Records: Mathematical Theory (Translations of Mathematical Monographs)

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Translations of

MATHEMATICAL MONOGRAPHS Volinuc 194

Records: Mathematical Theory Valero B. Nevzorov

AMCHcan malileIUULIcai JOCICLy

Selected Titles in This Series 194 Valery B. Nevzorov, Records: Mathematical theory, 2001 193 Toshio Nishino, Function theory in several complex variables, 2001 192 Yu. P. Solovyov and E. V. Troitsky, C'-algebras and elliptic operators in differential topology, 2001

191 Shun-ichi Amari and Hiroshi Nagaoka, Methods of information geometry, 2000 190 Alexander N. Starkov, Dynamical systems on homogeneous spaces, 2000 189 Mitsuru Ikawa, Hyperbolic partial differential equations and wave phenomena, 2000 188 V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, 2000

187 A. V. Fursikov, Optimal control of distributed systems. Theory and applications, 2000

186 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory 1: Fermat's dream, 2000 185 Kenji Ueno, Algebraic Geometry 1: From algebraic varieties to schemes, 1999

184 A. V. Mel'nikov, Financial markets, 1999 183 Hajime Sato, Algebraic topology: an intuitive approach, 1999 182 I. S. Krasil'shchik and A. M. Vinogradov, Editors, Symmetries and conservation laws for differential equations of mathematical physics, 1999

181 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 2, 1999 180 A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control, 1998

179 V. E. VoskresenskiT, Algebraic groups and their birational invariants, 1998 178 Mitsuo Morimoto, Analytic functionals on the sphere, 1998 177 Satoru Igari, Real analysis-with an introduction to wavelet theory, 1998 176 L. M. Lerman and Ya. L. Umanskiy, Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects), 1998 175 S. K. Godunov, Modern aspects of linear algebra, 1998 174 Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, 1998

173 Yu. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local properties of distributions of stochastic functionals, 1998

172 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 1, 1998 171 E. M. Landis, Second order equations of elliptic and parabolic type, 1998 170 Viktor Prasolov and Yuri Solovyev, Elliptic functions and elliptic integrals, 1997 169 S. K. Godunov, Ordinary differential equations with constant coefficient, 1997 168 Junjiro Noguchi, Introduction to complex analysis, 1998 167 Masaya Yamaguti, Masayoshi Hata, and Jun Kigami, Mathematics of fractals, 1997 166 Kenji Ueno, An introduction to algebraic geometry, 1997 165 V. V. Ishkhanov, B. B. Lur'e, and D. K. Faddeev, The embedding problem in Galois theory, 1997

164 E. I. Gordon, Nonstandard methods in commutative harmonic analysis, 1997

163 A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, and M. I. Yadrenko, Probability theory: Collection of problems, 1997

162 M. V. Boldin, G. I. Simonova, and Yu. N. Tyurin, Sign-based methods in linear statistical models, 1997 161 Michael Blank, Discreteness and continuity in problems of chaotic dynamics, 1997 160 V. G. OsmolovskiT, Linear and nonlinear perturbations of the operator div, 1997 159 S. Ya. Khavinson, Best approximation by linear superpositions (approximate nomography), 1997

(Continued in the back of this publication)

Translations of

MATHEMATICAL MONOGRAPHS Volume 194

Records: Mathematical Theory Valery B. Nevzorov

R American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair)

ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) B. B. Hes3opos PEKOPIIbI: MATEMATI4LIECKASI TEOP14A Translated from the Russian manuscript by D. M. Chibisov 2000 Mathematics Subject Classification. Primary 62-01, 62-02; Secondary 62E10, 62G20, 62G30. ABSTRACT. The book presents the theory of records which has been the subject of an intense research activity since 1952. It may be viewed as a research monograph with an extensive bibliography for probabilists and statisticians, actuarial mathematicians, meteorologists, hydrologists, reliability engineers, sports and market analysts. At the same time the book includes a wide variety of exercises and is written in a form which allows it to be used as a textbook for graduate students.

Library of Congress Cataloging-in-Publication Data Nevzorov, Valery B., 1946Records: mathematical theory / Valery B. Nevzorov; [translated from the Russian manuscript by D.M. Chibisov]. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282; v. 194) Includes bibliographical references. ISBN 0-8218-1945-3 (alk. paper) 1. Order statistics. I. Title. II. Series. QA278.7.N48 2000 519.5-dc2l 00-061821

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission®ams.org. © 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. Q The paper used in this book is acid-free and falls within the guidelines established to Nnsure permanence and durability.

Visit the AMS home page at URL: http://wv.ams.org/ 10987654321 060504030201

To my parents

Contents Preface

ix

Introduction. Lecture 1

1

Part 1. Order Statistics Lecture 2. Order Statistics and Their Distributions Lecture 3. Three Classical Representations for Order Statistics Lecture 4. Markov Property and Conditional Independence of Order Statistics Lecture 5. Order Statistics for Nonstationary Sequences Lecture 6. Moments of Order Statistics Lecture 7. Moments of Order Statistics (continued) Lecture 8. Asymptotic Distributions of the Middle Order Statistics Lecture 9. Asymptotic Distributions of the Middle Order Statistics (continued) Lecture 10. Asymptotic Distributions of Maxima Lecture 11. Domains of Attraction of the Limiting Distributions of Extremes Lecture 12. Some Topics Related to the Distribution of Extremes

Part 2. Record Times and Record Values Lecture 13. Maxima and Records Lecture 14. Distributions of Record Times Lecture 15. Distributions of Record Values Lecture 16. Shorrock's Representation for Discrete Records Lecture 17. Joint Distributions of Record Times and Record Values Lecture 18. Asymptotic Distributions of Record Values Lecture 19. The kth Records Lecture 20. Generating Function of the kth Record Times Lecture 21. Moment Characteristics of the kth Record Times Lecture 22. Tata's Representation and Its Generalizations Lecture 23. Correlation Coefficients for Records Lecture 24. Records for Nonstationary Sequences of Random Variables Lecture 25. Record Times in the Fe-Scheme Lecture 26. Independence of Record Indicators and Maxima Lecture 27. Asymptotic Distribution of Record Values in the Fa-Scheme Lecture 28. Records in Sequences of Dependent Random Variables Lecture 29. Records and the Secretary Problem Lecture 30. Statistical Procedures Related to Records vii

5 5 10 15 18 21 26

32 38

42

47 50 55 55 59 65 69 76 79 82 86 89 92 95 101 107 114 116 119 125 129

viii

CONTENTS

Appendix 1. Theory of Records: Historical Review

133

Appendix 2. Hints, Solutions, and Answers

137

Bibliography

153

Preface In 2002 the mathematical theory of records will mark its 50th anniversary. The first paper on this subject by Chandler (1952) attracted the attention of many researchers and inspired many new publications. It happens sometimes that, by novelty of its subject or brilliant results, a mathematical paper arouses and maintains interest of numerous researchers to a "fashionable" problem for several years or even decades, but as the "vein of gold" is exhausted this interest drops off sharply. This was not the case for records. The number of publications on this subject has been increasing exponentially, doubling about every 10 years (around 80 in 1977, about 160 in 1987, and more than 300 by the beginning of 1998). In the Introduction, we try to explain this phenomenon. Here we only point out that numerous models of records provide a convenient object for application of various mathemat-

