaces. We spent a little more space for an analysis of classes of integrable functions relative to vector measures of finite semi-variation and analyzed their structural properties to be used later. Also if µ is an operator measure in the strong (not necessarily the uniform operator) topology, some times a special argument is needed, and this is reflected in the treatment of Section 2. Thus the first two sections prepare the way to have the comparison of two vector measures defined on the same measurable class but with values in possibly different Banach spaces, and this aspect of the subject occupies much of Section 3. A Lebesgue type decomposition, using the concept of (strong or) s-boundedness of Rickart (1943) and some of its elaboration due to Traynor (1973) forms a basis to obtain Radon-Nikod´ ym type results which are needed in applications of geometric integration that is studied later on. See also Darst (1970) on related decompositions. The treatment on this “R-N differentiation” is through a sequence of steps, and is based on several results which in some form demand the Radon-Nikod´ ym property of Banach spaces, and it is always satisfied for reflexive B-spaces. The main result here is Theorem 3.10 in which the uniqueness of the R − N density does not generally hold, as shown by a simple counterexample. The treatment follows the author’s papers (1967, 1973) the second one of which uses an idea of Whitney’s (finite dimensional) geometric integration, suitably generalized. The L1 (µ)-space was introduced there and discussed briefly, the idea was analyzed by E.Thomas (1970) for general Radon vector measures and also for Lp (µ) spaces. (He told the author at the Oberwolfach Measure Theory Conference in June 1981 that his dissertation work was initially motivated by my 1967 paper noted above, and we had some interesting discussion on related problems.) A special case of this R − N vector measures differentiation was considered by Bogdonowicz and Kritt (1967) when both spaces X, Y are “uniformly smooth”. In this case the work simplifies as noted in the

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author’s (1973) paper, but excludes most applications of our problem and the intended geometric integration in the next section. The question of specializing the spaces to get a unique R-N derivative is solved for a class of Banach function spaces by J.D.M. Wright (1969), and some of his main results are included as Propositions 3.15, 3.17 and Theorem 3.18. It was contrasted by the (special) case of conditional measures. Also Wright’s approach has interest in the representation theory of certain random fields as illustrated in Exercise 6.8 later. (See also Problem 6.6 in that section to find how the situation can improve if the [measure] space is restricted to have a finer structure.) The next two sections deal with applications of the above theory to geometric integration and the consequent analysis of random currents which probably is considered in a book of this type for the first time. Section 4 is devoted to the (existing) geometric properties when the analysis is conducted on spaces having (or isomorphic with) the Euclidean structure. Hence, chains and simplexes (particularly of Lipschitz type) are the building blocks of the work, and many special properties from Whitney (1957) and Federer (1969) are recalled and utilized. (See also the recent somewhat elementary but also more readable book by Krantz and Parks (2008).) The results are obtained for the finite dimensional case, and then the sharp chains analysis is extended to certain infinite dimensional spaces using the concepts of direct and projective limits. The latter subject is treated in more detail in a recent paper of the author (cf., Rao (2011) which may be of particular interest in such extensions). For flat chains, in a finite dimensional case, the integral representation was an open problem in Whitney (1957, p.327). A satisfactory solution was obtained by Noltie (1986) for sharp chains, and also the corresponding (more difficult) solution for flat cochains in a finite dimensional Banach spaces was given by him (1975) which unfortunately was not published. Hence it is not widely known. Both of these aspects are detailed in the present chapter (Secs. 4-5). Recently further works, especially by Ambrosio and Kirchheim (2000) and White (1999), have also been devoted to different aspects of chains in metric spaces of high dimensions. The analysis is not yet ready for presenting integral representation of the (especially flat) cochains in infinite dimension. [Their works are more in the direction of Calculus of Variations, instead of integral representations needed in our study as described in the text.] The recent study by Adams (2008) is on sequences of flat chains in Banach spaces with support in compact sets so that the problem again goes back in essence to finite dimension. To proceed further into infinite dimensions, the use of direct and projective limits methods appear useful and perhaps even necessary. The approaches of Noltie and Adams seem complementary. All the work noted above, is for chains based on Banach spaces using its geometry but the range space is scalars. If the chains take values in a Hilbert space, especially L2 (P ) on a probability space then the resulting class is that of random currents. Using the inner products of the ranges of these chains

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more effectively, Itˆ o (1956) employing de Rham’s currents which are generalizations of Schwartz distributions, introduced the corresponding concept of random currents, as he (and independently Gel’fand) defined already in 1954, the generalized random distributions (or functions). For a detailed account of the latter see the monograph by Gel’fand and Vilenkin (1964). The related new objects are homogeneous and isotropic random currents. We included a brief account of this theory in Section 5 which uses much of the work in Section 4 on cochains and currents. This opens up a new and interesting area of applications especially in turbulence theory which was pioneered by Yaglom starting as early as 1948, giving a summary in his Berkeley Symposium account of 1961, with a comprehensive study included in his two volume research monograph (1987). In order not to extend this chapter into a monograph, sometimes the results and the details of proofs are only outlined, explaining the salient points. However, the material presented shows the great potential for a new and serious research. In connection with the general considerations, we record a broader result that subsumes the work of Theorem 6.1.1 and Proposition 6.1.2, although in a slightly less sharp form, which however is sufficient for our applications. Thus let ϕ : R → R+ be a left continuous nondecreasing mapping with ϕ(0) = 0 and extended to R by defining ϕ(−x) = ϕ(x), called a “ϕfunction”. If Lϕ (µ) is the space of scalar measurable functions on (Ω, Σ, µ) such that Z ϕ(α|f |)dµ < ∞ ρ(αf ) = Ω

for some α = αf > 0, define functional

kf kϕ = inf{k > 0 : ρ(

|f | ) ≤ k}. k

Then it can be verified that {Lϕ (µ), k · kϕ } is a complete linear metric space, also termed a generalized Orlicz space with the translation invariant metric k · kϕ. This is generally a non-locally convex metric space, also termed a Fr´echet function space, and not a Banach space unless ϕ is a symmetric convex function in addition. The above space includes the Lebesgue classes L0 (µ) and Lp (µ), p > |x| and ϕ(x) = |x|p , p > 0, respectively. It was noted before 0 when ϕ(x) = 1+|x| that if (Ω, Σ, µ) is the Lesbesgue unit interval, it was a classical result due p to M. M. Day that the adjoint space of 0 < p < 1 is trivial, having R L (µ), p only the zero functional, where kf kp = Ω |f |p dµ. However a change of metric here implies the following interesting result due to M. S. Steigerwalt and A. J. White (1971), namely if ϕ is a concave increasing ϕ-function (this is satisfied for the spaces Lp (µ), 0 ≤ p < 1) then it becomes a Banach space for the norm Z ′ kf kϕ = ϕ(λf (y))dy R+

where λy (y) = µ[ω : |f (ω)| > y]. The fact that k · kϕ is a norm under which the space becomes a Banach space is nontrivial. The details and

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8 Vector Measures and Integrals

related results are included in Rao and Ren (1991), Sec. 10.1, where the adjoint space is also described which is a space of measures that vanish on µ-null sets. The most useful fact is that in this new norm, the Bartle-DunfordSchwartz theorem applies and so a control measure as in Theorem 6.1.1 is available, and as a consequence the stochastic integration of functions relative to such Fr´echet space measures is well-defined. On the other hand in the earlier chapters we considered also integration of stochastic measures if they are infinitely divisible when the range space is more general. Thus the analyses of Chapters 4-6 and the above result complement each other in a sense. Moreover, in the earlier work the control measure is explicitly obtained whereas here using the Bartle-Dunford-Schwartz result, only its existence is asserted. The fact that a linear metric space can always be endowed with an equivalent translation invariant metric was first established by Kakutani in 1936 and it was extended to have the same property for its completion by Klee in 1952. These were referenced (without details) in Dunford-Schwartz (1958) and the complete account with some applications in S. Rolewicz (1972). These consequences are important for random measures. We have not studied random product integrals and measures, except briefly in Chapter 7 on a series representation. In the next and final chapter, we consider these problems to finish our account of the subject as it currently stands.

9 Random and Vector Multimeasures

This final chapter is devoted to a brief account of tensor and other kinds of product measures of both random as well as standard vector types collectively called multimeasures. The material here largely supplements the discussion and coverage of that of Chapter 7 as well as the bimeasure treatment of Chapter 4 and also parts of Chapter 8. Further their extensions to multimeasures which need not have finite variation will be of interest in stochastic applications, and so are discussed along with a brief indication of a noncommutative analog of these considerations. Also there are the tensor products of random (and vector) measures in the framework of multi-linear forms. These ideas have close relations to the multidimensional problems, extending the Morse-Transue theory, and they are also sketched at various parts rounding up our present study.

9.1 Bimeasures and Multiple Integrals In Chapter 2 we encountered bimeasures induced by second order (harmonizable/stationary) random process or fields and observed that they do not necessarily determine measures of finite variation on the product space. Hence it becomes necessary to use the (weaker) Morse-Transue integrals in the ensuing analysis. To begin, recall from Section 2.2 the concepts of bimeasures, bilinear forms and their connections involving integral relations. Thus if (S, S) is a measurable space and Z : S → L2 (P )(= L2 (Ω, Σ, P )) is a σ-additive map¯ ping (vector measure) and if β(A, B) = E(Z(A)Z(B)) = (Z(A), Z(B)), in inner product notation, then β(·, B) and β(A, ·) are σ-additive (i.e., scalar or ˜ × B) for a scalar σ-additive β˜ signed) measures, but β(A, B) need not be β(A on the ring S × S. It is just a bimeasure. Hence a weaker than the Lebesgue type definition of an integral for functions (f, g) : S × S → C relative to β [due to Morse and Transue (1956 and before) with some restriction but still weaker than Lebesgue’s method] is essential. This was already discussed. We state a special case for the present extensions and that includes more on the

448

9 Random and Vector Multimeasures

structure (and distinctions) of bimeasure integration since its extended discussion for multimeasures and thus the formulation of their integrals will become clearer. It may be recalled (from Chapter 2) that if (Si , Si ), i = 1, 2, are a pair of measurable spaces, the mapping β : S1 × S2 → C is a bimeasure if β(·, B) and β(A, ·) are scalar (or signed) measures for each pair (A, B) ∈ S1 × S2 but β(·, ·) need not define a similar (σ-additive) measure on S1 × S2 , the product ring. This fact presents a serious problem and prevents a direct application of many familiar measures as well as integration theorems. To understand the concept better we sketch an alternative form for bimeasures (cf. also Section 5.2). If Xi , i = 1, 2 are Banach spaces let X1 ⊗ X2 denote their tensor Pn x ⊗ x2i are product space defined as: xji ∈ Xi , j = 1, 2 and x = 1i i=1 formal sums, then x ∈ X1 ⊗ X2 so that the latter is the set of all such formal sums and it becomes a linear space. If there is a norm α on this product space satisfying kx1 ⊗ x2 kα = kx1 k1 kx2 k2 where xi ∈ (Xi , k · ki ), i = 1, 2, called a cross-norm, let X1 ⊗α X2 be the completion of X1 ⊗ X2 for k · kα . Although there exist several such cross-norms, those of interest here are the greatest and the least cross-norms denoted k · kγ and k · kλ defined as (for simplicity both are taken real or complex spaces): ) ( n n X X x1i ⊗ x2i , n ≥ 1 (1) kx1i k1 kx2i k2 : x = kxkγ = inf i=1

i=1

and if X⋆i is the adjoint space of Xi , then ( n ) X ⋆ ⋆ ⋆ ⋆ ⋆ kxkλ = sup x1 (x1i )x2 (x2i ) : xji ∈ Xj , kxji k1 ≤ 1, j = 1, 2, n ≥ 1 . i=1 (2) The corresponding completed (termed tensor product) spaces are denoted X1 ⊗γ X2 and X1 ⊗λ X2 . Only equivalence classes need be considered and the following relations are classical (cf. Schatten (1950)). To use them they are stated in the following form: Proposition 9.1.1 For the Banach spaces X1 , X2 their tensor products introduced above have the identifications: (a) (X1 ⊗γ X2 )⋆ ∼ = L(X2 , X⋆1 )), and = L(X1 , X⋆2 )(∼ ⋆ ⋆ (b) (X1 ⊗λ X2 ) ֒→ L(X1 , X2 ) ∼ denotes isometric equivalence in (a) and “֒→” is isometric embedwhere “=” ding in (b), L(X, Y) being the usual Banach space of bounded linear operators between X and Y. Both the details and extensions of this result are well-known and can be found in the tract (Schatten (1950), Chapter III). To obtain an alternative definition of a bimeasure, let Xi = C0 (Ωi ), i = 1, 2 where Ωi is locally compact and C0 (Ωi ) is the Banach space (under uniform norm) of continuous scalar functions vanishing at infinity. If V (Ω1 , Ω2 ) = X1 ⊗γ X2 then for the adjoint

9.1 Bimeasures and Multiple Integrals

449

space V (Ω1 , Ω2 )⋆ of (a) in the Proposition above, one has F ∈ V (Ω1 , Ω2 )⋆ representable with T : C0 (Ω1 ) → C0 (Ω1 )⋆ a bounded linear operator as follows: F (f ⊗ g) = T (f )g, g ∈ C0 (Ω2 ), f ∈ C0 (Ω1 ). (3) By Proposition 1, if we take B(f, g) = F (f ⊗ g), and using the classical Riesz-Markov theorem for an integral representation of T , one gets the following: Z g(ω2 )µf (dω2 ) B(f, g) = F (f ⊗ g) = (T f )(g) = Ω2 Z Z g(ω2 ) f (ω1 )µ(dω1 , ·) (dω2 ), (4) = Ω2

Ω1

R where µ represents T , i.e., (T f )(·) = Ω1 f (ω1 )µ(dω1 , ·) ∈ M (Ω2 ) while M (Ω2 ) = C0 (Ω2 )⋆ , the space of regular bounded (or Radon) scalar measures with total variation norm (cf. Dunford-Schwartz (1958) IV.7.3 and IV.9.9). Now in (4) above, can we express the right side as a double integral Z Z (f, g)(ω1 , ω2 )µ(dω1 , dω2 )? Ω2

Ω1

Note that the scalar µ : B(Ω2 ) × B(Ω1 ) → R(or C) is not necessarily σ-additive. So it is desirable to give a meaning for this (double) integral and use µ(·, ·) only as a bimeasure. The procedure to be used here actually involves (not the Lebesgue-Stieltjes but) the Morse-Transue integral ! We arrived at this result, by a different process, in Theorem 2.2.9 with some additional information established originally by Morse and Transue from “first principles”. The above procedure is based on the work of Varopoulos (1967) who terms the set V (Ω1 , Ω2 )⋆ the space of bimeasures, although he assumed Ωi to be compact (for his applications). As in Chapter 2, on using the definition of the MT-integration which is weaker than that of Lebesgue’s, we obtain: kF k = kBk = sup{|B(f, g)| : kf k∞ ≤ 1, kgk∞ ≤ 1} Z Z = sup (f, g)(ω1 , ω2 )µ(dω1 , dω2 ) : kf k∞ ≤ 1, kgk∞ ≤ 1 Ω2

Ω1

= kµk(Ω1 , Ω2 ), the semivariation of µ.

(5)

Since µ need not have an extension to be a (scalar) measure on Ω1 × Ω2 , it follows that V (Ω1 , Ω2 )⋆ % M (Ω1 × Ω2 )(= C0 (Ω1 × Ω2 )⋆ ) and the bimeasure integrals in (5) above can be replaced by the standard (Lebesgue) class provided µ (and hence B) has a finite Vitali variation. The above semi-variation is essentially the same as Fr´echet’s variation. The following result complements the discussion of Section 4.3:

450

9 Random and Vector Multimeasures

Theorem 9.1.2 Let (Si , Si ), i = 1, 2 be Borel measurable spaces where Si is a σ-compact space and let C0 (Si ) denote the space of continuous scalar functions (with uniform norm) which vanish at infinity. If B : C0 (S1 ) × C0 (S2 ) → X is a bounded bilinear form where X is a reflexive Banach space then there exists a unique vector bimeasure β : S1 × S2 → X such that, with MT-integration of (f1 , f2 ) by β (recalled again in Definition 9.1.3 below), one has: Z Z B(f1 , f2 ) = f1 (s1 )f¯2 (s2 )β(ds1 , ds2 ), fi ∈ C0 (Si ), i = 1, 2, (6) S1

S2

where kBk = kβk(S1 , S2 ), using the notation of (5). If X is merely a Banach space then the representation (6) holds provided B maps bounded sets of V (S1 , S2 ) = C0 (S1 ) ⊗γ C0 (S2 ) into conditionally compact subsets of X. This is an interesting analog of the classical Riesz-Markov theorem (cf., Dunford-Schwartz (1958), VI.7.3), and the details are given in Ylinen (1978). Definition 9.1.3 Let (Si , Si ), i = 1, 2 be a pair of measurable spaces and fi : Si → R (or C) be measurable functions. If β : S1 × S2 → C is a bimeasure [i.e., β(·, B), β(A, ·) are signed measures for each (A, B) ∈ S1 × S2 ] then (f1 , f2 ) is integrable relative to the bimeasure β provided (i) f1 is β(·, B) and f2 is f2 f1 β(A, ·) integrable for each R pair (A, B) ∈ S1 ×f S2 , so that Rβ (S1 , ·) and β (·, S2 ) f1 2 given by β (S1 , B) = S1 f1 β(ds, B) and β (A, S2 ) = S2 f2 β(A, ds2 ) are well defined for all A, B above, (ii) f1 is β f2 (·, S2 ) and f2 is β f1 (S1 , ·) integrable just fixed single R pair f(S1 , S2 ), and (iii) their integrals satisfy the relation R for the f2 f dβ 1 (S1 , ·). The common value in (iii) is then denoted (·, S ) = f dβ 2 S2 2 S1 R 1 R by S1 S2 (f1 , f2 )dβ.

Since (i) and (ii) are detailed enough one may ask if (iii) is an automatic consequence. However the following example [from Ylinen (1978)] shows that it need not be true in general. Example 9.1.4 n ≥ 1, with Si as power sets. P∞Let S1 = S2 = N, the integers + a < ∞, f : S → R , i = 1, 2 be given, with ci > 0, Let an > 0, i i n=1 n by: ∞ k X c1 1 X ci < ∞. ci , (k ≥ 2), f1 (1) = , f1 (k) = a1 ak i=1 i=1 P∞ Choose bi > 0 such that k=1 f1 (k)ak bk < ∞, and let f2 (n) = b1n . Consider β : S1 × S2 → R defined by: β({n}, {n}) = an bn , β({n}, {n + 1}) = −an bn+1 , n ≥ 1, P and set β({m}, {n}) =P0 for m P6= n 6= m + 1, so that m,n≥1 |β({m}, {n})| < ∞. Then β : (E, F ) 7→ m∈E n∈F β({m}, {n}) defines a bimeasure. A simple calculation shows that

9.1 Bimeasures and Multiple Integrals

Z

S1

451

f1 dβ(·, {n}) = bn cn , n ≥ 1,

and XX

m≥1 n≥1

|f1 (m)β({m}, {n})| =

∞ X

m=1

f1 (m)(bm + bm+1 ) ≤ 2

∞ X

m=1

f (m)an bm < ∞.

Consequently for any F ∈ S2 we have, by re-arranging the absolutely convergent series, Z ∞ X X X f1 dβ(·, F ) = f (m)β({m}, {n}) = b n cn . S1

n∈F m=1

n∈F

Similarly

XX

m≥1 n≥1

|f2 (n)β({m}, {n})| = a1 +

∞ X

(an + an−1 ) < ∞.

n=2

Thus f1 is β(·, F ) and similarly f2 is β(E, ·)-integrable. But Z Z ∞ X cn > 0. f2 dβ f1 (S1 , ·) = f1 dβ f2 (·, S2 ) = 0 and S1

S2

n=1

Hence only conditions (i) and (ii), but not (iii), are satisfied here. It should also be noted that the β-integral defined above is non absolute in the sense that (f1 , f2 ) is β-integrable does not imply that (|f1 |, |f2 |) is also β-integrable nor (f1 χA , f2 χB ) need be integrable relative to β shown earlier by an example (cf., Exercise 2.5.1). This implies that we need to restrict the above β-integral definition to obtain a “smooth” theory. Somewhat better behaved are a subclass of bimeasures β that are positive definite in the sense that β : S1 × S1 → C satisfies: [S1 = S2 = S, S1 = S2 = S are taken] n X n X ai a ¯j β(Ai , Aj ) ≥ 0, Ai ∈ S, ai ∈ C, n ≥ 1. (7) i=1 j=1

Such β’s appear, as seen in Section 2.2, from second order random measures Z : S1 → L2 (P ), where for A, B ∈ S1 , β(A, B) = E(Z(A)Z(B)) is a key example (E(·) is the expectation operator on (Ω, Σ, P ) of L2 (P )) and the general (nonpositive definite) one is obtained if the Zi : Si → L2 (P ), i = 1, 2 are distinct and β : S1 × S2 → C, is given by β(A, B) = E(Z1 (A)Z¯2 (B)), where β is just a bimeasure. Thus both are important for applications and the bimeasure integration has distinctly different properties not implied by the classical Lebesgue theory, the former being nonabsolute and the latter absolute. Some of the discussion of Section 2.2 already indicated this. But the following treatment is particularly useful for spectral analysis of second order random processes.

452

9 Random and Vector Multimeasures

There is an important representation of bounded bilinear forms on a product of a pair of continuous function spaces which we already mentioned in Section 2.2. It will be used here in discussing the “spectral domains” of second order random fields and to round out the treatment of Chapter 2. This is also due to Grothendieck which plays an important role in the study of Hilbertian operators. Theorem 9.1.5 (Grothendieck) Let (Si , Bi ), i = 1, 2 be Borelian spaces where Si is locally compact and C0 (Si ), the space of real continuous functions vanishing at infinity, i = 1, 2 with B : C0 (S1 ) × C0 (S2 ) → C as a bounded bilinear functional. Then there exists a bounded bimeasure β : B1 × B2 → C and a pair of probability measures µi : Bi → [0, 1] such that for some absolute constant KG one has Z Z B(f1 , f2 ) = (f1 , f2 )(s1 , s2 )β(ds1 , ds2 ), fi ∈ C0 (Sj ), i = 1, 2 (8) S1

S2

in which |B(f1 , f2 )| ≤ KG kBk · kf1 k2,µ1 kf2 k2,µ2

(9)

where kBk = kβk and the integral in (8) is in (strict) MT-sense. (The absolute constant K R G satisfies 0 < KG ≤ 1.782 and with (5) one has kBk = kβk, and kfi k22,µ = Si |fi |2 dµi , as usual.)

The representation (8) is a consequence of our work in Section 2.2 and the key inequality (9) is the famous Grothendieck representation relative to some probability measures. A complete proof of (9) is in Pisier (1986, p.55), Replacing µ1 , µ2 by 12 (µ1 +µ2 ) = µ, one can even take a single measure in (9) by increasing slightly the constant which is immaterial for our application here. (As seen already in Section 2.2 this inequality is related to the measure in Bochner’s boundedness principles.) We now derive some important consequences of this inequality when B is a strictly positive bilinear functional as defined in the classical book by M. H. Stone (1932, Def.2.14 on p.56). In this case B(·, ·) is actually a (bounded) sesqui-linear form on C0 (S1 ) × C0 (S2 ). Also it is known that when extended to L2 (S1 , µ1 ) × L2 (S2 , µ2 ) → C, this can be expressed as Z (T f )g dµ2 , f ∈ L2 (S1 , µ1 ), g ∈ L2 (S2 , µ2 ), (10) B(f, g) = (T f, g) = S2

for a unique bounded linear mapping T : L2 (S1 , µ1 ) → L2 (S2 , µ2 ) and using (9) one has (11) kT k = sup |B(f, g)| : kf k2,µ1 ≤ 1, kgk2,µ2 ≤ 1 ≤ KG kBk,

(cf., Riesz and Sz.Nagy (1955) p.202, and/or Taylor and Lay (1980), pp.342– 344, on the representation given above). Moreover, T is also positive for positive

9.1 Bimeasures and Multiple Integrals

453

definite B. Using these specializations, we may analyze the space of functions if S1 = S2 (= S (say)) with B(f, f ) < ∞. Further if B(f, f ) ≥ m > 0 (by Stone’s definition of strict positive definiteness) for all f ∈ L2 (S, µ), kf k2,µ = 1 where µ1 = µ2 = µ now and some fixed m > 0) then T −1 exists and is bounded. Under this hypothesis for any x⋆ ∈ L2 (µ), and B as above there is moreover a unique g ∈ L2 (µ) such that x⋆ (f ) = B(f, g), a result also termed the Lax-Milgram criterion (cf., Taylor and Lay (1980), p.342). These will be utilized as follows. Let B : L2 (µ) × L2 (µ) → C be a positive definite bilinear form which satisfies the “strict” condition, B(f, f ) ≥ m for f ∈ L2 (µ), kf k2 = 1, where the measure µ is given by the Grothendieck inequality as in (9) and (10) above (µ1 = µ2 = µ here). Consider the space L2 = {f ∈ CS : f is µ-measurable and B(f, f ) < ∞}.

(12)

Using (10) and the positive definiteness of B so that B(f, g) = (T f, g) where T is also a positive linear operator on L2 (µ) which p by the strictness condiB(f, f ) and the functional tion has a bounded inverse, define k · k : f 7→ 1 1 1 can be written as kf k2 = B(f, f ) = (T f, f ) = (T 2 f, T 2 f )L2 (µ) where T 2 1 denotes the square root of T which is positive and so kf k = kT 2 f k2,µ is a norm functional since k · k2,µ is a norm on L2 (µ). If {fn , n ≥ 1} is a Cauchy 1 sequence in the space (L2 , k · k) so that kfn − fm k = kT 2 (fn − fm )k2,µ → 0 2 as n, m → ∞, by the completeness of L (µ), there is a g ∈ L2 (µ) satisfy1 1 ing kT 2 fn − gk2,µ → 0. Let f = T 2 g. Since T −1 is a bounded linear oper1 ator in L2 (µ), f ∈ L2 (µ) and kf k = kT 2 f k2,µ = kgk2,µ < ∞, we deduce 1 1 1 that f ∈ L2 and kfn − f k = kT 2 fn − T 2 f k2,µ = kT 2 fn − gk2,µ → 0 as n → ∞. Thus {L2 , k · k} is a Hilbert space. (This could also be deduced directly from the Lax-Milgram criterion recalled above!) Now it should be noted in the preceding argument if B(·, ·) is simply positive definite without being “strict”, then in this analysis, one still has T : L2 (µ) → L2 (µ) to be nonnegative definite and L2 ⊂ L2 (µ) where k · k is a semi-norm. Since L2 (µ) of the Grothendieck space (and inequality) is a Hilbert function space, (L2 , k · k) is a semi-inner product function space contained in L2 (µ). Consequently its completion is (possibly) a subspace of L2 (µ), say (L20 , k · k). This completed space (L20 , k · k) plays the same role as the one above. Since the measure µ of Theorem 4 is not unique, a completion in the general case need not be the same whereas L2 is a dense subspace of all these completions. It is uniquely defined by the bilinear form B itself. Thus the preceding discussion can be summarized for reference as follows: Theorem 9.1.6 Let B : C0 (S1 ) × C0 (S2 ) → C be a positive definite bounded bilinear form where C0 (Si ), i = 1, 2, is the continuous function space on a locally compact set Si whose elements vanish at infinity and use uniform norms.

454

9 Random and Vector Multimeasures

Then there exist a probability measure µi on the Borel sigma algebra of Si and an absolute constant KG such that the representation (10) holds for a positive bounded linear operator on L2 (S1 , µ1 ) into L2 (S2 , µ2 ) satisfying the bound given in (11). If S1 = S2 = S, then taking µ = 21 (µ1 +µ2 ), one can have L2 (S, µ) where ˜ G ) in µ is again a probability measure with a suitable absolute constant K(≥ K (11). In this case if L2 is defined by (12) then the functional k · k2 : f 7→ B(f, f ) defines (L2 , k · k) ⊂ L2 (µ) and if B is “strict” (or satisfies the classical LaxMilgram) condition, then (L2 , k · k) itself is a Hilbert space. In general it will be a subspace of L2 (µ) for each Grothendieck measure µ and thus its completion is a function space contained in L2 (µ). It may be noted that for the Grothendieck inequality, B(·, ·) need not be positive definite and with L2 (µi )(= L2 (Si , µi )), i = 1, 2 exist. The above work can be extended to this case at least when it is given by a bimeasure through the MT-integrals using the results of Chapters 2 and 4 along with the generalized Jordan decomposition (cf., Corollary 4.3.8 and the related computation). The corresponding discussion will be omitted here. Since a bimeasure β(·, ·) always defines a bilinear form B(·, ·) on the space of β-integrable (complex) functions through MT-integration, the analysis of L2 for such B determined by a positive definite bimeasure β is of interest R R ⋆ here. This means B(f, g) = S S f (x)¯ g (y)β(dx, dy) as strict MT-integrals for all scalar bounded Borel functions f, g on a locally compact space S. As usual β(·, ·) is positive definite means that for bounded Borel sets Ai ⊂ S, n X n X ai a ¯j β(Ai , Aj ) ≥ 0, ai ∈ C, n ≥ 1, Ai ∈ B(S). (13) i=1 j=1

This implies that B(·, ·) is positive definite and the converse holds on using the (strict) MT-integral of Chapter of 2. (Here B = B(S) is the Borel σalgebra.) This gives a connection with random measures, namely there exists a probability space (Ω, Σ, P ) and a measure Z : B → L2 (P ) such that β(A, B) = E(Z(A)Z(B)) = (Z(A), Z(B)) the latter being the inner product of L2 (P ). The result opens up the possibility of two deep functional analytic results being employed to connect (actually dilate) harmonizable random fields with stationary ones and then analyze their spectral domains. Both these concepts are intrinsically related to Fourier analysis and as such the underlying measurable space (S, B) must be an LCA group (so S = G will be written). These are restated from Chapters 2 and 4 in a form that will be a convenient reference for the work here.

Definition 9.1.7 A mapping X : G → L20 (P )(= L20 (Ω, Σ, P )) with covariance ¯ 2 )) = (X(g1 ), X(g2 )) which is continuous (just mear(g1 , g2 ) = E(X(g1 )X(g surable is enough) is called (i) K-stationary (K for Khintchine) if r(g1 , g2 ) = r(g1 g2−1 , 0) where 0 ∈ G is the identity so that r is a function of one variable, say s, and by the Bochner (-Weil-Raikov) theorem

9.1 Bimeasures and Multiple Integrals

r(s) =

Z

ˆ G

hs, λi dµ(λ),

455

(14)

ˆ is the dual group of G and µ(·) is a positive bounded measure on the where G ˆ (ii) (a) harmonizable in the strong sense if r(g1 , g2 ) is continuσ-algebra of G, ous and is the Fourier transform of a bounded bimeasure β(·, ·) of finite Vitali variation so that Z Z (15) hg1 , λ1 ihg2 , λ2 i dβ(λ1 , λ2 ) r(g1 , g2 ) = ˆ G

ˆ G

¯ 2 , λ1 ) and the integral in (15) is where the bimeasure β satisfies β(λ1 , λ2 ) = β(λ in the Lebesgue-Stieltjes sense, (b) if β(·, ·) is only of finite Fr´echet variation so that now the integral in (15) is in Morse-Transue sense, and then {X(g), g ∈ G} is harmonizable in the weak sense. The work in Chapters 2, 4 and 5 on properties of bimeasure integrals translates to the mappings {X(g), g ∈ G}, called random fields. We briefly restate the previous work into this special class to discuss some related problems that are somewhat sharper and new in this context. After presenting a few structural results, deduced from the earlier analysis, we outline the corresponding elements that critically depend on the group structure of the set G. The analysis of Section 2 extends without difficulty to establish the following representation: Theorem 9.1.8 Let X : G → L20 (P ) be a random mapping where G is an LCA group so {X(g), g ∈ G} is termed a second order random field, with covariance r : G × G → C. If the random field is mean (or L2 (P ))-continuous and is either K-stationary (so (14) holds) or harmonizable (weakly or strongly so that (15) is true), then there exists a random measure Z : B(G) → L20 (P ) having orthogonal values on disjoint sets of B if (14) holds, so that (Z(A), Z(B))L20 (P ) = µ(A ∩ B), µ is as in (14); or in the case of (15) there is the corresponding Z such that (Z(A), Z(B)) = β(A, B) where β(·, ·) is a bimeasure of Vitali or Fr´echet variation respectively, so that in either case one has the representation Z hg, λi dZ(λ), g ∈ G, (16) X(g) = ˆ G

ˆ is the dual of G). the vector integral is in the Dunford-Schwartz sense (G

There is an important connection between the stationary and (both weak and strong) harmonizable random fields obtainable from their covariance structures (14) and (15), given for r. The fact that r is positive definite (and continuous) defined on a locally compact (abelian) group G brings in a well-defined structure theory (actively studied in Harmonic Analysis) and its influence on random fields, which even extends to the nonabelian case. Here some decomposition theory of group representations along with their correspondents in

456

9 Random and Vector Multimeasures

integration, resembling and generalizing the Itˆ o integral calculus presented in Chapter 7, will be restated to the extent needed here. An important consequence of the representation (16) is the following result. If T : L20 (P ) → L20 (P ) is a bounded linear operator, then {Y (g) = T X(g), g ∈ G} ⊂ L20 (P ) is a random field and with a classical result due to Hille, already used several times (proved in Dunford-Schwartz (1958), IV.10.8 (f)), it follows that Z hg, λi d(T ◦ Z)(λ), g ∈ G, (17) Y (g) = ˆ G

ˆ → C is a continuous character of G at g which is a homowhere hg, ·i : G ˆ → L20 (B) is a vector morphism, and one shows easily that Z˜ = T ◦ Z : B(G) (or random) measure that is not orthogonally valued even when Z(·) is. Consequently by the above theorem, {Y (g), g ∈ G} is a weakly harmonizable random field and its covariance has the representation given using the Morse-Transue (strict) integral as: Z Z ⋆ (18) hg1 , λihg2 , λ′ iβ(dλ, dλ′ ), rY (g1 , g2 ) = (Y (g1 ), Y (g2 ))L2 (P ) = ˆ G

ˆ G

ˆ i = 1, 2. where β(A1 , A2 ) = ((T ◦ Z)(A1 ), (T ◦ Z)(A2 ))L2 (P ) for Ai ∈ B(G), Here the integrals (17) and (18) are first established for simple functions and extended to the general case through a dominated convergence criterion which is not valid generally for MT-integrals but true for the strict MT-class. [Detailed discussion of these assertions (and counterexamples for non strict case) are given in Sections 4 and 5 of Chang and Rao (1986) for G = Rn , but the general LCA case follows the same lines of argument.] The representations (16), (17) and (18) have a deep significance for the analysis of stationary and harmonizable random fields as well as their extensions. We introduce the ideas here and analyze the mathematical consequences in the next section. First observe that the stationary random field {X(g), g ∈ G} ⊂ L20 (P ), G an LCA group, is shift invariant in the sense that ¯ 2 )) is uncharged if g1 , g2 are the covariance function r(g1 g2−1 ) = E(X(g1 )X(g replaced by (g1 g), (g2 g), a translation by g units. This implies that Ug : Xs 7→ Xsg defines {Ug , g ∈ G} a unitary group of operators on L2 (P ). This of course is false for the harmonizable family. But the significance of this fact is that Ug X(g1 ) = X(g1 g), which is stationary for g ∈ G, and the operator Ug : L20 (P ) → L20 (P ) is of a unitary family {Ug , g ∈ G} ⊂ B(L20 (P )) (first noted by Kolmogorov in 1941 with G = Z). It has great significance here. This follows from the fact that for g, g ′ ∈ G one has (observing Ug−1 = (Ug )∗ = Ug ∗ ) (Ug X(g1 ), Ug′ X(g2 )) = (X(g1 g), X(g2 g ′ )) = r(g1 gg ′−1 g2−1 ) = (X(g1 gg ′∗ ), X(g2 )) = (Ugg′∗ X(g1 ), X(g2 ))

9.2 Bimeasure Domination, Dilations and

Representations of Processes

457

so that Ug Ug∗′ = Ugg′⋆ ; Ugg′ = Ug Ug′ and Ug⋆ = Ug−1 follow. Here Ug is de2 fined on sp{X ¯ g , g ∈ G} ⊂ L0 (P ), and is taken as identity on the complement. Thus {Ug , g ∈ G} is a unitary group of operators and hence one can use Stone’s theorem to obtain an integral representation. This gives (16) since Xg = Ug X(0) and one has (with the above noted spectral theorem for unitary operators) Z Z Xg =

ˆ G

hg, λi dEλ X0 =

ˆ G

hg, λi dZλ

(19)

ˆ is the resolution of the identity (E 2 = Eλ E ⋆ = Eλ etc.), where {Eλ , λ ∈ G} λ λ ˆ is orthogonally valued, G ˆ being the dual of G. and {Zλ = Eλ X0 , λ ∈ G)} The group structure of the LCA group is essentially used here which is more familiar if G = R or Z as seen in Chapter 2. Comparing (17) and (19) it is seen that a (weakly or strongly) harmonizable field is obtained from a stationary one by means of a continuous linear transform and then defined as in (17). There is in fact a converse direction that any harmonizable field is a “projection” of a stationary field onto a subspace. Stated differently a harmonizable field can be “dilated” to a stationary field on a large or inclusive space L2 (P˜ ) and this fact lies deeper. We present the latter result and some related ones in the next section with the necessary mathematical preparation which again depends on a form of Grothendieck’s inequality and is termed a “domination problem” needing some additional analysis of “higher order semi-variations” of vector measures. Note that since G = Rn (or Zn ) is possible, and (14)-(19) already ˆ = Rn as well as Tn = include multiple integral representations such as G n (0, 2π] .

9.2 Bimeasure Domination, Dilations and Representations of Processes In studying the integration of scalar measurable functions on a measurable space (S, S) with respect to a vector measure ν : S → X where X is any Banach space (X = L20 (P ) in the above work most of the time) it was necessary to find a controlling measure µ ≥ 0 for ν, so limµ(A)→0Rkν(A)k ≤ limµ(A)→0 kνk(A) = 0. This measure µ plays a vital role in defining { A f dν, A ∈ S, f ∈ I} for a class of measurable functions I on (S, S). Here ν is of finite semi-variation which can have finer (or stronger) variational properties relative to µ. To understand this better we introduce higher order semi-variations: Definition 9.2.1 If ν : S → X is a vector measure with some controlling measure µ (and there always exists one by the Bartle-Dunford-Schwartz theorem) then its q-semi-variation (q ≥ 1) is defined as Z f dνk : f ∈ Lp (S, S, µ), kf kp,µ ≤ 1}, (1) kνkq (A) = sup{k A

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9 Random and Vector Multimeasures

where p−1 + q −1 = 1. [If q = 1 (so p = ∞) then this ‘one’ semi-variation kνk1 (·) is the usual semi-variation which does not depend on µ because L∞ (µ) can be replaced by the space of bounded scalar measurable functions.] It may be verified that the q-semi-variation implies the usual semi-variation but not conversely. Since µ(S) < ∞, for each q > 1, L1 (µ) ⊃ ∪q>1 Lq (µ), and this suggests that we may use Orlicz spaces in getting closer norm bounds than only using q-bounds. This is true. First we introduce the relevant concept and use it for such bounds. Definition 9.2.2 Let ϕ : R → R+ be a symmetric convex increasing function with ϕ(0) = 0. Then a vector measure ν : S → X with a controlling (positive finite) measure denoted µ, has ϕ-semi-variation kνkϕ,µ defined as

Z

kνkϕ,µ (A) = sup f dν : kf kψ,µ ≤ 1, measurable f , (2) A

R where ψ(x) = sup{|x|y − ϕ(y) : y ≥ 0} and kf kψ,µ = inf{α > 0 : S ψ( αf )dµ ≤ 1}. Hence ψ(·) is also symmetric, convex, satisfying |xy| ≤ ϕ(x)+ψ(y), x, y ∈ R, called the Young inequality, and ψ the complementary (Young) function of ϕ.

The concept is motivated by a classical C.J. de la Vall´ee Poussin result on characterizing sets of uniform integrability for µ. Its application in stochastic analysis is well-known (cf., e.g., Rao (1981), Sec 1.4.). The fact p that k · kψ,µ is a norm can be verified and if ϕ(x) = |x|p then ψ(x) = |x|q q

so that ϕ(1) + ψ(1) = 1. The following useful result will be employed later:

Theorem 9.2.3 Let (S, S) be a measurable space, ν : S → X, (a Banach space) be a vector measure with control measure µ : S → R+ . Then there exists a symmetric convex function ϕ, ϕ(x) x ↑ ∞ as x ↑ ∞, such that kνkϕ,µ (S) < ∞; and it has ϕ-semivariation finite (implying also the finite semi-variation). Proof. Since weak and strong σ-additivities of a Banach valued measure are equivalent, x⋆ ◦ ν : S → C is σ-additive and for any {An , n ≥ 1} ⊂ S, a disjoint sequence, it follows from the σ-additivity of ν, 0 = lim kν n→∞

∞ [

n=1

n

X

An − ν(Ak ) k=1

n ∞ [ X = lim sup{|(x⋆ ◦ ν) (x⋆ ◦ ν)(Ak )| : kx⋆ k ≤ 1}. An − n→∞

n=1

(3)

k=1

Hence the set {x⋆ ◦ ν : kx⋆ k ≤ 1} ⊂ ca(S, S) is uniformly σ-additive and x ◦ ν is µ-continuous where µ is a (finite) control measure of ν, as noted above. ⋆

9.2 Bimeasure Domination, Dilations and

Representations of Processes

459

⋆

ym derivative of x⋆ ◦ ν relative to µ, then If gx⋆ = d(xdµ◦ν) is the Radon-Nikod´ (3) implies Z ⋆ 0 = lim |(x ◦ ν)(A)| = lim gx⋆ dµ µ(A)→0

µ(A)→0

A

uniformly in x , kx k ≤ 1. Hence {gx⋆ : kx k ≤ 1} ⊂ L1 (µ) is bounded and uniformly µ-integrable since µ(S) < ∞. Next by the classical de la Vall´ee Poussin theorem (cf., e.g., Rao (1981), Theorem I.4.4), there exists a convex function ϕ R as described in the statement and S ϕ(|gx⋆ |) dµ ≤ K0 < ∞, a uniform bound. If ψ is as given by the function in Definition 2 above, then one has, using (2),

Z

kνkϕ (S) = sup f dν : kf k ≤ 1 ψ,µ

S Z d(x⋆ ◦ ν ⋆ : kx k ≤ 1 : kf kψ,µ ≤ 1} = sup{sup f dµ S ≤ 2 sup{kgx⋆ kϕ,µ : kx⋆ k ≤ 1} ≤ 2K0 < ∞. (4) ⋆

⋆

⋆

Then by the H¨ older inequality for Orlicz spaces, we deduce that ν has finite ϕ-semi-variation. 2 The last one is a sharper inequality than stating that ν-has a finite semivariation (= 1-semi-variation), and may not be replaced by q-semi-variation for q > 1, since ϕ(x) may grow slower than |x|q . However for a class of Banach spaces including those which are isomorphic to Hilbert spaces a refined type of (4) can be obtained, and this is a form of Grothendieck’s inequality and exemplifies the role of the latter in this work. We now isolate some spaces analyzed by Lindenstrauss and Pelczy´ nski (1968), defined as follows. Unfortunately no motivation can be offered for introducing this class. Definition 9.2.4 For a pair of isomorphic Banach spaces E, F , denote d(E, F ) = inf{kT k kT −1 k : T ∈ B(E, F ), an isomorphism }, inf{∅} = +∞. The d(·, ·) is called the “Banach-Mazur” functional and one has d(E, F ) ˜ ·) can be considered as a distance ≤ d(E, G)d(G, F ) so that log d(·, ·) = d(·, S functional. A Banach space X is termed an Lp,λ -space if X = i∈I Xi , {Xi , i ∈ I} being a directed set (by inclusion) of subspaces of X of finite dimension for each i ∈ I, whence for any i ∈ I, d(Xi , ℓm p ) ≤ λ and for an m-dimensional Lebesgue sequence space ℓm , λ ≥ 1. Further, X is an Lp -space if it is an Lp,λ -space for p some λ ≥ 1. It can be verified that the Lebesgue space Lp (µ′ ), on some measure space, is an Lp,λ -space for every λ > 1 and an abstract (M )-space (hence C0 (Ω), on a locally compact Ω) is an L∞,λ -space for all λ > 1. Further an L2 -space coincides with a Banach space isomorphic to some Hilbert space. For instance a Banach space and its adjoint both have smooth norms (i.e., twice “Fr´echet differentiable” norms) are (not necessarily Hilbert but) L2 -spaces. [These can be

460

9 Random and Vector Multimeasures

independently shown, see e.g., the note (Rao (1967)).] Thus the (large enough) class of Lp -spaces will be of interest, and Grothendieck’s theorem is extended to these spaces by Lindenstrauss and Pelczy´ nski (1968) and although it was cited many times before, we now establish the result in the following form: Theorem 9.2.5 Let (S, S) be a measurable space and ν : S → X be a vector measure where X is an Lp -space 1 ≤ p ≤ 2 in the sense of Definition 4. Then there exists a measure µ : S → R+ dominating (or controlling) ν such that for each f ∈ B(S, S), the space of scalar S-measurable bounded functions, the following inequality holds:

Z

(5)

f dν ≤ kf k2,µ , Lp

S

and hence ν has 2-semivariation finite relative to µ.

Proof. Let B(S, S) be the space of bounded S-measurable scalar functions on R S and consider the mapping T : f 7→ S f dν, f ∈ B(S, S) where the integral is well-defined in the sense of Dunford and Schwartz ((1958), IV.10), which utilizes a controlling measure in showing that T f is uniquely determined by ν. [This is important since Lp is not necessarily an adjoint Banach space.] In the general case, (5) is established using a form of Grothendieck’s theorem as follows, presented in three steps for convenience. I. First consider C(S), (instead of B(S, S)) the continuous (real) function space on a compact Hausdorff space S, and let ℓs : C(S) → R be the evaluation functional, ℓs (f ) = f (s), (ℓs ∈ C(S)⋆ , the adjoint space). If K denotes the set of extreme points of the unit ball of C(S)⋆ , then by the Krein-Milman theorem (cf., Dunford-Schwartz (1958), pp.440–442) the set K is closed (in the dual topology of C(S)), and its elements are of the form αℓs , |α| = 1. Hence by the Grothendieck-Pietsch inequality in the form given by Lindenstrauss and Pelczy´ nski ((1968), Cor.2 to Thm.4.3 and Prop.3.1), there exists a regular probability measure µ0 on K, and an absolute constant C1 such that Z 2 kT f kLp ≤ C1 (6) |f (s)|2 dµ0 (s) = C1 kf k2,µ0 . S

For the complex case C(S) = Cr (S) + iCr (S) so that (6) holds if C2 = 2C1 . Then (5) obtains if we take µ = C2 µ0 in (6). This key inequality will now be extended from C(S) to B(S, Σ) in the following two steps. II. Let B(S)(= B(S, S)) be the space of all bounded S-measurable scalar functions on S. Then by the Stone isomorphism theorem (Dunford-Schwartz (1958), IV.6.8) there exists a compact (extremally disconnected) Hausdorff ˆ space Sˆ and an isometric algebraic isomorphism I between B(S) and C(S) ˜ which respects the order and complex (conjugate) relations so that T = ˆ → Y is a well-defined mapping satisfying the conditions of Step T ◦ I −1 : C(S) I above. Hence there is a regular positive Borel measure µ on Sˆ such that (6) now becomes:

9.2 Bimeasure Domination, Dilations and

kT˜f kLp ≤ kf k2,µ1 ,

Representations of Processes

ˆ f ∈ C(S).

461

(7)

ˆ and (7) gives Let f˜ = I(f ), f ∈ B(S) so f˜ ∈ C(S), Z 2 2 2 ˜ ˜ ˜ |f˜|2 dµ1 kT f kLp = kT f kLp ≤ kf k2,µ1 = ˆ S

¯˜ = hf˜f, µ1 i = hI(f˜)I(f ⋆ ), µ1 i = hI(f f¯), µ1 i with h·, ·i and duality pairing, = hf f¯, I ⋆ (µ1 )i, I ⋆ is adjoint of I, Z |f |2 dµ2 , µ2 = I ⋆ (µ1 ), =

(8)

S

where µ2 is a bounded additive set function in B(S)⋆ = ba(S, S), the space with total variation as norm. Here the integral relative to finitely additive µ is as in (Dunford-Schwartz (1958), p.108 ff). It remains to show that in (8) µ2 can be replaced by a suitable σ-additive measure. ˜ be the class of III. Let µ ˜ be the measure generated by (S, µ2 ) and let S µ ˜-measurable sets. Then it is well known (cf., Sion (1965), p.67, Thm.5.4, ˜ ⊃ S, µ ˜ Moreover or Rao (2004), p.43, Thm.10) that S ˜ is σ-additive on S. µ ˜(A) ≤ µ2 (A), P A ∈ S, with equality if and only if µ2 is σ-additive. ˜ (and Let f = m i=1 ai χAi , Ai ∈ S, disjoint, ai 6= 0. Now by definition of µ ε boundedness of µ2 ) given ε > 0, there exist Aεin ∈ S such that Ai ⊂ ∪∞ n=1 Ain , and ∞ X ε µ ˜(Ai ) + > µ2 (Aεin ). (9) m|ai |2 n=1 ε ε ε ˜ Replacing may take Ai = ∪∞ n=1 Ain . Let Pm Ain by Ai ∩ Ain εin S if necessary, we ε ε fN = i=1 ai χ∪N ε , so f N ∈ B(S, S), and fN → f pointwise and boundedly k=1 Aik as N → ∞. Now (8) becomes Z ε ε 2 |fN (s)|2 dµ2 (s) kT fN kLp ≤ S

=

m X i=1

|ai |

2

N X

µ2 (Aεik ), since µ2 is additive.

k=1

Letting N → ∞ here and using a standard result (c.f., Dunford-Schwartz (1958), IV.10.10), we get kT f k2L2p

≤ =

m X i=1

Z

S

|ai |

2

ε µ ˜(Ai ) + , by (9), m|ai |2

|f |2 d˜ µ + ε.

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9 Random and Vector Multimeasures

Since simple functions are dense in B(S, S), and ε > 0 is arbitrary, this holds for all f and so (5) follows. 2 It should be noted that the µ ˜ obtained above, although ‘controlling’ ν, R is not the usual control measure that is needed to define T f = S f dν unambiguously. However it can be utilized for our purpose. If ν : S → X, an adjoint space (i.e., X = Z⋆ for some normed linear or Banach space Z), then this ν has (the smallest) dominating measure, which is of interest here when the L2 spaces are considered, (although it excludes the vector measure generated by the BM process which has infinite variation). This fact will be included here to understand the above inequality better. It is taken from Dinculeanu (2000, p.65). This gives a simplification for adjoint space valued vector measures ν, if the definition of ‘semi-variation’ is strengthened, in contrast to that used in Proposition 1.2.4 and throughout our work in this book. Proposition 9.2.6 Let X(= Z⋆ ) be an adjoint space and (S, S), a measurable couple with ν : S → X a vector measure in the norm topology of X. Then ν has the finite semi-variation if and only if it has finite variation, so both are equal on S. (Thus the variation being σ-additive, it can be a controlling ν, but if X = L2 (P ), then, (ν(A), ν(B)) = β(A, B) gives β just as a bimeasure). Proof. Since it is well-known that kνk(A) ≤ |ν|(A), A ∈ S, it suffices to prove the reverse inequality with kνk(A) < ∞ and the fact that X = Z⋆ . For any A ∈ S, consider the trace σ-algebra S(A) and let Ak ∈ S(A), k = 1, . . . , n. If ε > 0 is given, since ν(Ai ) ∈ X⋆ , choose xi here such that one has kν(Ai )k ≤ ν(Ai )(xi ) + nε where ν(Ai )(xi ) = hν(Ai ), xi i and use the duality pairing for some xi ∈ Z, kxi k = 1, by definition of norm (and the HahnBanach theorem). So [this is semi (not Fr´echet-variation, the latter has xi as scalars] n X i=1

kν(Ai )k ≤

n X

ν(Ai )(xi ) + ε

i=1 n X

=|

ν(Ai )(xi )| + ε.

i=1

Hence taking the supremum over i, and remembering the definition of (semi-) variation, one has from the arbitrariness of ε > 0, that |ν|(A) = (var ν)(A) ≤ (semi − var (ν))(A) = kνk(A), A ∈ S. Thus the statement follows. 2 A simple consequence of this result is that a vector measure ν : S → L20 (P ) having finite semi-variation (since L22 is a reflexive hence adjoint space) denoted |ν|(·), satisfies ν ≪ |ν|. By the known vector measure analysis (cf., e.g., Uhl and dν Diestel (1977), p.76) L22 has the Radon-Nikod´ ym property so that g = d|ν| (·), the (R-N) derivative exists. Hence one has

Z

2 Z

2 Z Z

2

= f g d|ν| ≤ kT f k22 = f dν |g|2 d|ν|, |f | d|ν| ·

S

S

S

S

(Bochner integral),

= kf k22,µ

9.2 Bimeasure Domination, Dilations and

Representations of Processes

463

if µ = |ν| · C where C = kgk22>0 , giving (5). The result of Theorem 5 above is better suited for our work. In fact the vector measure ν : S → Lp arises as a random measure in the representation: Z eisλ dν(λ), s ∈ S Xs = ˆ S

where Sˆ is a different space than S (e.g. for the stationary random fields S is a locally compact (abelian) group with Sˆ as its dual) if S is non-abelian then Sˆ is a “dual object” in which case the situation is more involved. (This latter problem was discussed briefly in Rao (1989) when G is a separable locally compact “type I” group and the result admits further extension (see Section 4 below) in which it is necessary to use C ⋆ - algebras etc.) When the results ˆ must from Fourier analysis are invoked it is essential that the spaces S (and S) have group (or at least semi-group) structure. This is the special aspect of the present study and we consider some dilation results using the preceding work related to Theorem 5 above. The following detailed analysis illustrates these statements. Recall that {Xg , g ∈ G}, a random process valued in L20 (P ), and indexed by an LCA group G, is K-stationary if its covariance r(·, ·) given by r(g1 , g2 ) = (Xg1 , Xg2 ) is continuous and r(g1 , g2 ) = r˜(g1 g2−1 ), to be written as r˜(g1 − g2 ). It is also representable as, (Bochner-Weil theorem), Z hg, λi dF (λ), g ∈ G, (10) r˜(g) = ˆ G

ˆ is the dual group of G and F is a bounded positive function (measure) where G ˆ so that (G, ˆ F ) is a (bounded) Borel measure space. Denote hg, λi ˆ G) ˆ G, on (G, as ϕg (λ) so that {ϕg , g ∈ G} is a family of (continuous) F -integrable functions, and then Xg is representable as Z ϕg (λ) dZ(λ), g ∈ G (11) Xg = ˆ G

ˆ → L2 (P ), which has moreover ˆ relative to a vector measure Z : B(G)(= G) 0 orthogonal values when Xg is K-stationary so (Z(A), Z(B)) = F (A ∩ B). If T : L20 (P ) → L20 (P ) is a bounded linear operator, then by a classical ˜ g = T Xg is representable theorem of Hille (used already in our work) X as: Z Z ˜ ˜g = ϕg (λ) dZ(λ), (12) ϕg (λ)d(T ◦ Z)(λ) = X ˆ G

ˆ G

ˆ = B(G) ˆ → L2 (P ) is a vector measure but no longer orthogonally where Z˜ : G 0 valued. (Here T can go past ϕg (·) since the latter function is bounded and this ˜ ˜ ˆ then need not hold in general.) So if F˜ (A, B) = (Z(A), Z(B)), A, B ∈ B(G), ˜ F (·, ·) is a bimeasure and (10) becomes in this case

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9 Random and Vector Multimeasures

˜ g2 ) = ˜ g1 , X r˜(g1 , g2 ) = (X

Z Z ˆ G

ˆ G

ϕg1 (λ)ϕg2 (λ′ )F˜ (dλ, dλ′ ),

(13)

ˆ×G ˆ is in the (strict) Morse-Transue sense. A class where the integral on G ˜ of the type {Xg , g ∈ G} ⊂ L20 (P ) whose covariance r˜ satisfies (13) without assuming that it comes as an image defined by (12) but given abstractly is said to be of (weak) Cram´er class. Here F˜ will only have the Frech´et variation finite. In case it has finite Vitali variation then it is said to be of strong (or classical) Cram´er class and then the integral of (13) is in the Lebesgue-Stieltjes sense. This distinction is essential. If in some cases Z˜ is orthogonally valued, or equivalently F˜ (A, B) = F˜ (A ∩ B), then the resulting ˜ g , g ∈ G} is said to be of Karhunen class. We now discuss with Theorem 5 {X above the possibility that a Cram´er random field may be dilated to a Karhunen field. First we remark that using the classical Kolmogorov existence theorem it is possible to show that a process (or field) with covariance given by (13) always exists on some probability space (Ω, Σ, P ) i.e, in L20 (P ). Further it admits an integral representation for a vector measure Z˜ as: Z ˜ ˜g = ϕg (λ) dZ(λ), g ∈ G, (14) X ˆ G

ˆ → L20 (P ) has orthogonal increments, whence (Z(A), ˜ ˜ where Z˜ : B(G) Z(B)) = ˆ are two sets (not necessarily groups and need not F˜ (A ∩ B). Here G and G have any relation as long as (13) hence (14) holds). Then for a random field ˜ g , g ∈ G} given by (14) for an L20 (P )-valued vector measure by Theorem {X 5 (see also Proposition 6) with L2 (= L20 (P )) there exists a measure µ on the ˆ such that (G ˆ = B(G)) ˆ G) ˆ measurable space (G,

Z

ϕg (λ)dZ(λ) ˜ ≤ kϕg k2,µ , g ∈ G. (15)

ˆ G

L20 (P )

We assumed for (15) that ϕg be bounded. But the result (seen in the proof) ˜ integrable ϕg and in particular for all ϕg ∈ L2 (G, ˆ µ). Hence holds for all Z(·) ˜ ˜ (15) implies with F (·, ·) as the bimeasure of Z(·), Z

2

Z

˜ ϕg (λ) dZ(λ) |ϕg (λ)|2 dµ − 0≤

ˆ ˆ G G Z Z Z Z ϕg (λ)ϕg (λ′ )F˜ (dλ, dλ′ ). (16) ϕg (λ)ϕg (λ′ ) dµ(λ) dµ(λ′ ) − = ˆ G

ˆ G

ˆ G

ˆ G

If β(A, B) = µ(A)µ(B) − F˜ (A, B), then (16) implies β is positive definite and hence is a covariance bimeasure. Also it defines a positive hermitian bilinear form, say I, given by Z Z f (λ)g(λ′ )β(dλ, dλ′ ), f, g ∈ L2 (µ), I(f, g) = (17) ˆ G

ˆ G

9.2 Bimeasure Domination, Dilations and

Representations of Processes

465

using the MT-integration here. It is possible to construct H, with the Aronszajn (or reproducing kernel Hilbert) space determined by I(·, ·). A concrete method involving a probability triple is desirable in this study and it can be described as follows. Define an inner product [·, ·]′ : L2 (µ) × L2 (µ) → C by the relation [f, g]′ = I(f, g) which by (17) is a semi-inner product on L2 (µ). Let N0 = {f : [f, f ] = 0} and if H1 = L2 (µ)/N0 , the factor space, and consider [·, ·] : H1 × H1 → C, as usual by letting [(f ), (g)] = [f, g]′ where (f ) is the equivalence class of f so that [·, ·] is the inner product on the (quotient) space H1 . Let H0 be the completion of the resulting space and π0 : L2 (µ) → H0 be the canonical mapping onto H0 . Consider two orthogonal subspaces of H0 defined as 2 ′′ ′ H′ = sp{X ¯ = sp{X ¯ g , g ∈ G} ⊂ L0 (P ) and H g = π0 (ϕg ), g ∈ G} 2 which is not necessarily in L0 (P ). Consider the direct sum H = H′ ⊕ H′′ (⊂ H0 ), and endow the inner product in H as the sum of H′ and H′′ where H′ ⊕ {0} and {0} ⊕ H′′ are identified as subspaces of H which are orthogonal: H′ ⊂ L20 (P ) = L20 (Ω, Σ, P ). With the Kolmogorov existence theorem we can find a probability space (Ω ′ , Σ ′ , P ′ ) so that H′′ ⊂ L20 (P ′ ). Let ˜ Σ, ˜ P˜ ) = (Ω, Σ, P ) ⊗ (Ω ′ , Σ ′ , P ′ ) and then realize H as a subspace of L2 (P˜ ) (Ω, 0 in which both Xg , Xg′ (or their equivalence classes) lie. Now if Yg = Xg + Xg′ then it is asserted that {Yg , g ∈ G} is a Karhunen process (or field) in L20 (P˜ ) which is a dilation of the Xg -field. Since by construction Yg ∈ L20 (P˜ ) and Xg = proj (Yg ) it suffices to verify that {Yg , g ∈ G} is a Karhunen field. It then implies that the Xg -field in L20 (P ) is dilated in the super Hilbert space L20 (P˜ ). From the construction, L20 (P ) can be identified as a closed subspace of 2 ˜ L0 (P ). Let Q be an orthogonal projection on L20 (P˜ ) onto H′ (⊂ L20 (P˜ )). Since Xg ∈ H′ for all g ∈ G, we have QYg = Xg , g ∈ G and the covariance of the Yg -field is given by, since Xg2 ⊥ Xg′ 2 , r˜(g1 , g2 ) = (Yg1 , Yg2 )H = (Xg1 , Xg2 )H2′ + (Xg′ 1 , Xg′ 2 )H′′ Z Z Z Z ′ ˜ ′ ϕg1 (λ)ϕ¯g1 (λ′ ) dβ(λ, λ′ ) ϕg1 (λ)ϕ¯g2 (λ )F (dλ, dλ ) + = ˆ G ˆ ˆ G ˆ G G Z Z Z ϕg1 (λ)ϕg2 (λ)dµ(λ). = ϕg1 (λ)ϕ¯g2 (λ′ ) dµ(λ) dµ(λ′ ) = ˆ G

ˆ G

ˆ G

Hence the {Yg , g ∈ G} is a Karhunen field as desired. The preceding outline can be summarized precisely as follows. It will be termed a dilation theorem. Theorem 9.2.7 Let {Xg , g ∈ G} ⊂ L20 (P ) be a process or field representable by a vector integral (14) relative to a set {ϕg , g ∈ G}, ϕg being ˜ individually bounded. Here Z(·) is a measure, valued in L20 (P ). Then there 2 ˜ exists a super space L0 (P ) ⊃ L20 (P ) as a subspace, and a Karhunen field ˜ g , g ∈ G} in the dilated space such that Xg = QX ˜g , g ∈ or process {X

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9 Random and Vector Multimeasures

G where Q is an orthogonal projection of L20 (P˜ ) onto the space sp{X ¯ g, g ∈ G}. The following specialization is a frequently used result in our applications. Here one chooses the random field {Xg , g ∈ G} to be (weakly or strongly) harˆ its dual group and hϕg , λi = ϕg (λ) are monizable with G as an LCA group, G the characters of G which are continuous and uniformly bounded. [If G = Rn , then ϕs (λ) = eihs,λi , the exponentials, and in the stationary case the measure F˜ is positive and bounded, by the Bochner-Weil theorem. In the Cram´er case this becomes the Karhunen field.] The above dilation theorem implies the following important specialization: Corollary 9.2.8 [Stationary dilation of harmonizability.] Let {Xg , g ∈ G} be a (weakly or strongly) harmonizable random field in L20 (Ω, Σ, P ). Then ˜ Σ, ˜ P˜ ) and a (weakly) stationary ranthere exists a super Hilbert space L20 (Ω, ˜ g , g ∈ G} in the latter space such that it is a dilation of the dom field {X ˜ g , g ∈ G where Q(L2 (P˜ )) = L2 (P ) and harmonizable field, i.e, Xg = QX 0 0 Q is the orthogonal projection whose range can and will be taken to be sp{X ¯ g , g ∈ G}. It is interesting to observe that the preceding theorem (and its Corollary) may be used to deduce an important result due to Na˘ımark (cf., Sz.Nagy (1955)) concerning dilation of a ‘symmetric’ to a ‘self-adjoint’ operator. Recall that a linear operator T in a Hilbert space H is symmetric if (T x, y) = (x, T y), x, y ∈ DT , domain of T , and it is self-adjoint if D(T ) ⊂ H is also dense so that the adjoint T ⋆ is defined and T = T ⋆ . Thus a bounded operator T is symmetric if and only if it is self-adjoint (s.a). Now the classical results of spectral analysis in Hilbert space imply that the s.a. R operator A has its spectrum to be a subset of the real line and A = R λ dEλ (by Stone’s theorem) where {Eλ , −∞ < λ < ∞} is a resolution of the identity, (i.e., Eλ⋆ = Eλ = Eλ2 and Eλ Eλ′ = Emin(λ,λ′ ) ) and then the operational calculus says that for each boundedR Borel function f : R → R, the mapping f (A) is defined and f (A)x = R f (λ) dEλ x for x ∈ D(A) where E(·) x is now a vector measure. We present a connection between this and Cram´er- Karhunen class given by Theorem 7 (and Corollary 8) above, and sketch the steps. The key observation here is that a Karhunen process (or field) Xg has its covariance given in (13) [and (14)] which becomes Z ˜ ˜ (18) ϕg1 (λ)ϕ¯g2 (λ) dF˜ (λ) r(g1 , g2 ) = (Xg1 , Xg2 )L20 (P ) = ˆ G

ˆ If now G = Rn where ϕg ∈ L (F˜ ), F (·) being a bounded positive measure in G. ˆ Rn ) as its dual, then (14) becomes an LCA group and G(= Z ˜ ˜0 ˜t = ϕ(t, λ) dZ(λ) = ϕ(t, A)X (19) X 2

Rn

9.2 Bimeasure Domination, Dilations and

Representations of Processes

467

˜ 0 reduces to X ˜ 0 . Here A is just the where ϕ(0, λ) = 1, all λ, so that ϕ(0, A)X multiplication A ←→ λ, and so is only a closed densely defined (but generally an unbounded) operator. The fact that A is s.a., multiplication Ai f = λi f and that its spectrum is a subset of R are crucial. The necessary details are given in Getoor (1956, Thm.3A). Then by the spectral theorem for Ai and from the fact that a Cram´er field Xt with bounded ϕt has a dilation to a Karhunen field, one has the relation: Z ˜ ϕ(t, λ)(Q ◦ E)(λ)Y (20) Xt = QYt = 0 = ϕ(t, A)Y (0), R

˜λ , −∞ < λ < +∞} is the resolution of the identity for A so whereR {E ˜λ )(·). In this case {Eλ , −∞ < λ < ∞} is A = R λdEλ with Eλ (·) = (Q ◦ E no longer a resolution of the identity, although Xt = QYt is valid. Here Eλ is “increasing”, i.e., Eλ ≤ Eλ′ so Eλ Eλ′ = Eλ′ Eλ = Eλ , for λ ≤ λ′ , Eλ+0 = Eλ . Also Eλ → 0 as λ → −∞ (strongly) with Eλ → Q as λ → ∞. This is termed a generalized spectral family. It was shown by Na˘imark in 1943 that the Eλ family has the projection property in a larger Hilbert space containing the given one as a subspace, i.e., the family can be dilated. This result is obtained now as a consequence of our Theorem 7 above. In particu˜ itA ˜ lar if ϕt (λ) = eitλ , all the hypotheses are satisfied = Ut R itλ and ϕt (A) = e ˜ ˜ = e (Q ◦ E)(dλ), having the is a unitary operator with Tt = Qϕt (A) R additional property that {T , t ∈ R} is positive definite in the sense that t Pn Pn ⋆ j=1 (Tti −tj xi , xj ) ≥ 0, (T−t = Tt ), with each kTt k ≤ 1. Thus we have i=1 the following:

Theorem 9.2.9 Let A be a symmetric densely defined operator in H, and {ϕt , t ∈ R} be a class of bounded Borel functions with ϕ0 (λ) = 1. Then {Tt = ϕt ◦ A, t ∈ R} is a family of bounded Hilbert space operators for which there is a super Hilbert space K ⊃ H and a self-adjoint operator A˜ ˜ with Q : K → H being an on K extending A, such that Tt = Q(ϕt ◦ A) orthogonal projection. In particular if the Tt -family is contractive and positive definite in H with T0 = identity, then there is a super Hilbert space K ⊃ H, and a family of weakly continuous unitary group of operators {Ut , t ∈ R} on K such that Tt = QUt where Q is the orthogonal projection as before. This connection between operator families of contractions, and more generally individually bounded classes satisfying certain simple conditions and dilations of certain Cram´er classes with Karhunen families are close.These facts are deduced from different points of view. This is quite significant. It should also be observed that in dilating a Cram´er class into a Karhunen field, we assumed that the “kernels” ϕ(t, ·) of (13) were bounded. Some preliminary analysis indicates that {ϕ(t, ·), t ∈ R or G} must satisfy some integrability (or growth) condition, and it will be interesting to obtain optimal conditions on this family in order that such a dilation exists. We move on to study some other aspects of these ϕ’s in the related covariance structure.

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9 Random and Vector Multimeasures

9.3 Spectral Analysis of Second Order Fields and Bimeasures Let X = {Xg , g ∈ G} ⊂ L20 (P ) where G is an LCA group, be a stationary field so that its covariance r(g1 , g2 ) = (Xg1 , Xg2 ) satisfies r(g1 , g2 ) = r˜(g1 g2−1 ) and where r˜(·) is a continuous scalar function. It is also positive definite and by the Bochner-Weil theorem, writing g for g1 g2−1 , Z (1) r˜(g) = hg, λi dµ(λ), g ∈ G, ˆ G

for a unique bounded regular measure µ ≥ 0, often called the spectral measure ˆ being the dual group of G. of the random field X, G n ˆ IfPG = R (so G = Rn too), hg, λi = eiht,λi with t ∈ Rn and ht, λi = n ¯ j }, µ(·) being a finite Borel measure on G(= ˆ Rn here). In this exp{ j=1 tj λ case µ is usually called the spectral measure of r(·), or of the stationary field. It is of interest to present a basis for this terminology, and then extend the idea for nonstationary processes and fields. Briefly, let {Xt , t ∈ R} be in L20 (P ) and be K-stationary so that the mapping Ut : Xs → Xs+t is well-defined on 2 sp{X ¯ t , t ∈ R} = L ⊂ L0 (P ) satisfying kUt Xs k22 = kXs+t k22 = (Xs+t , Xs+t ) = r(s + t, s + t) = r(s, s) = kXs k22 and that kUt Xs k2 = kXs k2 . Thus Ut is an isometry and for h ∈ R, (Ut Xs , Xh )L20 (P ) = (Xs+t Xh )L20 (P ) = r(s + t, h) = r˜(s + t − h), stationarity, = r˜(s − (h − t)) = r(s, h − t) = (Xs , Xh−t )L20 (P ) = (Xs , U−t Xh )L20 (P ) , t, h, s ∈ R.

(2)

Hence Ut⋆ = U−t , Ut⋆ being the adjoint of Ut . This shows that {Ut , t ∈ R} is a weakly continuous unitary group on L and may be extended to L20 (P ) by setting it to be identity on L⊥ (= L20 (P ) ⊖ L). Then Xt = Ut X0 and by the well-known (and often recalled) Stone spectral representation, there exists a unique resolution of the identity {Et , −∞ < t < ∞} such that Z Z itλ eitλ dZ(λ) (3) e E(dλ)X0 = Xt = U t X0 = R

R

where X0 can be brought inside the integral by a classical theorem used many times. Since Eλ is a resolution, the measure Z(·) is orthogonally valued. It follows from (3) that by the familiar (operational) calculus Z Z Z r˜(s − t) = r(s, t) = eisλ dZ(λ), eitλ dZ(λ) = ei(s−t)λ dµ(λ) (4) R

R

R

where µ(A ∩ B) = (Z(A), Z(B)) by orthogonal increments of Z(·). This Z(·) is a random (orthogonally valued) measure and µ(·), determined uniquely by

9.3 Spectral Analysis of Second Order Fields

and Bimeasures

469

Z(·), is the spectral measure (of the Xt -process). Motivated by this, the relation (13) of the above section (i.e., (2.13)) of the non-stationary process determines the function F˜ (·, ·), the bimeasure which is “nonnegative definite”. It is also called a spectral bimeasure although it does not generally determine a scalar measure. We now introduce spectral domains in both cases and discuss their structural analyses, with a somewhat compressed account on details. Definition 9.3.1 Let {Xg , g ∈ G} ⊂ L20 (P ) be a stationary random field with G as an LCA group so that it has an integral representation Z hg, λi dZ(λ), g ∈ G, (5) Xg = ˆ G

ˆ → L2 (P ) has orthogonal values on diswhere the vector measure Z : B(G) 0 ˆ joint sets, and let µ : B(G) → R+ be its spectral measure. Then the space ˆ µ) is called the spectral domain of the random field {Xg , g ∈ L2 (G, B(G), G}. The interest in this class L2 (µ) of (measurable) functions is its relation with 2 the space H = sp{X ¯ g , g ∈ G} ⊂ L0 (P ) of random elements as described in the following: Proposition 9.3.2 There is an isometric isomorphism of H onto L20 (µ), so that for each h ∈ H there is a unique f ∈ L2 (µ) such that when H ⊂ L20 (P ), one has: Z Z 2 |f |2 dµ. (6) |h| dP = ˆ G

Ω

ˆ = R, (Rn ). Sketch ofPProof. We discuss the case that G = R(or P Rn ) so that G n n itλj If hn = i=1 ai Xti (∈ H), then by (3), for fn (t) = j=1 aj e , using (5) we get: Z X n n − X aj eitj ,λ dZ(λ) dP aj eitj λ dZ(λ) E(|hn |2 ) = (hn , hn )L20 (P ) = ˆ G

i=1

j=1

Z X n 2 aj eitj λ dµ(λ) = (fn , fn )L2 (µ) . = ˆ G

j=1

Thus (6) holds for simple functions. Recalling the fact that µ is a regular ˆ and that trigonometric polynomials are dense in L2 (µ), it measure on G, follows that the mapping τ : H → L2 (µ) with τ hn = fn , extends to all of H into L2 (µ). But this mapping is actually onto as our earlier analysis indicates. So we can conclude that H and L2 (µ) are isometrically isomorphic. 2 It is clear that one has to use some nontrivial properties from harmonic analysis here. This can be extended for vector valued stationary random fields, but it involves an additional twist which is now discussed.

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9 Random and Vector Multimeasures

Nn 2 An n-dimensional (or vector) random field X : g → i=1 (L0 (P ))i , where 2 (L0 (P ))i being the same for all i, is termed stationary if for each vector a = (a1 , . . . , an ) ∈ Rn (Cn ), the scalar product a · X : g → L20 (P ) is simply stationary in the previous sense. This implies that each component is stationary and that the processes Xi and Xj are ‘stationarily correlated’, in the sense that (Xj (g1 + h), Xk (g2 + h))L20 (P ) = (Xj (g1 ), Xk (g2 ))L20 (P ) for any g1 , g2 , h ∈ G. One can follow the previous analysis and establish an analog of (5) as : Z hg, λi dZ(λ), g ∈ G, (7) Xg = ˆ G

ˆ → Nn (L2 (P ))i , Z(·) being an n-vector of where Z = (Z1 , . . . , Zn ) : B(G) 0 i=1 an orthogonally valued measure with (Zk (A), Zℓ (B))L20 (B) = 0 if A ∩ B = ˆ is ∅ for 1 ≤ k, ℓ ≤ n, and Zk (dλ) = E(dλ)(Xk (0)) where {E(λ), λ ∈ G} the resolution of the identity as in (3). The details are quite similar to the previous case, but the vector (or matrix) multiplication presents some new problems. The details are discussed in Rozanov ((1967), Section 4 of Chapter 1) and we omit them but consider the (nontrivial) analog of Proposition 2 just given above. It should be noted that the vector measure Z = (Z1 , . . . , Zn ) with n-components as Xg = (X1g , . . . , Xng ), the n-vector field, and a = (a1 , . . . , an ) ∈ Cn , satisfy (for (6), (with a dot product of Cn )) Z hg, λi d(a · Z)(λ), g ∈ G, (8) a · Xg = ˆ G

ˆ → C has orthogonal increments. Moreover, for A1 , A2 ∈ B(G), ˆ and a · Z : B(G) one has n X ai a ¯j (Zi (A1 ), Zj (A2 )) (a · Z(A1 ), a · (A2 )) = = =

i,j=1 n X

i,j=1 n X

i,j=1

ai a ¯j Bij (A1 , A2 )) ai a ¯j Bij (A1 ∩ A2 ),

(9)

where Bij (·, ·) is the bimeasure determined by Zi (·) and Zj (·), and the vector having orthogonal increments as well as Xig and Xjg are stationarily correlated. This implies that the matrix (Bij (·)) is positive definite as well, since (9) holds for all vectors a ∈ Cn . It follows that the covariance matrix (Bij (·)) of the stationary n-vector random field is a nonnegative Hermitian function representable as Z (Bij (g)) = hg, λiµij (dλ), 1 ≤ i, j ≤ n (10) ˆ G

9.3 Spectral Analysis of Second Order Fields

and Bimeasures

471

where (µij (A)) is such that µij (A) = µji (−A), when G = Rn (or Zn , the integer n-lattice). Since µii (A) = E(Zi (A)Z¯i (A)) = E(|Zi (A)|2 ) ≥ 0 one has the inequality (from (10)) |Bij (g)|2 ≤ Bij (0)Bij (0),

1 ≤ i, j ≤ n.

(11)

A similar inequality holds for the measures µij using the CBS-inequality for the E(|Zi (A1 )Zj (A2 )|) P to conclude that the Hermitian Pnmatrix measure n (µij (A)) is dominated by i=1 µii (A). Thus if α(A) = i=1 µii (A), then µij ≪ α. But (µij (A)) is an n × n positive definite matrix and has an esdµ ym derivasentially unique square root, or if dαij = γij , is the Radon-Nikod´ d tive, then dα (µij ) = (γij ), as a positive definite matrix, having a unique (relative to α) square root function (νij ) and we can R consider a Borel measurable n× n matrix function B(·) to define the integral Gˆ (B(λ)ν(λ)(ν(λ)B(λ))⋆ dα(λ) which is positive definite and using the (matrix or vector) norm one can consider Z Z k(B(λ)ν(λ))(ν ⋆ (λ)B ⋆ (λ))k dα(λ), (12) kB(λ)ν(λ)k2 dα(λ) = ˆ G

ˆ G

R

and denote it as Gˆ B(λ)ν(λ)2 B ⋆ (λ)dα(λ) when it is finite. Symbolically we express the integral of B(·) relative to µ to be defined if (12) holds and express it as a matrix of integrals: Z Z ⋆ B(λ)ν(λ)(B(λ)ν(λ))⋆ dα(λ). B(λ)dµ(λ)B (λ) = ˆ G

ˆ G

R

Now define k · k22,µ : B 7→ k Gˆ B(λ) dµ(λ)B ⋆ (λ)k = ((B, B))2,µ = kBk22,µ . Let L2 (µ) = {B : kBk2,µ < ∞} and if B1 , B2 ∈ L2 (ν), we set the matrix inner product between them as: Z Z ((B1 , B2 ))µ = B1 (λ)dµ(λ)B2⋆ (λ) = (B1 (λ)ν(λ))(B2 (λ)ν(λ))⋆ dα(λ) ˆ G

ˆ G

(13) and easily verify that {L2 (µ), ((·, ·))} is a semi-inner product space over C. Since (B1 ν) : λ 7→ B1 (λ)ν(λ) is measurable and square integrable for the positive measure α(·) strongly or in Bochner’s sense, the classical vector space of ˆ B(G), ˆ α) is complete and it is a Bochner square integrable functions on (G, Hilbert space when functions kB1 − B2 k2,µ = 0 are identified modulo ν. Note that α(·) is just a controlling measure of µ(·), considered as a vector (or matrix valued in Cn × Cn (identified as a C2n ) measure. Any other nonnegative measure α ˜ dominating µ(·) will do as well. In fact the inner product ((·, ·))µ does not depend on the auxiliary dominating or controlling measures α or α ˜ and is uniquely defined (just as the Hellinger distance between probability measures P and Q effectively used by Kakutani in (1948)). [See Exercise 2 for related questions with further details on this point.] We may summarize the above facts as follows:

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9 Random and Vector Multimeasures

Theorem 9.3.3 The space {L2 (µ), ((·, ·))µ } of n × n matrix valued meaˆ B(G), ˆ µ) where µ is a positive definite Hermitian surable functions on (G, measure, is complete and is a Hilbert space when equivalent functions relative to a controlling measure are identified, and moreover such control alPn 2 ways exists, µ i=1 ii being one such. [Here the scalar field for L (µ) is Cn . This result, of some independent interest, will be used to obtain the ndimensional analog of Proposition 2 above. It has been originally established by Rozanov (1967) and by Rosenberg (1964) using somewhat different arguR ments. An important observation is that the integral Gˆ B(λ) dµ(λ)B ⋆ (λ) of (13) is an n × n matrix valued object, and it is not definable as n2 individually (component-wise) defined integrals. The completeness of Theorem 3 will not hold for such a simplistic approach. This was noted by Rosenberg (1964) and we shall include a counterexample (on incompleteness) in the complements section. [In this connection see Exercise 2(a).] ˆ → (L2 (P ))n is an orthogonally valued measure, so that Z = If Z : B(G) 0 (Z1 , . . . , Zn ) is an n-component orthogonal function, and as defined before, it has a controlling measure (α(·) say), by being Ra vector measure, (the classical Bartle-Dunford-Schwartz theorem) and hence Gˆ B(λ) dZ(λ) is well-defined, for all measurable B(·) which are limits of simple functions as an element of (L2p (P ))n , with matrix inner product is given by Z Z Z B1 dµB2⋆ , B2 dZ = B1 dZ, ˆ G

ˆ G

ˆ G

and also using the fact that µ(A) = (E(Zi (A)Z¯j (A)), i, j = 1, . . . , n) is a Hermitian positive definite matrix measure so that

R R one defines the norm as kBk2,µ = ˆ B(λ) dZ(λ) for the elements ˆ B dZ ∈ (L20 (P ))n = H, G

2,P

G

ˆ of all Z(·)-integrable m × n matrix valued B(G)-measurable deterministic (= nonrandom) functions. The same argument of Proposition 2 above extends with standard work and establishes the following:

Proposition 9.3.4 There is an isometric isomorphism of the Hilbert space H ˆ of m × n-matrix valued B(G)-measurable Z-integrable functions in (L20 (P ))n , 2 ˆ and the Hilbert space (L (µ), k · k2,µ ) of the m × n-matrix valued B(G)measurable functions, integrable for the positive definite n × n-measure µ with norm k · k2,µ , denoted by Z B1 (λ) dZ(λ), B2 (λ) dZ(λ (14) ˆ G

L2 (µ)

where kB1 k22,µ = ((B1 , B1 ))L2 (µ) . ˆ and the stationarity What are the special roles played by the spaces (G, G) ˆ of the random field {Xg , g ∈ G} ⊂ L20 (P ) compared with the spaces (S, S)

9.3 Spectral Analysis of Second Order Fields

and Bimeasures

473

and the random field {Xt , t ∈ G} ⊂ L20 (P ) which is of Karhunen type relative ˆ S, ν) where ν is the spectral to some family {ϕ(t, ·), t ∈ S} contained in L2 (S, ˆ S, ν) is dense and measure of the Xt field? If the set {ϕ(t, ·), t ∈ S} ⊂ L2 (S, ˆ ˆ (S, S, ν) is a regular (finite R or σ-finite) measure space2 (S is a locally compact space) and H = sp{X ¯ t = S ϕ(t, λ) dZ(λ), t ∈ S} ⊂ L (P ), is the Hilbert space of the random field, then we can assert an isometric isomorphism between H and L2 (ν). Then we obtain an extension (straightforward but with some work) to the vector case as in Propositions 2 and 4 above. A precise version will be included as a problem in the complements section (cf., Exercise 2(b)-2(c)) later. An interesting adjunct to the above results, also due to Kolmogorov and Rozanov, is to characterize two n-dimensional stationary vector valued random fields one of which is a linear transform of the other. A very simple (but nontrivial) example of practical importance with some applications will also be presented in the complements section later. It illustrates the existence of a certain linear operator T such that Y = T X is (not merely harmonizable but) in fact once again stationary when X is a stationary process (cf., Exercise 3). Thus the problem now is to find a class of such linear operators preserving stationarity. If X : G → (L20 (P ))n is a stationary random field, hence having the representation (12) with µx as its special matrix measure and Z x (·) = (Z1x (·), . . . , Znx (·)) as the (vector) stochastic measure with orthogonal values, consider the transformed Y with similar properties, and µy , Z y as its representing measures. Thus Z Z hg, λ′ i dZ y (λ′ ), g ∈ G, (15) hg, λi dZ x (λ), and Yg = Xg = ˆ G

ˆ G

whose covariances are given by Z x xx (Bij (g)) = hg, λiµij (dλ) , ˆ G

as:

y (Bij (g)

=

Z

ˆ G

hg, λiµyy ij (dλ)

.

(16)

But Y = T X and T is a bounded linear operator. So one can express Y Z Z hg, λiZ y (dλ), g ∈ G. (17) hg, λiT ◦ Z x (dλ) = Yg = ˆ G

ˆ G

y

By the present hypothesis Z (·) also has orthogonal values. Considering T as a suitable matrix function T (λ) = (tij (λ), 1 ≤ i, j ≤ n) acting on Z x (·) and using the classical uniqueness theorem of Fourier transforms (extended easily to the present situation) we can also conclude: Z y x ˆ Z (A) = tij (λ)Zj (dλ), i, j = 1, . . . , n , A ∈ B(G). (18) A

This relation can be translated to their spectral functions as:

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9 Random and Vector Multimeasures

µyy ij (A) =

Z

A

∗ ti (λ)µxx ij (dλ)tj (λ),

ti (λ) = (ti1 (λ), . . . , tin (λ).

(19)

Writing the matrix T (λ) = (tij (λ), i = 1, . . . , n, j = 1, . . . , n), (19) becomes Z yy T (λ)µxx (dλ)T ⋆ (λ), (20) µ (A) = A

the linear transform T = T xy represented as a function is called the spectral ′ ′ characteristic, and since (µxj (dλ)¯ µyk (dλ′ )) = µxx jk (dλ), if λ = λ , and = 0 if λ 6= λ for all j, k, the processes X, Y are stationarily correlated. With these conditions on T (·), the Kolmogorov-Rozanov characterization of such processes (i.e., random fields) can be given as: Theorem 9.3.5 Let X, Y be n- and m-dimensional stationary random fields that are also stationarily correlated. Then Y = T X, with T xy = (txy ij , 1 ≤ i ≤ n, 1 ≤ j ≤ m) as the spectral characteristic of a linear transformation T , if and only if their spectral measures µxij , µyij satisfy the relations (written in matrix form): µyy (dλ) = T xy (λ)µxx (dλ)(T xy (λ))⋆ ,

µyx (dλ) = T xy (λ)µxx (dλ).

(21)

Proof. By stationarity of X, we have as in (15) Z Z ⋆ ′ hg, λi dZ (λ), and if Yg = hg, λiT xy (λ)Z x (dλ). Xg = ˆ G

ˆ G

The latter integral is well-defined by hypothesis, since Z Z µyy (dλ) T xy (λ)µxx (dλ)(T xy (λ))⋆ = ˆ G

ˆ G

(22)

holds. On the other hand this implies (21) and Z ′ ′ ′ ′ ′ hgg ′−1 , λiµyjjy′ (dλ) = (Yj (g), Yj ′ (g ′ )). (Yj (g), Yj ′ (g )) = ˆ G

So the correspondence Yg ←→ Yg′ is isometric. But it is well-known (and we used it before) that every isometric operator in Hilbert space is necessarily linear. So Yg and Yg′ are indistinguishable. Thus Yg = Yg′ , a.e., g ∈ G may be identified. The converse is obtained by retracing the steps (and even from just juggling the definition). 2 Remark 9.1 Let X be a discrete stationary process (G = Z, integers) and Y be aPstationary process obtained linearly with a spectral characteristic P −iλn |C(n)| < ∞. Then we get T (λ) = ∞ C(n)e and n n=−∞ Z π ∞ X Y (n) = eiλn T (λ)Z(dλ) = C(n − m)Z(m), −π

m=−∞

which is known as a moving average process. Thus a linear transformation T , under appropriate specialized conditions, in (21), leads to useful and interesting applications.

9.3 Spectral Analysis of Second Order Fields

and Bimeasures

475

It was already seen that for general bounded linear mappings we automatically leave the (weakly) stationary family but will end up in the class of weakly harmonizable classes. This entails that the representation of its covariance B(·, ·) is given by Z Z ⋆ (23) hg1 , λ1 i(hg2 , λ2 i)µ(dλ1 , dλ2 ) B(g1 , g2 ) = (Xg1 , Xg2 ) = ˆ G

ˆ G

where µ(·, ·) is a positive definite bimeasure and the integral is generally in the (strict) Morse-Transue sense with µ(·, ·) having a finite Fr´echet variation. Even if µ had finite Vitali variation, so that the process is strongly harmonizable, the integral in (23) is not a product Lebesgue integral. So some well-known results with product measures will not hold. [See Rao (2004), Sec.6.3 on non cartesian product integrals for (different and) additional conditions for even specialized problems with representation of linear mappings, differentiation and the like.] Here we present a new approach to obtain analogs of Propositions 2 and 4 above, sometimes converting the problem from bimeasures to bilinear forms and back. It was shown in the preceding sections (cf., Theorems 1.5 and 1.6) that a bounded bilinear functional B : C0 (S) × C0 (S) → C can be represented (via Grothendieck’s inequality) as: Z 2 B(f, g) = (T f, g)L (µ) = (T f )¯ g dµ, f, g ∈ C0 (S), (24) S

relative to a probability measure µ on the Borel measurable space (S, S), S being a locally compact set and B(·, ·) a Hermitian positive definite functional, so that T is a positive linear operator. Moreover if B(·, ·) is strictly positive definite in the sense of M.H. Stone, then |B(f, g)| ≥ mkf k2,µ kgk2,µ, m > 0, and hence T of (24) is positive (T = V 2 ) and has a bounded inverse. Let B(f, g) = (V 2 f, g)L2 (µ) = (V f, V g)L2 (µ) , V = V ⋆ . The set V = {f : B(f, f ) < ∞} = {f : kV f k2,µ < ∞}(⊂ L2 (µ)) is called a spectral domain of the covariance bilinear functional B and , as seen in Theorem 1.6 above, is a closed subspace, of a Hilbert space in its norm kf k′2 = kV f k2,µ when equivalent classes of elements are identified. Note also that the measure µ here is related to Grothendieck’s inequality of the bilinear functional B(·, ·) and this is a key and nontrivial result. To extend the above statement for more general processes (and fields) we use the dilation property and work on the larger (or super) space and then relate it to the original formulation as given in the following sketch. Let {Xt , t ∈ G} ⊂ (L20 (P ))n ∼ = L20 (P, Cn ) be a vector harmonizable field and consider its stationary dilation {Yt , t ∈ G} ⊂ L20 (P˜ ), Cn ) where the latter ˜ be the orthogonal projection is the dilated (or super) Hilbert space. Let Q ˜ : L20 (P˜ , Cn ) → L20 (P, Cn ) such that Xt = QY ˜ t , t ∈ G, the LCA index group. Q If Z and Z˜ are the corresponding stochastic measures representing the X and Y fields, then for each t ∈ G one has

476

9 Random and Vector Multimeasures

Z

ˆ G

˜ t=Q ˜ ht, λi dZ(λ) = Xt = QY

Z

ˆ G

˜ ′ ), ht, λ′ i dZ(λ

t ∈ G,

(25)

with the corresponding bimeasures β(A, B) = (Z(A), Z(B))L20 (P,Ck ), and also ˜ ˜ ∩ B) = (Z(A), ˜ ˜ β(A, B) = β(A Z(B)) L20 (P˜ ,Ck ) , in accordance with the dilation result. Now with the Kolmogorov-Rozanov theorem (cf., Rozanov (1967), p.33) ˜ Ck ) of the there is an orthogonal projection Π on the “frequency” space L2 (β, 2 ˜ k L0 (P , C ) such that Z Z ′ ′ ˜ ′ ˜ ˜ Q Π (Iht, λi)(λ′ )Z(dλ ), t ∈ G (26) ht, λ iZ(dλ ) = ˆ G

ˆ G

where I is the identity n × n-matrix. If ft (λ′ ) = Π(Iht, ·i)(λ′ ) then (25) and (26) give the matrix bilinear form as Z Z ⋆ ht, λiβ(dλ, dλ′ )ht′ , λ′ i = r(t, t′ ) = (Xt , Xt′ ) = (QYt , QYt′ ) ˆ ˆ G G Z ˜ = ft (λ)β(dλ) f¯t⋆′ (λ′ ) (27) ˆ G

where β˜ is the hermitian n × n-matrix measure of the Yt -field and the left is the strict integral for the (matrix) bimeasure β of the Xt -field. We need to extend the left integral for “L20 (β, Ck )”-space, for which we have to recall the special concept with care! We should consider the left side [hence ˜ ˆ to the right side which is easier since β(·) is a (matrix) measure on β(G) replace ht, ·i by a trigonometric polynomial and with appropriate limits] as follows: Z Z f (λ)β(dλ, dλ′ )f ⋆ (λ′ ) lim n1 ,m1 →∞

(λ : f (λ)∈An1 )

(λ : f ⋆ (λ′ )∈Bm1 )

exists for all increasing Borel sets An1 , Bm1 ⊂ Cn and this will give the left side of (26) with the ht, ·i replaced by limits of trigonometric polynomials f (·) ˜ where Π(f ) is β-integrable. This implies (27) is valid in this generality so that we have Z Z ⋆ Z ⋆ (λ′ ) ˜ f (λ)β(dλ, dλ′ )g(λ′ )⋆ = (Πf )(λ)β(dλ)(Πg) (28) ˆ G

ˆ G

ˆ G

holding for all (matrix) trigonometric polynomials and their limits such as ˜ f (·) for which (Πf )(·) is β(·)-integrable. (This is an exact correspondence of Cram´er’s (1951, p.336) when β(·, ·) is a bimeasure of finite variation, and the general case is approached via MT-integrals.) The fact that trigonometric poly˜ and hence for β is needed. The positive definiteness of nomials are dense, for β(·) ˜ implies that the integrals in (28) are nonnegative Hermiβ(·, ·), and hence of β(·) R R⋆ tian matrices when f = g there. So ((f, g))β = trace ( Gˆ Gˆ f (λ)β(dλ, dλ′ )¯ g (λ⋆ ) 2 gives a semi-inner product on the left for L (β), the functions f for which

9.3 Spectral Analysis of Second Order Fields

and Bimeasures

477

((f, f )) < ∞ for the left of (28), and ((f, g))β˜ is similarly defined for the right side of (28). We can now discuss some consequences of this construction. 2 n Let HX = sp{X ¯ ¯ t , t ∈ G} ⊂ t , t ∈ G} ⊂ L0 (P, C ) and HY = sp{Y 2 ˜ k L0 (P , C ), be the closed linear spans so that they are Hilbert spaces. Also ˜ and extended the mapping τ1 : Yt → ht, ·i is an isometry from HY into L2 (β) linearly. Now the dilated field {Yt , t ∈ G} is stationary and it follows from the well known Rozanov’s (1967, p.32) account (cf. also Rosenberg (1964, p.297)) ˜ By the dilation theorem, Q(H ˜ Y ) = HX . These that τ1 is onto τ1 (HY ) = L2 (β). relations are important. From (27) , if τ2 : Xt → ht, ·i, then we have on taking traces Z Z ⋆ ′ ′ ⋆ ((Xs , Xt )) = tr(Xs , Xt ) = tr (r(s, t)) = tr hs, λiβ(dλ, dλ )ht, λ i ˆ G

ˆ G

= ((hs, ·i, ht, ·i)) = ((τ2 Xs , τ2 Xt )).

(29)

Extend τ2 linearly to all of HX , and denote it by the same symbol. Recalling the strictness condition discussed after (28), the trigonometric polynomials are in L2 (β) and also are dense in its norm. (If the strictness condition is not imposed this statement need not hold!) ˜ is an orthogonal projection, it follows Since from the dilation result Q ˜ is also a similar projection from (25) and (26) that its image Π on L2 (β) ˜ which as a range of Π is a closed suboperator and let M = Π(L2 (β)) R space. Moreover if h ∈ L2 (β), by (25) we see that Gˆ h(λ)Z(dλ) ∈ HX and R 2 ˜ n ˜ ˆ (Πh)(λ) dZ(λ) exists and is in L0 (P , C ). Also Πh ∈ M. Further using (29) G we have Z Z khk22,β = tr h(λ)Z(dλ) h(λ) dZ(λ), ˆ ˆ G ZG Z ˜ ˜ = tr (Πh)(λ) dZ(λ) (Πh)(λ)dZ(λ), =

ˆ G 2 kΠhk2,β˜,

ˆ G

(30)

so that the mapping τ : h 7→ Π(h), is an isometry τ (L2 (β)) ⊂ M. If the strictness is not used as discussed after (28), then the completion of L2 (β) is just contained in the function space M, and then its closure will have to be used as a spectral domain space of the random field. But with the strictness condition we now show that the embedding τ is onto so that L2 (β) itself must be complete, and hence is a Hilbert space. ˜ where β˜ is a regular (matrix) measure, and then Let h ∈ M ⊂ L2 (β) trigonometric polynomials (with matrix coefficients) are dense in the latter space. So for given ε > 0, we can find such a (trigonometric) polyno˜ satisfying kh − pn k ˜ < ε. Hence Π(pn ) ∈ M and, since mial pn ∈ L2 (β) 2,β h = Πh, kh − Π(pn )k2,β˜ = kΠ(h − pn )k2,β˜ ≤ kh − pn k2,β˜ < ε. But by the 2 Rstrictness hypothesis the trigonometric polynomial pn ∈ L (β) above, and since ˜ ˆ pn (λ) dZ(λ) ∈ HX = QHY , we have G

478

9 Random and Vector Multimeasures

Z

ˆ G

˜ pn (λ) dZ(λ) = Q

Z

ˆ G

Z ˜ ˜ pn (λ) dZ(λ) = (Πpn )(λ) dZ(λ). ˆ G

(31)

˜ Now pn → h in β-norm, and pointwise for a subsequence, so we can take limits in (31), for the Dunford-Schwartz integrals, and interchange the limit and integral. R R If follows that Gˆ h dZ = Q Gˆ g dZ˜ whence h is strictly β-integrable. This gives us Z Z Z ⋆ Z ′ ¯⋆ ′ h dZ h dZ, h(λ)β(dλ, dλ )h (λ ) = ˆ G

2

ˆ G

ˆ G

2

ˆ G 2

so that h ∈ L (β) and τ (L (β)) = M. It follows that L (β) is complete. We may summarize the above analysis as the following:

Theorem 9.3.6 Let {Xt , t ∈ G} ⊂ L20 (P, Cn ) be a weakly or strongly harmonizable random field, G being an LCA group. Then its spectral domain L2 (β) defined as the space of strictly MT-square integrable complex (matrix) functions, is complete for the resulting inner product ((·, ·))2,L2 (β) following the definition of (28). If the strictness hypothesis is relaxed, then it is an inner product space whose completion will then also be a function space. It is to be noted that even if β(·, ·) has finite variation so that the process is strongly harmonizable, then also the bimeasure integral has to be defined separately without which L2 (β) need not be complete. The strictness condition is a kind of regularity of the bimeasures since they are not product measures in general. When G = Z the above result was discussed in Mehlman (1991 and 1992). Under a stronger (Lebesgue-Bochner type) condition on the bimeasure integrals even when they are valued in a Banach space (not necessarily of finite dimension as we assumed here), a detailed analysis of the bimeasure functional spaces are considered by Dinculeanu and Muthiah (2000). (A special case (scalar version) is in Exercise 1.) This is of interest since it may also be used in generalizing the analysis for stochastic integration and some other results. An extension of the preceding analysis to Cram´er and Karhunen processes and fields can be given. Here the “kernels” f (t, ·) instead of ht, ·i (the characters of G) should be suitably restricted to ensure the “regularity” properties of the bimeasures since the index set is not a commutative group or a semi-group. But this can evidently be done. It will not be discussed here, the precise details are not yet available, and also the point of view of this work is different. In the next section we discuss some problems with multi- (or poly-) measures that will extend the bimeasure analysis.

9.4 Multimeasures and Multilinear Forms The preceding analysis generalized for multi or polymeasures and related multilinear forms are included here to round up the present discussion. Integral

9.4 Multimeasures and Multilinear Forms

479

representation of a multilinear operator is also considered as it constitutes an extension of the classical Riesz representation. It was seen early in Section 2.4 that some important applications of random measures whose products, having the required moments, lead to multimeasures of the following type. Let Z : S0 → Lp (P ), p ≥ k ≥ 2, be a random measure and consider β(A1 , . . . , Ak ) = E(Z(A1 ) · · · Z(Ak )),

A1 , . . . , Ak ∈ S0 .

(1)

Since Z(·) is σ-additive and k th product moment is assumed to exist, it is clear that B(A1 , . . . , Aj−1 , ·, Aj+1 , . . . , Ak ) is a signed measure for each j = 2, . . . , k − 1. Such a β(·, . . . , ·) is called a multi- (or poly-) measure. If k = 1, then this is a signed measure so that it is bounded, and if k = 2 we have a bimeasure. In this case (1) defines a positive definite set function. Moreover sup{|β(A, B)| : A, B ∈ S0 } is seen to be bounded, but this is already We conclude, with our earlier work, that B(f, g) = R R a nontrivial result. ′ ′ S S f (λ)g(λ )β(dλ, dλ ) is a continuous bilinear form and its boundedness follows from the Grothendieck theorem, which is after all a nontrivial result. What can one say about the multimeasure β(·, · · · , ·) of (1)? The relevant aspects answering this direct question, leading to multilinear forms, will now be our first object to analyze. For this, it is desirable to restate the Vitali and Fr´echet variations from bimeasures to multimeasures, and then discuss their use in the present analysis. Definition 9.4.1 Let (Si , Si ), i = 1, . . . , n be measurable spaces and β : S1 × · · · × Sn → C (or R) be such that β(A1 , . . . , Aj−1 , ·, Aj+1 , . . . , An ) is σadditive for j = 2, . . . , n − 1, for each fixed (A1 , . . . , Aj−1 , Aj+1 , . . . , An ). It is termed a multimeasure. Other names include polymeasure and pseudomeasure. This is an abstractly stated concept, not necessarily obtained as in (1). Then β (i) has finite Vitali variation if ( N X |β(Ai1 1 , . . . , Ain n )| : Aij k ∈ S; disjoint, k = 1, . . . , nj , |β| = sup i1 ,...,in =1

j = 1, . . . , n, N > 0

)

< ∞,

and (ii) has finite Fr´echet variation if ( N X β(Ai1 1 , . . . , Ain n )ai1 · · · ain : |aij | ≤ 1, ai ∈ C, kβk = sup i1 ,...,in =1 ) Aij k ∈ S, disjoint, k = 1, . . . , nj , j = 1, . . . , n, N > 0

< ∞.

(2)

(3)

480

9 Random and Vector Multimeasures

It is clear that kβk ≤ |β| ≤ ∞. Only |β| ≤ ∞ can be asserted as the bimeasure case shows (although for n = 1, both variations (2) and (3) are equal and finite), and the inequality prevails. The relation between these variations in the multimeasure case is similar to that of bimeasures result as seen from the following: Theorem 9.4.2 The multimeasure β : S1 × · · · × Sn → C always satisfies kβk < ∞, so its Fr´echet variation is finite, whereas the Vitali variation can be infinite. Thus, as in the case of bimeasures, the Fr´echet variation of a multimeasure β on the product σ-algebra S1 ⊗ · · · ⊗ Sn is finite. Sketch of Proof. If n = 1, this is a classical result asserting that a signed measure is bounded, and if n = 2, the case of bimeasures, the result was established in Section 2.2 where it is obtained from a random measure. In the general case the argument once again starts with n = 2. For a bimeasure ˜ = β(A, ·) for β : S1 × S2 → C, consider the vector measure β˜ defined as β(·) any fixed A ∈ S1 , as an element of X = ca(S, S), the Banach space of scalar measures with the total variation norm k · k. But then since a vector mea˜ sure is bounded (Dunford-Schwartz (1958), IV.10.4), so that sup{kβ(A, ·)k : ˜ A ∈ S1 } < ∞. Now kβ(A, ·)k = variation of the scalar measure β(A, ·), and sup{|β(A, B)| : B ∈ S2 } = 4KA < ∞. Hence sup{|β(A, B)| : A ∈ ∞ by the classical S1 , B ∈ S2 } ≤ 4 supA∈S1 KA = 4C < ∞, sup KA < P n analysis (Dunford-Schwartz (1958), III 1.5.). Let µn (·) = i=1 ai β(Ai , ·). But n this is a scalar measure and so for a partition S = ∪j=1 Bj , and ai ∈ C, |ai | ≤ 1, n

X

ai β(Ai , ·) kµn (·)k = i=1

m m o n X [ ¯bj µn (Bj ) : |bj | ≤ 1, Bj = S 2 = sup j=1

j=1

m n X m o n X [ as above . ai¯bj β(Ai , Bj ) : |bi | ≤ 1, bj ∈ C, = sup i6=1 j=1

j=1

The supremum of the left side as Ai vary forming a partition of S1 with |ai | ≤ 1, this gives kβk(S1 , S2 ) which is finite by the above quoted result. But the right side (the last expression) with these suprema is the Fr´echet variation of β on S1 × S2 . This shows that the result holds for bimeasures. For multimeasures, we use induction assuming it for n − 1, and obtain the result for n. This is not entirely simple, but not difficult, needing a careful application of the Fr´echet variation as given in (3). We leave it to the reader to fill in the details. 2 Remark 9.2 An indirect argument, again using induction and a procedure of establishing the boundedness of scalar measures, which is presented nicely in

9.4 Multimeasures and Multilinear Forms

481

Rudin ((1974), Theorem 6.4), is extended by Blei ((1985), Theorem 4.3) in establishing the above result. The reader may consider that procedure which is also interesting and gives an alternative perspective. We next consider the multilinear forms and their integral representations on C0 (S), the space of continuous scalar functions vanishing at infinity on S, a locally compact space, as an application of the above result and its place in the random measure analysis. Recall the classical Riesz representation theorem for continuous linear mappings T : C0 (S) → X, a Banach space. Thus one has the characterization: Z f (s) dµ(s), f ∈ C0 (S) (4) Tf = S

where kT k = sup{kT f kX : kf k ≤ 1} is given by kT k = kµk(S) with kµk(S) as the semivariation of the vector measure µ : S → X if and only if T is a weakly compact operator, (cf., Dunford-Schwartz (1958, p.493)) with a unique µ as a “regular” σ-additive measure. In the above reference this result is proved if S is compact, but it holds as stated here with a simple modification, noted with necessary details in my paper (cf., Rao (1982), p.136). Here this will be used for a locally compact space S. It may be noted that, in the classical Riesz theorem, X = R or C and this extends to reflexive Banach spaces X without additional conditions, but in general µ : S → X⋆⋆ when T is not weakly compact (or X does have an isometric copy of c0 , the space of sequences converging to zero under the uniform norm and this is a purely technical condition). Here we present a multilinear version of the Riesz theorem relative to a multi- (or pseudo- or poly-) measure, after recalling the respective integrals and the consequent multilinear forms, to indicate the point of view of Dobrakov’s work. The bimeasure integration considered can be extended to the multimeasures case as follows. If f1 , . . . , fn are scalar bounded and measurable maps on (Si , Si ), i = 1, . . . , n, let µ(·, . . . , ·) be a multimeasure and Ai ∈ Si (each Si is a locally compact space with Si determined by its Borel subsets). Consider the (Lebesgue) integral of f1 relative to µ(·, A1 , . . . , An ). Put Z f1 (s)µ(ds, A2 , . . . , An ) (5) µf1 (A2 , . . . , An ) = S1

which is well-defined and satisfies exactly the same conditions as in the bimeasure case. Then: µf1 (·, . . . , ·) : S2 × · · · × Sn → C is an (n − 1) multimeasure and kµf1 k(n−1) ≤ kf1 k∞ kµkn

(6)

where k · k(n−1) is the Fr´erchet variation of µf1 and k · kn that of µ (perhaps kµk(n) is more symmetrical). This is verified as in the bimeasure case and the fact that kµkn−1 is dominated by kµkn . From this using induction (with some care) one can verify the following:

482

9 Random and Vector Multimeasures

Proposition 9.4.3 Let fi : Si → C, i = 1, . . . , n be bounded and Si -measurable on (Si , Si ). If µ : S1 × · · · × Sn → C is a multimeasure, then the following multilinear integral of (f1 , . . . , fn ) is well-defined and the result is a multilinear form for which one has the bound: Z Z Qn (f1 (s), . . . , fn (s))dµ(s1 . . . . , sn ) ≤ ( i=1 kfi k∞ )kµk (7) ··· S1

Sn

where kµk is the Fr´echet variation of µ on S1 × · · · × Sn . The same bound holds if fi and (Si , Si ) are permuted in all ways.

Some details of this construction may be found in Blei(1985) who also studies an extension of this for “fractional dimensions”. There n is replaced by a n , defining and analyzing it needing some new ideas. rational m An extension of multilinear forms can also be given using (5) if their range is a Banach space X which is general (and not necessarily reflexive). But then the form must be weakly compact (or X has no isometric copy of c0 ). We present the result in the following form which has a somewhat more difficult converse part: Theorem 9.4.4 Let (Si , Si ), i = 1, . . . , n, be Borelian measurable spaces where each Si is locally compact. If C0 (Si ) is the space of continuous scalar functions vanishing at infinity, and T : ×ni=1 C0 (Si ) → X is an n-linear form which is weakly compact (hence bounded), then there exists a unique ‘regular’ multimeasure µ : ×ni=1 S1 → X such that Z Z (f1 (s1 ), . . . , fn (sn ))µ(ds1 , . . . , dsn ), fi ∈ C0 (Si ) ··· T (f1 , . . . , fn ) = S1

Sn

(8) for which kT kX = kµk where kµk is the Fr´echet variation of µ, and kT kX = sup{kT (f1, . . . , fn )kX : kfi k∞ ≤ 1, 1 ≤ i ≤ n}. The representation is unique. On the other hand the multiple integral of (8) always defines a multilinear form T. [The same holds if T is a bounded multilinear from on ×ni=1 C0 (Si ) → X if X again does not have a copy of c0 isometrically.] This is a straightforward extension of the classical Riesz theorem given for Banach spaces as in Dunford-Schwartz (1958, VI.7.3) noted above, but the nontrivial details are to be filled in carefully. A complete discussion of it is also given by Dobrakov and Panchapagesan (1995, Theorem 4.1). These will be omitted here. If X = R (or C), this is sketched in Blei (1985, p.55) who then extended the work for “fractional dimensions”. It needs some additional analysis. Application of the above result to random measures and multiple stochastic integrals, generalizing some work of Chapter 7, are also possible, but the details are not yet available. In some aspects of stochastic analysis, the Si , i = 1, . . . , n arise as indexing sets of random fields {Xs , s ∈ S} ⊂ Lk0 (P ), 2 ≤ k ≤ p so that the k th moment is given by

9.4 Multimeasures and Multilinear Forms

β(s1 , . . . , sk ) = E(Xs1 · · · Xsk ) =

Z

Ω

(k)

Z

Ω

483

(Xs1 (ω) · · · Xsk (ω))dP (ω), si ∈ S,

(9) and the function β(·, . . . , ·) is a k th order moment and its structure is of interest (if k = 2, it becomes a covariance function discussed before). Taking fi = χAi , Ai ∈ B(Si ), in (9) after extending it to B0 (Si ), the space of bounded Borel functions vanishing at infinity, in the usual way we get T (χA1 , · · · , χAn ) = µ(A1 , · · · , An ), T is uniquely determined by the multimeasure µ and conversely. This implies that there is an exact relation between multilinear functionals on ×ni=1 B(Si ) and multimeasures on ×ni=1 B(Si ) so that if n = 2, T˜(·, ·) is a bilinear functional, µ ˜ (·, ·) is a bimeasure (T˜ = T |B0 (S1 ) × B(S2 ), and µ ˜ is similarly identified). But in applications to stochastic analysis, one starts with the stationary or harmonizable random families which assume that the indexing set S to be a locally compact group or at least a semi-group, and the analysis is based on second moments which bring in the geometry of the Hilbert space as seen in the preceding chapters. This leads to noncommutative harmonic analysis. Here a brief indication of the possible directions of developments will be sketched which is restricted further to separable unimodular locally compact groups. A few facts from abstract harmonic analysis are recalled as needed for this discussion, following Mautner (1955) and others (all references and background are in the author’s paper Rao (1989)), to show what is really involved in such an extension. For a group G, the set of scalar multiplicative functions called characters, is again a group with identity function 1 and f −1 = f¯, the complex ˆ conjugate. Thus if G = Rn then the space of characters denoted G(= Rn ) it·x is the set of functions {e , t ∈ G, x ∈ G} and if G = [0, 2π] (addition ˆ can be identified with Z, all the integers. If G is an LCA mod 2π) this G ˆ is also an LCA group (generalizing the preceding two examgroup, then G ples), but for locally compact nonabelian groups G, there may be no nontrivial (i.e., other than the constant 1) characters, and the theory of such nonabelian groups gets more complicated, and yet we need some aspects of that theory for an analysis of random fields valued in spaces Lp (P ). Here we outline a few facts for G that is separable, locally compact and whose right and left invariant measures are the same and unimodular which still generalize the case G = Rn (or [0, 2π]n ) and many others. The characters {eit·λ , t ∈ Rn } are now replaced by (C goes into a Hilbert space H) unitary mappings {Ug , g ∈ G} into an algebra A ⊂ B(H), the space of all bounded linear mappings on H, which has many topologies other than the norm (or uniform) one. These mappings are called representations of G into H. We now include a sketch of what is needed, referring the reader for details to the two volume work by Hewitt and Ross (1963, 1970). This gives a better view of the subject. An LC group G is called type I if the weakly closed self adjoint algebra A(i.e. with each of its elements, their adjoints are also included) generated by {Ug , g ∈ G}) contained in B(H)

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9 Random and Vector Multimeasures

has the property that its commutator A′ is abelian. (Here A′ stands for the algebra from B(H) each of whose elements commutes with all members of A.) It is known that each separable unimodular group G is of type I (or type II) having the following properties that are of interest in our application. ˜ be the set of all irreducible (strongly continuous) unitary represen(i) Let G tations of an LC group G into a Hilbert space H (irreducible T means that for ˜ an operator on H, only {0} and H are invariant subspaces), then one T ∈ G, ˜ relative to which it becomes a locally compact can introduce a topology in G Hausdorff space. If µ is a Haar measure on G then there exists a unique regular ˜ such that (G, ˜ µ (= Radon) measure µ ˜ on G ˜) becomes a dual object (or gauge) and(the most important one here) a Plancherel formula (given in (iii) below) holds for it. (ii) The representation space H = L2 (G, dµ) can be taken where µ is a Haar L measure of the unimodular G, and H may be expressed as a direct sum ˜ H = ˜ Hy where Hy is a representations space for each y ∈ G and one y∈G has Z ⊕ Hy d˜ µ(y), (direct integral) (10) L2 (G, µ) = ˜ G

−1

and if (La f )(x) = f (a x), x ∈ G, then letting A be the weakly closed self adjoint algebra generated by {La , a ∈ G} ⊂ B(H), one has a direct sum decomposition of A by Ay similarly defined by {Uy (g), g ∈ G}y∈G˜ and one has for each f ∈ L1 (G, µ) ∩ L2 (G, µ), the following integral: Z ˜ Uy (g)f (g)dµ(g), Uy (g) ∈ Ay , y ∈ G, (11) fˆ(y) = G

which defines a bounded mapping on H. Also fˆ may be extended uniquely to a dense subspace containing L1 (G, µ) ∩ L2 (G, µ) in H and the resulting function again denoted fˆ, is the (generalized) Fourier transform of f. Using this new definition we have the following key property: (iii) There is a linear functional called trace on each Ay denoted by τy : Ay → C, which is positive, normal (i.e., τy (Bα ) ↑ τy (B) for B, Bα ∈ Ay , 0 ≤ Bα ↑ B and α in a directed indexing), faithful (τy (B) > 0 if B 6= 0), and semi-finite (extending σ-finiteness), in terms of which the following form of the Plancherel formula is valid (note that τy is a trace which is a linear functional on Ay and is defined for each y preserving measurability in y): Z Z τy (fˆ1 )(y)fˆ2⋆ (y)˜ µ(dy), f1 , f2 ∈ L2 (G, µ), (12) f1 (g)f2 (g) dµ(g) = G

˜ G

and fˆi⋆ is the adjoint of the operator function fˆi given by (11). This is an essential and important result that is needed here, taken from Mautner (1955). A different argument is also found in Segal (1950). The preceding discussion enables us to introduce the following:

9.4 Multimeasures and Multilinear Forms

485

Definition 9.4.5 A random field {Xg , g ∈ G} ⊂ L20 (P ), G being a locally compact separable unimodular group, is weakly harmonizable if it is weakly continuous (as in earlier treatment) and the set: nZ o Xg f (g) dµ(g) : kfˆk∞ = sup kfˆ(y)k ≤ 1, f ∈ (L1 ∩ L2 )(G, µ) (13) ˜ y∈G

G

is bounded in L20 (P ), fˆ being the (generalized) Fourier transform of f . All the above long-winded discussion is needed to introduce the concept of harmonizability using a form of the V -boundedness implied by (13). With this an integral representation of the field {Xg , g ∈ G} can be given as follows: Theorem 9.4.6 Let {Xg , g ∈ G} ⊂ L20 (P ) be a weakly harmonizable random field with G as a separable unimodular (locally compact) type I group. Then there exists a weakly σ-additive regular operator valued (stochastic) measure ˜ the dual object of G, having a regular (Plancherel) M(·) on Borel sets of G measure µ ˜ such that ( M(·) being the random spectral measure of Xg on ˆ → B(B(H), L2 (P )) B(G) 0 Z Xg = τy (ug (y)M(dy)), g ∈ G, (14) ˜ G

where {ug (·), g ∈ G} is the representation as in (11) and τy (·) is a trace functional (a linear, normal, faithful mapping as in (12)) so that X(·) is continuous in the (strong) or norm topology of L20 (P ). Conversely, a weakly continuous X : g 7→ Xg defined by (14) is weakly harmonizable. Moreover, the random field defined by (14) has the covariance function r(·, ·) given by: Z Z (15) τy1 ⊗ τy2 {(ug1 (y1 ) ⊗ ug2 (y2 ))β(dy1 , dy2 )}, r(g1 , g2 ) = ˜ G

˜ G

˜ × B(G), ˜ and B(G), ˜ the where β(·, ·) is an operator valued bimeasure on B(G) ˜ Borel σ-algebra of the dual space G of G. Outline of Proof. The argument will be outlined using several results from abstract harmonic analysis to indicate how advanced tools are needed. For f ∈ L1 (G) ∩ L2 (G), by (11) fˆ is defined and is a measurable operator function. Its boundedness follows from considering H as that given by the R⊕ ‘direct sum’ H = G˜ Hy µ ˜(dy), embedding Hy in H, treating Hy as a closed subspace. Then uy (g) = u(g, y) from B(Hy ) can be extended as µ ˜(g, y) to be u(g, y) on Hy and as identity on Hy⊥ so that {˜ u(g, y), g ∈ G} is a set of unitary operators in B(H). If in (11), u is replaced by u ˜ the resulting operator is mea˜ surable and will be denoted fˆ again. Also A(H) = {fˆ : fˆ(y) ∈ B(Hy ), y ∈ G} ˆ can be regarded as a subalgebra of B(H). If T : f 7→ f , then T is one-to-one

486

9 Random and Vector Multimeasures

from the Fourier analysis discussed earlier, and is norm nonincreasing which follows from:

Z

kfˆ(y)kop = f (g)˜ u(g, y) dµ(g) (k · kop is operator norm), op Z G ≤ |f (g)| k˜ u(g, y)kop dµ(g), by a property of such integrals, ZG |f (g)| dµ(g) = kf k1 , since u ˜(g, y) is unitary. ≤ G

ˆ ˜ Here sup{kf(y)k op : y ∈ G} ≤ kf k1 < ∞ is used. Since X(·) is weakly harmonizable, Z (16) f (g)Xg dµ(g) ∈ L20 (P ), f ∈ L1 (G, µ) ∩ L2 (G, µ), T1 (f ) = G

so that T1 is bounded and T˜ = T1 ◦ T −1 gives T˜(fˆ) = T1 (f ) for all f ∈ L1 (G, µ) ∩ L2 (G, µ) and kT˜(fˆ)k2 ≤ Ckf k2 , for some constant C > 0. By the general theorem discussed earlier, T˜ can be expressed as a direct sum of opera˜ Since the range space of the bounded T˜ is tors from A(Hy ) into L20 (P ), y ∈ G. reflexive, the operator is weakly compact, and it can be given an integral representation, by the Riesz-Markov theorem (a specialization of Theorem 4 above). The separability of G and other conditions of the hypotheses are needed here (cf. also Na˘ımark (1964), Chapter 8, Sec.4)), and there exists a regular weakly ˜ into the space B(A(H), L20 (P )) such σ-additive operator measure M on B(G) that Z ˜ ˆ τy (fˆ(y)M(dy)), fˆ ∈ A(H), (17) T (f ) = ˜ G

˜ → L20 (P ) is σ- addiwhere τy is a (linear) trace on A(Hy ), and M(·)X : B(G) tive and regular for each X ∈ A(H)[M(·) itself need not be σ-additive in the uniform topology]. From this analysis we have Z

Z

τy (fˆ(g)M(dy)) hZ i τy f (g)˜ u(g, y) dµ(g) M(dy) = ˜ G ZG Z τy [˜ u(g, y)M(dy)] dµ(g) = f (g)

f (g)Xg dµ(g) =

˜

ZG

G

G

˜ G

since f is scalar and τy is linear commuting with the integral over G, and a form of the Fubini type reasoning applies. This gives Z Z τy [˜ u(g, y)M(dy)] dµ(g) = 0. (18) f (g) Xg − G

˜ G

9.4 Multimeasures and Multilinear Forms

487

Since f ∈ L1 (G, µ) ∩ L2 (G, µ) is arbitrary and the set is dense in L1 (G, µ) (as well as in L2 (G, µ)) the (continuous) function inside the parenthesis must ˜ by U which is possible, from (18) we obtain (14). vanish. Now replacing U For the converse, let (15) hold and if f ∈ L1 (G, µ) ∩ L2 (G, µ), then one can use a standard argument to get (using the linearity of the trace τy ): Z Z hZ i f (g) Xg f (g) dµ(g) = τy (u(g, y)M(dy) dµ(g) ˜ G ZG Z G f (g)u(g, y)dµ(g) M(dy) τy = ˜ G ZG ˆ τy [f (g)M(dy)]. = ˜ G

From this we can deduce

Z

˜ Xg f (g) dµ(g) ≤ kfˆkop kMk(G)

(19)

G

˜ < ∞ in (19) where kMk(·) is the semi-variation of M and if we set C = kMk(G) it follows from Definition 5 above that {Xg , g ∈ G} is weakly harmonizable, and is also weakly continuous. Finally (15) can be obtained on using a generalization of the MT-integrals employed before, to the vector case in the following way: r(g1 , g2 ) = (Xg1 , Xg2 )L20 (P ) Z Z ⋆ = u(g2 , y2 )M(dy2 )) τy2 (˜ u(g1 , y1 )M(dy1 )), τy1 (˜ ˜ ˜ G Z G Z u(g1 , y1 ) ⊗ (˜ u(g2 , y2 ))⋆ }E(M(dy1 ) ⊗ M⋆ (dy2 )) τy1 ⊗ τy2 {(˜ = ˜ G

˜ G

where E(·) is the expectation in L20 (P ), Z Z u(g1 , y1 ) ⊗ u˜(g2 , y2 )⋆ ]β(dy1 , dy2 ), = τy1 ⊗ τy2 [˜ ˜ G

˜ G

β(·, ·)being the operator valued positive definite bimeasure, Z Z u(g1 , y1 )β(dy1 , dy2 )˜ u(g2 , y2 )⋆ ]. τy1 ⊗ τy2 [˜ = ˜ G

˜ G

This is (15) in a different form.

2

Remark 9.3 The preceding result for type I groups G (essentially the same as those admitted for Theorem 6) was considered by Yaglom (1960) for stationary random fields which simplifies the discussion, and the corresponding MT-integration can be avoided. Details and some serious applications, in the stationary case, were outlined by him in the interesting work published in his Berkeley Symposium paper (1961).

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9 Random and Vector Multimeasures

Remark 9.4 The measures M(·) of representation (13) need not be σ-additive in the uniform operator topology. However in the case of LCA groups G, this problem does not arise since then H = C and the weak and strong σ-additivities coincide by a classical theorem due to Pettis. Remark 9.5 If G is not separable, then properties of type I group and the related work cannot be invoked, and the methods with general C ⋆ -algebras have to be employed, as was done by Ylinen (1975). However our separable groups are known to cover all type I, LCA, compact as well as semisimple and algebraic Lie groups (cf., Hewitt and Ross (1970) and Yaglom (1960)). The preceding work is included not only to show the great potential of these considerations that employ some deep parts of harmonic analysis, but have interesting applications related to several problems. We conclude this chapter with one such application to a linear filter question, which deals with a k-dimensional random field. Let G be an LCA group and X, Y : G → L20 (P, Ck ) be the Hilbert space of k-vector random fields. Suppose τh is a translation (or shift) operator so that (τh X)(g) = X(g + h). A linear mapping Λ acting on these operator fields is called a filter if τh (ΛX)(g) = Λ(τh X)(g) for all g, h ∈ G. An equation of the form (ΛX)(g) = Y (g), g ∈ G (20) is called a filter equation where X(·) is an “input” and Y (·) “output”. The problem is to find conditions on the filter Λ in order that, for a given output Y -process (or field), a unique input field X may be found where Λ can be, for instance, an integro-differential operator (so may be unbounded). The problem was formulated and solved by Bochner (1956) if Y is desired to be a stationary process when X is stationary and Λ is an integral operator, under some reasonable conditions. Here we present an analogous result if the Y and X fields are, weakly harmonizable to indicate some possibilities of the types of problems under the filter equations, if (ΛX)(g) = Y (g) (or ΛX = Y formally) when G is an LCA group. Theorem 9.4.7 Consider the linear (integral) filter equation: Z A(s)X(g − s) ds, Y (g) = (ΛX)(g) =

(21)

G

where ‘ds’ is a Haar measure on the LCA group G,and X, Y are Ck -valued random fields 1 ≤ k ≤ ∞ with βy (·, ·) as the k ×k matrix of ds-integrable functions. Then there is a weakly harmonizable input X if and only if the following two conditions hold: R R⋆ (i) D D (I − F F −1 )(λ)βy (dλ, dλ′ )(I − F F −1 )⋆ (λ′ ) = 0, R R⋆ (ii) Gˆ Gˆ F (λ)−1 βy (dλ, dλ′ )(F (λ′ )−1 )⋆ exists,

9.5 Complements and Exercises

489

ˆ is an arbitrary Borel set, F = A, ˜ is the Fourier transform of A, where D ⊂ G [also termed the spectral characteristic of the filter Λ], F −1 is the “generalized inverse” of the matrix F (·), and ‘⋆ ’ denote the adjoint operation of the matrix, the integrals being in the (strict) MT-sense. When these conditions are met, the input can be given by: Z hg, λiF −1 (λ)Zy (dλ), g ∈ G, (22) Xg = ˆ G

ˆ → L20 (P, Ck ) is the random measure representing Y. The soluwhere Zy : B(G) tion will be unique whenever F (λ) is nonsingular for all λ ∈ G. This result shows how concrete applications of the general theory demand further detailed, and often advanced tools. The above theorem is proved in detail if G = Rn and further illustrated if F is specialized with particular A’s in (Chang and Rao (1986), pp.91-95) and the extension to the LCA group G can be obtained from it without much difficulty. The details will be omitted here. Many other aspects of these results can be studied. A few adjuncts will be given in the final complements section. Thus this chapter and the book will be concluded at this point.

9.5 Complements and Exercises 1. In Definition 1.3 introducing the Morse-Transue integral, we have formulated three conditions for the MT integral [for a bimeasure with finite Fr´echet variation), to exist and also saw in Example 1.4 that only the first two conditions do not imply the third. If we restrict the class of functions then the first two conditions determine the third. We indicate this strengthened formulation now which is discussed by Dinculeanu and Muthiah (2000): Let β : S1 × S2 → C be a bimeasure of finite Vitali variation, i.e.,   n X  |β|(S1 , S2 ) = sup |β(Ai , Bj )| : Ai ∈ S1 , Bj ∈ S2 , disjoint, n ≥ 1   i,j=1

< ∞.

¯ + , is again a bimeasure where the Si are (i) Verify that |β| : S1 × S2 → R σ-rings of Si , i = 1, 2 (i.e., |β|(·, S1 ), |β|(S1 , ·) are σ-additive). Moreover if β1 (·) = β(·, S2 ) and β2 (·) = β(S1 , ·), then verify that |β1 |(A) ≤ |β|(A, S2 ) and |β2 |(B) ≤ |β|(S1 , B). (ii) Let fi : Si → C, i = 1, 2 be Si -measurable and β : S1 × S2 → C be a bimeasure of finite Vitali variation. We say that (f1 , f2 ) is β-integrable if (a) fi is |βi | integrable where βi (·) is the positive finite measure defined as in the above part. One says that f1 is |β|f2 and (b) f2 is |β|f1 integrable using the notation of Definition 1.3 for these variation measures. Verify

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9 Random and Vector Multimeasures

that the resulting integrals are equal so that (f1 , f2 ) is |β|-integrable. This means the conditions (i) and (ii) of Definition 1.3 for this strengthened hypothesis imply that the resulting integrals are equal. [Even with this stringent condition, the work is not very simple. Note that this stronger condition is satisfied for strongly (but not weakly) harmonizable processes. A vector valued version of this result is also detailed in Dinculeanu-Muthiah (2000)]. 2(a) If H : S → L20 (P ) × L20 (P ) is a Hermitian nonnegative definite n × n matrix of scalar σ-additive functions on the σ-algebra S of the Borel sets of S(= [0, 1] for the example below), let ϕ : S → (L20 (P )) be a p × n matrix and ψ : S → L20 (P ) beR a q × n-matrix each of whose elements is measurable and bounded. Define S ϕ · dH · ψ ⋆ element-wise to obtain a p × q-matrix of integrals using the Lebesgue-Stieltjes definition. Verify that this gives a matrix integral which is linear and for p = q, using the trace functional, get a (semi-) norm. The thus obtained space of matrix functions of p × n-order is linear and normed, but not complete. The last property is verified by the following example. Let S = (0, 1), p = 1, q = 2, and a sequence ϕP n , ψn be n given by : ψk ( k1 ) = (−k, k), and ψk (s) = 0 if s 6= k1 . Let ϕn = k=1 ψk . So ϕn is 1 × 2 (i.e., a row), ψn is 2 × 2 matrix. Now take H(x) to be 1 1 where these vector functions are nonvanishing only H( k1 ) = k12 1 1 + k −2 on the countable set (1, 12 , 13 , . . .) of S. RVerify that {ϕn , n ≥ 1} is a Cauchy sequence in the norm given by kϕk22 = S ϕ · dH · ϕ⋆ . But the limit of this sequence does not belong to the space L2 (H) so that it is not complete when the coordinate-wise definition of norm is used. Note that {ϕn , n ≥ 1} is even a ‘simple type’ Cauchy sequence and it has no limit in the space! Thus a more subtle definition of norm as given in the text is needed. (See Rosenberg (1964) on this example.) (b) In our definition of (L2 (µ), ((·, ·))µ ) of Theorem 3.3 above, the associated norm of a measurable function B(·) is given by Z Z 1 1 kBk2H = tr B(λ) dµ(λ)B ⋆ (λ) = tr (B(λ)µ′ (λ) 2 )(B(λ)µ′ (λ) 2 )⋆ dν ˆ G

ˆ G

where ν is the trace of µ [or one can use any other dominating (positive) mea1 ym derivasure, with (µ′ (·)) 2 being the positive square root of the Radon-Nikod´ 1 tive µ′ (·) relative to ν]. This may not be of full rank. But kBkH = kBµ′ 2 k2,ν ,and the right side is the standard Bochner-norm of L2 (H, ν). Then the Cauchy sequence of {Bn , n ≥ 1} in k · kH -norm is equivalent to that in k · k2,ν and H

1

1

1

Bn → B, i.e., Bn (µ′ ) 2 → B(µ′ ) 2 in L2 (ν). Since (µ′ ) 2 need not be invertible we need an additional argument with the generalized inverse of µ′ . We sketch it as follows. In the finite dimensional case, an operator is essentially a matrix, say A(n × m) and its generalized inverse matrix G is an m × n matrix (by definition) that satisfies the conditions AGA = A, GAG =

9.5 Complements and Exercises

491

G, (AG)′ = AG and (GA)′ = GA where prime ‘′ ’ denotes matrix transposition. It is known that for any such A, of rank r, there is a unique matrix G also of rank r. Note that AG is an orthogonal projection. Moreover since A is of rank r, it can be expressed as A = BC where B is n × r and C is r × m matrices each of rank r and from this one can show that the generalized inverse G of A is computable as G = C ′ (B ′ B)−1 B ′ . [An elementary treatment, including the derivation of these facts and a few applications are in Chipman and Rao (1964). If A is an n × m matrix of measurable scalar functions, then it follows from the above that the generalized inverse also consists of measurable functions and the same holds for continuous functions of A as well as of G.] To apply this 1 ˜ in L2 (ν). Is there idea, let Bn → B in H which means that Bn (µ′ ) 2 → B 1 ˜ in the above? If ((µ′ ) 12 )− is the ¯ such that B(µ ¯ ′ ) 2 is B an operator function B 1 ¯ = B·((µ′ ) 12 )− , with the generalized generalized inverse of (µ′ ) 2 and if we take B ¯ ′ ) 12 = B((µ′ ) 21 − (µ′ ) 12 = B · M where M is a inverse, then it is ν-measurable B(µ 1 ¯ ′ ) 12 k2,ν → 0 with projection of norm 1, kBkH < ∞, and this gives kBm µ′ 2 − B(µ ¯ ∈ L2 (ν) as the desired element. This is where we needed and used the finite B dimensionality of µ to obtain the Radon-Nikod´ ym (RN) derivative µ′ = dµ dν for the B(H)-valued measure µ (i.e., in B(H)). In the infinite dimensional case if µ takes values in a reflexive space or the one with the “RN-property”, the same argument works. (The measurability of the generalized inverse also has to be verified, but this is known and standard.) Thus if µ is Hilbert-Schmidt (or HS) operator (the HS space is reflexive) or is of trace class (this is a separable adjoint Banach space and so both have the RN-property) the whole argument works for µ of finite Vitali variation. Here we need to use some results from Diestel and Uhl(1977). The infinite dimensional case is also available and discussed in Kakihara ((1997), Secs. 3.4 and 3.5). Thus our discussion can be completed. 3. Let {Xs , s ∈ S} be a Karhunen family relative to a collection {gs (·), s ∈ S} ˆ ν) where ν is the spectral measure of the ˆ S, of functions on a measure space (S, R ˆ ν) is a dense set. If Xs -field i.e., Xs = Sˆ gs (u) dν(u), and {gs , s ∈ S} ⊂ L2 (S, 2 ˆ 2 ˆ ˆ L = sp{X ¯ s , s ∈ S} ⊂ L (S, ν) show that for any Y ∈ L there is an h ∈ L (S, ν) R R ˆ such that Y = Sˆ h(u) dZ(u) where Xs = S gs (u) dZ(u), (Z(A), Z(B))L2 (P ) = ν(A, B). If the process or field is an n-vector, with respect to a suitable n × n matrix valued g(t, ·) Rand an n-vector orthogonally valued representing measure Z(·), so that Xt = Sˆ g(t, u) dZ(u), is well-defined, let Ya ∈ sp{X ¯ t , t ∈ S} = D ⊂ H. Here the space is determined by the class {Xt , t ∈ S} withR linear combinations having n × n matrix coefficients. Show that Ya = Sˆ h(a, u) dZ(u) with the matrix function h(a, ·), called a spectral characteristic of the Y field, ˜ µ), µ(A ∩ B) = (Z(A), Z(B))H a Hermitian mawhich lies in the space L2 (S, trix measure. Show further that for a linear mapping T : D → D, there ex˜ µ) → L2 (S, ˜ µ) such that T˜ha ∈ L2 (S, ˜ µ) and ists a linear operator T˜ : L2 (S,

492

9 Random and Vector Multimeasures

R T Ya = Sˆ (T˜h(a, ·))(u) dZ(u). In the case that this X : R → L20 (P, Cn ) process is K-stationary so that g(t, u) = I · eiht,ui the result was established by Rozanov (1967, p.33), and his argument extends to this case. Consider a further extension of the result to harmonizable fields following the ideas of Theorem 3.3. (See also Kakihara (1997) and Sec 3.4 where Cn is replaced by a Hilbert space, as in Exercise 2 above.) 4(a) It was seen in Chapter 5 (see Section 5.2) that a convolution can be defined between a pair of bimeasures on G × G where G = Rn . In higher dimensions there are difficulties to extend this type of analysis. In particular for multimeasures, or multilinear forms the following difficulty appears. By Theorem 4.4 above, each continuous n-linear operator T : ×ni=1 C0 (Si ) → X (a Banach space), is representable by an integral with multimeasure µ (cf. (8) there) and kT kX = kµk, where the latter is the Fr´echet variation of µ. However, if S1 , S2 , . . . , Sn are LCA groups which are not discrete, e.g., Rn , then Tj (eit1 x , . . . , eitn xn ) gives the Fourier transform of the corresponding µj if X = C, j = 1, 2. But for n ≥ 3, considering T1 , T2 and for the related measures µ ˆ1 , µ ˆ2 their product µ ˆ1 µ ˆ2 is not necessarily the Fourier transform of a measure, contrary to the case of n = 1, 2. (Such a result was pointed out by Graham and Schreiber (1988) with n = 3.) This can be avoided by considering large subspaces of the space of multilinear mappings with a stronger topology, but still weaker than that obtainable by demanding the vector measure µ to have finite Vitali variation. It is called the completely boundedness condition defined as follows (we take X = C for simplicity and also X = Mp the space of all p × p matrices of complex values). The tensor product C0 (S) ⊗ Mp can and will be identified with C0 (Si , Mp ), the algebra of Mp -valued continuous functions vanishing at infinity [i.e., X = Mp with a Banach or Euclidean norm, so A ∈ Mp iff kAkB = sup{|Ax| : kxk ≤ 1} Pp Pp 1 or respectively kAkE = ( i=1 j=1 |mij |2 ) 2 which clearly satisfies, kAkB ≤ √ kAkE ≤ p kAkB and we can take either of these norms, and so one can identify C0 (S) ⊗ Mp as C0 (S, Mp ), noted above]. If Φp : ×ni=1 C0 (Si , Mp ) → Mp , a linear function or operator, then for Fi ∈ C0 (Si , Mp ), one can define Φp (F1 , . . . , Fn ) using nth order matrix multiplication in Mpn to be in Mp again. The matrix norm kΦp k on ×ni=1 C0 (Si , Mp ) is well-defined for each p ≥ 1, changing with p. If supp≥1 kΦp k < ∞, then Φ is called completely bounded, and its norm is denoted by kΦkCB . Verify that the relation between Vitali, Fr´echet and completely bounded norms satisfy the inequalities kΦk ≤ kΦkCB ≤ kΦkV , the first being Fr´echet’s and the last Vitali’s, which may be infinite while the others are finite. Try an example to verify this fact. (b) To see how these concepts apply to random fields and measures, recall that a measurable mapping X : S → L20 (P ) = H, on (S, S) is a random field and taking S = G, a locally compact group, it is termed weakly harmonizable if it is the Fourier transform of some (necessarily unique) random measure Z : B(G) → H. In the general case (i.e., G is not necessarily separable or unimodular) there are no known integral representations as

9.5 Complements and Exercises

493

given above. The Fourier analysis now proceeds through C ⋆ -algebra methods. [We recall that a C ⋆ -algebra is a Banach algebra with an involution. This is isometric and isomorphic to an algebra B(H) of bounded operators on a suitable (complex) Hilbert space H according to a renowned theorem of Gel’fand and Na˘ımark(cf., Loomis (1953, p.28) or Na˘ımark (1964, Chapter II), and so one can think of a C ⋆ - algebra just as a B(H) for some H.] Thus let H = L20 (P ) on some probability space, and ϕ : G → H be a measurable mapping which is termed a second order random field. It is called weakly harmonizable if ϕ is the Fourier transform of a Hilbert space valued bounded linear mapping Φ of some C ⋆ -algebra C ⋆ (G)[⊃ L1 (G, α), with α as ˆ and Φˆ is a continua (left) Haar measure, embeded densely] so that ϕ = Φ, ous bounded function which determines Φ uniquely. Since the left and right Haar measures on G are distinct, one will have left and right stationarity of random fields, also called homogeneous fields when they both agree (as in the unimodular case). In general a field ϕ : G → H(= L20 (P )) is termed hemi-homogeneous if there exist a pair of continuous positive definite functions ρi : G → C such that the covariance function R : (s, t) → (ϕ(s), ϕ(t))H satisfies R(s, t) = ρ1 (t−1 s) + ρ2 (st−1 )), and thus for two sided homogeneous case one finds R(s, t) = ρ(st−1 )(= ρ(t−1 s)). The random field ϕ is strongly harmonizable if R(s, t−1 ) is the Fourier-Stieltjes transform of a bounded regular bimeasure on G× G so that it determines a scalar measure. If on the other hand R(s, t−1 ) is the Fourier transformation of a completely bounded bilinear form on C ⋆ (G) × C ⋆ (G) as defined in (a) then ϕ will be called a completely bounded field. With this long introduction we assert that the following classification holds for a second order random field ϕ: namely ϕ is homogeneous ⇒ ϕ is hemihomogeneous ⇒ ϕ is strongly harmonizable ⇒ ϕ is completely bounded ⇒ ϕ is weakly harmonizable. None of these implications is reversible. One has also a related dilation property for completely bounded random fields ϕ : G → H(= L20 (P )). There is a super Hilbert space K ⊃ H and a continuous (right) homogeneous random field ψ : G → K such that ϕ = PH ψ where PH is the orthogonal projection onto H from K. (Many needed details, to be filled in, can be found in Ylinen (1988) who has been working on these generalized problems in various publications and referenced in the above paper. Investigate if there is a staircase of dilations between homogeneous and weakly harmonizable families motivated by the above set of inequalities. These are of interest to readers studying general harmonizable fields. The basic references here are Christensen and Sinclair (1987, 1989). These are open but interesting and inviting problems.) We close our discussion of these general considerations with the observation that a study of random measures is benefited from an abstract vector valued measures view-point and that on the other hand the latter are considerably enriched in breadth and depth from a knowledge of random measures which naturally engender new ideas having considerable interest.

494

9 Random and Vector Measures

Bibliographical Notes This final chapter is devoted to a selective study of random measures determined by integral representations of second order fields of interest in applications. They employ the point of view of vector measures into Banach spaces and some further developments of the latter which are motivated by the former. The main concern here is to develop the integration theory for bimeasures, and an extension to multimeasures, in such a way that it is applicable to stochastic analysis without imposing restrictive conditions that impair the probabilistic significance or consequences. Many of the standard Lebesgue limit theorems and multiple integrals of Fourier type may not hold. For instance, if (Si , Si ), i = 1, 2 are measurable spaces and (S1 × S2 , S1 × S2 ) is their (Cartesian) product, f : S1 × S2 → C is S1 ⊗ S2 -measurable, withP β : S1P × S2 → C as a bimeasure, n n let fn → f pointwise as n → ∞ where fn = i=1 j=1 aij χEi χEj is a simple R R Pn Pn function. Let A B fn (ω1 , ω2 )β(dω1 , dω2 ) = i=1 j=1 aij β(A ∩ Ei , B ∩ Fj ), for (A, B) ∈ S1 × S2 which is well-defined. If the family of these integrals forms a Cauchy sequence in C for each (A, B) ∈ S1 × S2 let the limit be denoted as {Cf (A, B), (A, B) ∈ S1 × S2 }. Then one can verify that Cf (·, ·) is a bimeasure on S1 × S2 and if β has a finite Vitali variation, the same is true of Cf (·, ·). However, the basic Lebesgue dominated (and Vitali) convergence theorems do not generally hold for such an extension. So it becomes a nontrivial task to consider double and multiple integrals for bimeasures (and multimeasures). The basic analysis of Morse and Transue seems to be needed. Assuming Vitali variation for a bimeasure β and restricting the integration of functions (now |β|(·, ·) is again a bimeasure) but allowing β to take values in a Banach space, Dinculeanu and Muthiah (2000) have given conditions and developed a substantial theory so that the usual limit theorems and change of variable formulas are still valid. See also Ylinen (1978) and some related work in Chang and Rao (1983 and 1986) on similar formulations and consequences. Still it is not easy to translate these results for stochastic analysis. Mostly one considers f (x, y) = f1 (x)f2 (y) in this work which can be treated more generally in the above definition. It was suggested by Kluv´anek (1981) to study the f (x, y) which need not be products. This is not pursued in the literature because the usual limit theorems do not hold for it and has other limitations. At present one uses the MT-integration and its slightly restricted version, called the strict MT-integrals suggested in Chang and Rao (1983). This is largely needed in this book and elsewhere in stochastic analysis if weakly harmonizable classes are to be covered. The Grothendieck inequality, especially in the completed form by Lindenstrauss and Pelczy´ nski (1968) plays a key role in Section 2 to describe a related integration when the (vector) measure does not have finite (Vitali) variation. This allows us to obtain integral representations of random processes relative to general (random) measures, and the detailed work here will be of particular interest. Also the dilation of harmonizable processes (or fields) to stationary

Bibliographical Notes

495

classes on a super Hilbert space is seen to be of fundamental importance in this analysis. This work highlights the deep connection between random measures of stochastic theory and harmonic analysis based on different types of vector measures. Section 2 shows a crucial connection between random measures and spectral measures of classical functional analysis. Also Karhunen class shows how numerous second order random fields use and motivate newer types of operator analysis of non-selfadjoint classes (see Mizel and Rao (2009) for several such cases). The spectral methods and applications in Section 3 are of particular interest when vector valued fields are considered. We first treated the finite dimensional vector random processes (fields) and focused on the basic structure of the problems to see how the Bochner integrals have to be extended to analyze the spectral domain of stationary vector processes. Here the works by Rozanov (1967) and Rosenberg (1964) are important which then may be extended to some infinite dimensional versions as in Kakihara (1997). Gramian Hilbert analysis appears to be convenient in such an extended study. Finally multilinear forms and multimeasures of Section 4 show the promise of several new directions for continued studies with new problems that are not seen in the works of bimeasures. This is the case with operator valued measures which are of finite Fr´echet variation, as Blei’s (1985) analysis shows, giving a generalized Riesz type representation but also raising new questions. Here convolutions of algebraic structures or multiplication of Fourier transforms not being of the same kind are new difficulties when we deal with more general objects than bimeasures, i.e. for multi- (or poly-) measures. It seems natural in this context to study completely bounded classes so that an algebraic structure may be introduced. Also the weak (and strong) harmonizability, completely bounded (sub) classes of fields, including the stationary ones are to be studied. See also Blei (1985, 1993) for another approach to an aspect of multiple stochastic analysis with less probability preparation. It shows the difficulty of bringing in Fubini type argument into this analysis. The above ideas are discussed at length in Section 4 and some are given as complements in Section 5. We have included easily available references when many of the details could not be given in the text for space reasons. They are noted at appropriate places in the text. Ylinen (1988) has studied the Fourier transforms of functions and processes which play an important role as shown in Exercise 4 here. It connects the operator theoretical analysis depending on C ⋆ -algebra methods and random fields on general locally compact groups. Even our particular treatment of a similar topic on separable unimodular groups is still dependent on Mautner-Segal and others works, showing the need of new tools. With all the preceding analysis and directions of multivariate stochastic fields for future studies, we conclude this account after showing several interesting and expanding areas involving vector valued as well as random measures in both the theory and applications.

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Notation Index

[Same symbols are sometimes used in different chapters to minimize proliferations of special notations.]

Chapter 1 |µ|(·) – Vitali variation of a vector measure µ, 4 (Ω, Σ, P ) – probability space, 3

kµk(·) – Fr´echet (or semi-) variation of a vector measure µ, 7

L0 (P ) – space of scalar random variables, 12 D – delta ring, 16 dp (·) – metric on Lp (µ), p ≥ 0, 18

Chapter 2 (S, S) – measurable space, 23 β(·, ·) – bimeasure, 25

L1 (β) – space of M T -integrable functions for β, 33 Z(·) – random measure, 25 B(·, ·) – bilinear form, 35 τx – shift operator, 38

f ∗ g – convolution of f, g, 47

µ ˜ – Plancherel measure associated with µ, 47

524

Notation Index

G/H – quotient space of group G, by subgroup H, 49 k1 , k2 , kn – cumulants, 52 Lk (·, ·) – Bochner K-transform on L20 (P ), 57

Chapter 3 ϕZ(A) – characteristic function, 62 ∧, ∨ – min, max symbols, 68 E(Z(A)|Z(B)) – conditional expectation, 69 L = L(X) – linear span of a process X = {Xt , t ∈ T }, 83

L1 (Z) – space of Z-integrable scalar functions, 92

L1 (Z) ≃ L2 (β), isometric isomorphism of spaces, 100 {Ut , t ∈ R}, unitary operators, 106

⊕ – direct sum, 105

T1 ⌣ T2 – the operators T1 , T2 commute, 108 ˆ – delta ring on G, ˆ 102 B0 (G)

Chapter 4 B(R) – bounded scalar Borel functions on R, 126 K – Schwartz space on Rn , 127 ρ(·, ·) – tempered measure, 128

∆ – gradient operator, 139

L2,2 – boundedness principle, 147 lA (·) – local functional, 151 sgn – signum function, 158 Z – group of all integers, 162

Notation Index

525

Chapter 5 X1 ⊗ X2 – tensor product of the spaces, 170

ˆ 2 (X1 ⊗X ˘ 2 ) – projective (injective) tensor product, 171 X1 ⊗X

B1 ∗ B2 – convolution of bilinear forms, 180 ˆ1, G ˆ 2 ) – set of Fourier transforms of bilinear forms on LCA groups (G1 , , G2 ), S(G 182 Z1 ∗ Z2 – Convolution of random measures, 187

∆2 – condition for Orlicz spaces, 206

Lq (P, X) –Lebesgue space of X-valued qth power integrable funtions, 213

Chapter 6 C(S) – real continuous functions on a compact S, 219 E A (Y)(= E(Y|A) – conditional expectation of Y, given A, 222 P ⊂ B(I) ⊗ Σ – predictable σ-algebra, 228

µx ((s, t] × A) = E(χA (Xt − Xs )) – Dol´eans-Dade measure, 230

[X, Y ] – quadratic covariation of processes X, Y , 233 DB (X) – debut of a process X into a set B, 237 [T1 , T2 ) – stochastic intervals, 239

B(H) – space of all bounded linear operatives on H, 243 dS.dA – symbolic differential classes, 261 Hn (x) – Hermite polynomial, 266

Chapter 7 d(X) – Frech´et metric, 273 M(F) – locally bounded adapted processes to a filtration F, 273 Cd (Tg ) – c´ adl´ ag (c´agl´ ad) processes, 273 ∆Xs – semimartingale increment, 285 r Im (f ) – multiple Itˆ o-Wiener integral, 290

δm,n – Dirac delta function, 290 Kn (t1 , . . . , tn ) – Volterra kernel, 295

526

Notation Index

Ij ≺ Ij+1 – ordering intervals, 295

ES – a σ-algebra of ordered sets, 295 Kn (u, t) – Poisson-Charliar polynomial, 303 ≺≺ – strict linear ordering, 304

GS – the collection of grids of a set S in R2 , 304 Kv (KA ) bound for V − (A−) martingale process, 305

f

D′ (D′ L) – extended Dirichlet (local extended Dirichlet) class, 314 f(f) – a different (strict type II) order in R2 , 318 R f dX1M – line integral, 320 T2 A ≺ B, A f B – are different orders of sets in R2 , 323

µ(x, y) – M¨ obious function, 335 Dn – difference operator, 336 A(≻σ) – diagonal set, 342 ∆n (A) – diagonal measure, 344

{ n2 } – integral part of the fraction, 345

(H∗ , B(H∗ ), µ) – White noise space (BM or Poisson), 353

f ⋄ g – Wick product, 357

Chapter 8 µ ≺ λ (or µ << λ) – the vector measure µ is absolutely continuous for λ, 375 kνkϕ – Orlicz norm on vector measure ν, 379 Rw A f dµ – weak integral, 381

V(Z, λ) – space of Z valued measures with R − N property for λ, 392

L1X (µ) – scalar functions integrable for X-valued measures µ, 399 k · k# , (k · kb ) – sharp (flat) norms on chains, 406

{Yα = Z Xα , α ∈ I} – projective system of function spaces, 415 lim← (Yα , f˜αβ ), [lim→ ](Yα , fαβ )] – projective (direct) limit spaces, 417 GS k (X) – general flat or sharp chain space, 418 Hkα (·) – Hausdorff α-dimensional measure on Rk , 421

Fk (Rn )∗ – space of flat k-cochins, 427

Notation Index

∗λ = λ0 – Hedge’s star operator, 430

dφp (·) – exterior derivative of p-covector, 431

δφp – codifferential, 431 E(U, V ) – space of differential forms, 431 Vi (ϕ), ϕ ∈ D, – random current, 435

Chapter 9 X1 ⊗α X2 – tensor cross-space with cross-norm α, 448 kνkϕ,µ – ϕ-semivariation of ν with µ as control, 458

d(E, F ) – Banach-Mazur functional, 459 kBk2,µ – matrix norm, 472

((·, ·))L2 (µ) – inner product for matrix functions, 487 β(A1 , . . . , Ak ) – multi (or poly) measure, 483

M(·) – strongly σ-additive operator measure, 487 ΛX = Y – filter (operator) equation, 488

527

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Author Index

Abraham, R., 335 Adams, T., 436, 444 Alexandroff, A.D., 75 Almgren, F.J., 420 Ambrosio, L., 436, 444 Argabright, L., 44, 59 Aronszajn, N., 141, 173 Bakry, D., 309 Balakrishnan, A.V., 47 Bartle, R.G., 14, 21, 74, 438 Bass, R.F., 250, 268 Berg, C., 27, 29 Berger, M.A., 300 Bernstein, S., 118 Bessaga, C., 438 Bogdonowicz, W.M., 443 Berezansky, Yu. M., 367 Bichteler, K., 236 Bishop, R.L., 431 Blei, R.C., 143, 165, 482, 495 Bochner, S.,23, 27, 39, 41, 56, 60, 74, 78, 86, 92, 99, 115, 122, 132, 141, 163, 165, 168, 189, 195, 210, 224, 236, 267, 272, 309, 333, 360, 369, 388, 404, 439, 466, 488, 495, Bourbaki, N., 49, 411 Bram, J., 110

Brennan, M.D., 268, 269, 277, 304, 311, 363 Bretagnolle, J., 85, 93 Brillinger, D.R., 51, 53, 54, 60 Brooks, J.K., 387 Burkholder, D.L., 372 Cairoli, R., 269, 304, 313, 323, 328, 369 Cartier, P., 2, 34, 36, 349 Chang, D.K., 55, 489, 494 Chipman, J.S., 491 Chobanyan, S.A., 161 Christensen, E., 493 Chung, K.L., 18 Cram´er, H., 155 Cuculescu, I., 227, 238, 244, 248, 267 Cudia, D.F., 394 Daley, D.J., 48, 51, 54, 60 Darst, R.B., 443 Daucunha-Castelle, D., 85 Day, M.M., 394 Dellacherie, C., 236, 249, 251, 260 De Rham, G., 407, 432, 433, 445 Devinatz, A., 163, 165 Dharmadhikari, S.W., 77 Diestel, J., 8, 21, 392, 427, 439, 462

530

Author Index

Dinculeanu, N., 8, 17, 21, 22, 29, 59, 123, 159, 374, 381, 386, 393, 443, 478, 489, 490, 494 Dobrakov, I., 17, 60, 374, 378, 381, 443, 481 Doleans-Dade, C., 230, 264, 267 Doob, J.L., 24, 43, 62, 69, 223, 250, 259, 261, 271, 306, 330, 362, 366 Dozzi, M., 333, 335, 370 Dugundji, J., 414, 415 Dunford, N., 8, 14, 20, 21, 22, 24, 26, 31, 59, 65, 72, 75, 93, 100, 108, 113, 120, 126, 146, 151, 169, 175, 212, 220, 239, 296, 359, 374, 375, 378, 381, 387, 395, 428, 443, 449, 457, 472, 478

Graham, C.C., 58, 59, 178, 182, 215 Green, M.L., 269, 304, 309, 311, 313, 314, 317, 319, 323, 330, 369 Grenander, U., 166 Gross, L., 368, 371 Grothendieck, A., 104, 107, 141, 142, 145, 149, 157, 164, 170, 174, 177, 180, 186, 207, 226, 227, 232, 272, 250, 368, 447, 448, 450, 453, 455, 456, 471, 496

208, 312, 333,

123, 161, 182, 266, 449, 475,

Hahn, M.G., 114 Hardin, Jr. C.D., 85, 89, 90, 91, 122 Hardy, G.H., 12 Edwards, R.E., 102 Harris, T.E., 24 Engel, D.D., 292, 293, 296, 303, 369 Hausner, M., 367 Hewitt, E., 17, 146, 483 Federer, H., 423, 425 Hida, T., 209, 267, 289, 350, 353, 364, Feller, W., 77, 205 370 Flanders, H., 431 Hille, E., 21, 27, 60, 98, 102, 103, 106, Floyd, E.E., 400 111, 112, 121, 359, 451, 459, 464, Fock, V.A., 364 487 Foia¸s, C., 162, 166, 183 Hitsuda, M., 251, 267, 289, 358, 359, Fremlin, D.H., 423 370 Friedman, N.A., 153 Holden, H., 358, 368, 370 Hosoya, Y., 94 Gangolli, R., 349, 365 Huang, Z.Y., 360 Garling, D.J.A., 204, 205, 207, 216, 221 H¨ urzeler, H.E., 336, 337, 338, 369 Geitz, R.F., 439 Gel’fand, I.M., 39, 41, 43, 59, 120, Ikeda, N., 260 127, 132, 136, 150, 164, 219, Ionescu Tulcea, A., and C., 54, 56, 203, 427 352, 358, 366, 369, 432, 435, 445, Isaacson, D., 263 488 Itˆ o , K., 127, 188, 223, 227, 257, 261, Getoor, R.K., 110, 112 268, 289, 361, 432, 433, 435, 442, Gil de Lamadrid, J., 44, 59 445 Gilbert, J.E., 215 Ito, T., 162, 215 Gin´e, E., 114 Gnedenko, B.V., 76, 439 Jacod, J., 288, 370 Goldberg, R.R., 44 Jagers, P., 24 Goldberg, S.I., 431 Jameson, G.J.O., 157, 165 Gould, G.G., 9, 14, 21, 367, 375

Author Index

Kakihara, Y., 491, 495 Kakutani, S., 19, 267, 268, 356, 369, 405, 446, 471 Kallianpur, G., 209 Kadˇe, H.I., 395, 396 Kalton, N.T., 218 Kanter, M., 84, 88, 93, 122 Kaplansky, I., 441 Karhunen, K., 96, 99, 106, 110, 113, 120, 122, 125, 140, 154–156, 165 Katznelson, Y., 170 Kawata, T., 57, 58, 59 Khintchine, A.I., 71, 122 Kibble, W.F., 72, 117 Kingman, J.F.C., 204, 216 Kirchheim, B., 436, 444 Klee, V.L., 395, 396, 446 Klotz, L., 440 Kluv´anek, I., 10, 29, 92, 375, 494 Kolmogorov, A.N., 76, 107, 122, 202, 439 Krantz S.G., 444 Kritt, B., 443 Krivine, J.L., 85, 93, 172 Kuelbs, J., 84, 86, 115, 122 Kunita, H., 256 Kuo, H.H., 359, 260, 368, 371 Lay, D.C., 452, 453 Leonov, V.P., 52, 60 L´evy, P., 18, 20, 22, 58, 62, 63, 64, 66, 67, 68, 71, 72–75, 84, 87, 113, 116, 122, 155, 189, 201, 204, 214, 223, 228, 237, 309, 317, 320, 338, 348, 349, 363, 364, 368 Lewis, D.R., 382, 397, 443 Lindenstrauss, J., 104, 107, 165, 226, 459, 460, 494 Lin, V.Ja., 48 Littlewood, J.E., 12 Lo`eve, M., 18, 164, 242 Loomis, L.H., 120, 157, 239, 493 Mackey, G.W., 53, 60 Mandrekar, V., 209

531

Marcus, D.J., 67, 114 Marsden, J.E., 335 Masani, P., 122 Maurey, B., 219 Mautner, F.I., 483, 484, 495 Maynard, H.B., 396, 397 McKean, H.P., 268, 370 McShane, E.J., 218, 237, 252, 255, 256, 267, 270, 369 Mehlman, M.H., 478 M´etivier, M., 251 Meyer, P.A., 211, 227, 250, 251, 256, 258, 260, 261, 267, 271, 288, 305, 313, 314, 315, 329, 362, 368, 370 Minlos, R.A., 352, 354, 359, 363 Mizel, V.J., 300, 353, 495 Mlak, W., 164, 166 Morse, M., 29, 35, 43, 96, 121, 128, 140, 159, 164, 447, 489 Musielak, J., 20, 206, 216 Muthiah, M., 59, 123, 159, 478, 489, 490, 494 Na˘imark, M.A., 36, 105, 120, 156, 160, 163, 219, 486 Neveu, J., 365, 384 Niemi, H., 165 Noltie, S.V., 405, 407, 411, 419, 421, 425, 427, 436, 442, 444 Nualart, D., 267 Okazaki, Y., 114, 115, 116, 193, 195, 200, 209, 212, 215, 216 Orey, S., 274 Orlicz, W., 20, 206, 216 Oxtoby, J.C., 28 Palay, R.E.A.C., 56 Panchapagesan, T.V., 374, 386, 399, 438, 443, 482 Parks, H.R., 444 Parvardah, A., 122 Peck, N.T., 218

532

Author Index

Pelezy´ nski, A., 104, 107, 165, 226, 459, 460, 494 Pellaumail, J., 230, 231 Peter, F., 181 Pettis, B.J., 11, 15, 17, 59, 195, 244, 245, 359 Phillips, R.S., 21, 60, 92, 111, 112, 359, 392, 400 Pietsch, A., 104 Pisier, G., 122, 123, 165, 214 P´ olya, G., 12, 63 Ramachandran, B., 72, 73, 122 Rao, C.R., 72, 73, 122 Rao, K.M., 281 Ratiu, T., 334 Redfern, M., 361 Reiter, H., 49 Ren, Z.D., 206, 379, 438 Rickart, C.E., 386, 385, 437, 443 Riesz, F., 7, 27, 43, 45, 56, 92, 100, 101, 126, 132, 134, 137, 142, 147, 150, 151, 168, 206, 209, 210, 211, 235, 238, 262, 274, 383, 395, 400, 427, 428, 438, 445, 446, 452, 474, 476, 477, 478, 482, 490 Roberts, J.W., 218 Rogers, C.A., 422 Rolewicz, S., 445 Rosenberg, M., 472, 477, 490, 495 Ross, K.A., 17, 483 Rota, G.-C., 336, 338, 341, 344, 347, 349, 370 Royden, H.L., 85, 87 Rozanov, Yu.A., 470, 472, 477, 495 Rudin, W., 179, 187, 481 Russek, A., 371 Rva˘ceva, E.L., 84 Rybakov, V.I., 438 Ryll-Nardzewski, 437 Samorodnitsky, G., 114, 123 Sard, A., 389

Sazonov, V.V., 439 Schatten, R., 408, 448 Schilder, M., 68, 84, 85, 122 Schoenberg, I.J., 191, 197, 349 Schreiber, B.M., 58, 59, 178, 182, 208, 215 Schriber, M., 93 Schwartz, J.T., 8, 14, 20, 21, 22, 24, 26, 31, 39, 65, 74, 77, 92, 94, 97, 100, 103, 108, 109, 111, 113, 121, 122, 129, 131, 133, 151, 164, 165, 169, 175, 195, 196, 198, 211, 214, 238, 239, 288, 296, 348, 349, 350, 350, 351, 352, 353, 354, 356, 367, 369, 372, 376, 379, 380, 388, 389, 393, 403, 426, 427, 429, 430, 435, 440, 442, 445, 446, 451, 453, 456, 456, 457, 467, 473, 476, 477, 488 Schwartz, L., 17, 195, 196, 407, 429 Segal, I.E., 484, 495 Shiryaev, A.N., 52, 60 Sinclair, A.M., 493 Sion, M., 461 Skorokhod, A.V., 359 Soltani, A.R, 122 Sreehari, H., 77 Stone, M.H., 19, 100, 102, 209, 288, 400, 452 Stratonovich, R.L., 188, 254, 256, 260 Swift, R.J., 140 Szeg˝ o, G., 166, 302 Sz.-Nagy, B., 105, 111, 147, 160, 162, 165, 166, 170, 183, 452 Talagrand, M., 218, 360, 393, 439 Taqqu, M.S, 114, 123 Tarieladze, V.I., 161 Taylor, A.E., 442, 453 Thomas, E., 443 Thornett, M.L., 41, 48, 59 Tong, A.E., 153 Transue, W., 29, 35, 43, 96, 447, 489 Traynor, T., 386, 387, 388, 437, 443

Author Index

Trotter, H.F., 75, 76 Tweddle, I., 436

533

Whitney, H., 385, 390, 393, 403, 404, 406, 407, 408, 409, 410, 419, 420, 423, 424, 444 Uhl, Jr, J.J., 8, 21, 392, 393, 427, Wick, G.C., 357 439, 462, 491 Wiener, N., 56, 288 Urbanik, K., 20, 205 Wijman, R.A., 50 Woyczy´ nski, W.A., 20 Vakhania, N.N., 161 Wong, E., 309, 432, 435, 442 Vere-Jones, D., 48, 51, 54, 60 Wright, J.D.M., 19, 204, 401 403, 441, Vilenkin, N.Ya.., 39, 41, 43, 127, 132, 444 136, 150, 164, 352, 367, 435, 445 Ville, J., 223 Yaglom, A.M., 41, 133, 134, 136, 140, Voropolus, N.T., 60, 171, 449 164, 445, 488 Yan, J.A., 360 Walsh, B.J., 438 Ylinen, K., 59, 145, 148, 165, 215, Walsh, J.B., 269, 304, 314, 318, 323, 450, 488, 493, 495 328, 334, 369 Yosida, K., 146 Wallstrom, T.C., 336, 338, 341, 344, Young, L.C., 24, 188, 237, 267, 276, 347, 370 369 Watanabe, S., 256, 257, 260, 261, 268 Zakai, M., 309, 432, 435, 442 Weil, A., 53 Zolotarev, V.M., 193 Weyl, H., 181 Zygmund, A., 56 White, B., 444

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Subject Index

A Abstract(L-) space, 19 Adapted to filtration, 224 Ample measure, 403 A-process, 305 Aronszajn space, 33 A-sub (super) martingale, 306 B Bimeasure integral, 30 Bochner integrable, 392 Bochner-Minlos theorem, 352 Bochner-Schwartz theorem, 39 Boundary of a chain, 423

Conditional probability function, 202 Condition (M), 419 Condition (S) on semi-martingales, 284 Conservative process, 162 Continuity of a bimeasure, 230 Control of random measures, 13 Cram´er class, 464 Countable chain condition, 402 Cross-norm, 171 Cross-term domination, 326 Cumulant, 60 Currents, 435 D

C C` adl` ag process, 232 C` agl` ad process, 273 Characteristic functional, 352 Class (C)-bilinear functional, 128 Coboundary of a chain, 406 Comass, 407 Completely bounded field, 493 Completely monotone, 117 Completely nonunitary, 166 Completely random measure, 204

Debut of a set 237 Diagonal measure, 344 Difference operator, 336 Dilation theorem, 465 Direct limits, 407 Discrete bimeasure, 208 Dissipative process, 162 Doob-Dynkin lemma, 69 D-S (Dunford-Schwartz) integral, 26 (D′ ) and (D′ L) conditions, 314

536

Subject Index

E ε-chain of stopping times, 277 F Filtration of σ-algebras, 224 Filter equation, 488 Flat chain, 406 Fourier-Hermite polynomial, 356 Fr´echet variation of Multimeasures, 492 Fundamental theorem of stochastic calculus, 262 G Generalized ∆2 -condition, 206 Generalized Orlicz (=Musielak-Orlicz) space, 206 Generalized random field, 127 Generalized semi-stable, 72 Generalized spectral family, 156 Generalized Young function, 207 General sharp (flat) chains, 406 Greatest cross-norm, 171 Grid, 304 Grothendieck constant, 143 H Hahn-Banach property, 10 Hellinger-Hahn theorem, 209 Hermite generating function, 266 Hitsuda-Skorokhod integral, 252 Hitting time, 237 Hodge star operator, 430 Homogeneous random current, 432 H-process, 305 I Incidence algebra, 335 Interior product, 429 Isotropic random current, 434 J Jordan decomposition, 7

Jordan decomposition for quasimartingales, 251 Jordan type decomposition of bimeasures, 149 K Kaplansky-Hilbert module, 441 Karhunen class, 97 K-stationary, 51 Kolmogorov-Rozanov characterization, 474 K-stationary, 128 L Lax-Milgram criterion, 453 Least-cross norm, 171 L2,2 -boundedness, 141 L´evy-BM, 309 L´evy-Khintchine representation, 62 Linear regression for random measures, 67 Local classes of random fields, 133 Local functional, 151 Lifting Theorem, 203 Line (stochastic) integral, 321 Lipschitz (co-) mass, 408 Local (semi-) martingale, 238 Locally L2,2 -bounded process, 295 M Martingale (sub), 224 Mass (and comas), 406 McShane stochastic integral, 255 Measure Kernel, 60 Minimally positive definite, 77 Measure kernel, 205 Minlos theorem, 352 M¨ obius function, 335 M¨ obius inversion formula, 335 Modular measure, 402 Modular space, 206 Moment stationarity, 50

Subject Index

Moving average process, 474 Multi (or Poly) measure, 479 Multiplication (convolution) of random measures, 187 Multiplicity of a process, 209 Multiple Wiener-Itˆ o integral, 289 N Nightingale(=nigh-martingale), 24 Na˘imark’s theorem, 219 O ‘Odot’ operation, 177 Optional σ-algebra, 228 Orthogonal random measure, 59 P Parameter of p-stability, 190 Pettis integrable, 393 Phyllips R − N theorem, 392 Planar Martingale, 305 Plancherel measure pair, 47 Point processes, 204 Pointwise convergence topology, 414 Poisson white noise, 353 Polymeasure, 479 Poset, 335 Positive definite bimeasure, 148 Positive definite operator, 207 Predictable σ-algebra, 228 Predictable stopping time, 239 Predictable transform, 249 Progressively measurable, 237 Pseudomeasure, 479 Purely nondeterministic, 166 Q Quadratic covariation of a process, 233 Quadratic variation of a process, 257 Quasi-martingale, 251

537

R Random curent, 432 Random integrator, 60 Random measure, 2 Reduced process, 277 Regular bimeasure, 28 Reproducing kernels, 172 RKHS (=Aronszajn) space, 33 R-N property, 225 S Schur’s lemma, 172 Semi-invariant, 51 Semi-martingale, 231 Semi-ring, 3 Semi-stable, 72 Semi-variation, 7 Sharp (co-) chain, 406 Shift invariant bimeasure, 38 Shift transformation, 110 Signed measure, 4 Simplex m-vector, 404 Spectral characteristic, 474 Spectral distribution, 95 Spectral measure of a random field, 485 Spherical image map, 394 Stable class, 63 Stable random measure, 63 Standard filtration, 224 Stationary dilation of harmonizability, 466 Strictly harmonizable, 94 Stochastic Fubini theorem, 327 Stochastic Green formula, 315 Stochastic Green-Gauss theorem, 331 Stochastic integrator, 250 Stochastic interval, 239 Stochastic stokes theorem, 314 Stratonovich integral, 254 Strict Morse-Transue integral, 33 Stochastic partials, 330 Stone algebra, 403

538

Subject Index

Stopping line, 231 Stopping time, 237 Strongly µ-measurable, 195 Subnormal operator, 111 Superposition of process, 237 Symbolic calculus for semimartingales, 260 Symmetric (stochastic) integral, 254 Symmetric stable measure, 64 T Tempered measure, 38 Topology of pointwise convergence, 414 Trace algebra, 4 Triangular (finite) product set, 342

V -martingale, 305 V -process, 305 Vector measure, 1 V -sub (super) martingale, 306 W Weak direct limit, 414 Weak (planar) martingale, 309 Weak semi-martingale, 280 Weakly of class (C), 97 Weakly measurable function, 392 Weakly stable, 77 Weakly stable characterization, 98 Weakly stationary, 101 Weil-Mackey-Bruhat extension, 60 White noise space, 354 Wick product, 357 Wold-decomposition, 166

V Variation (=Vitali’s), 4 Variation bound of a process, 270 V -boundedness, 26

X-Y-Z Yosida-Hewitt decomposition, 146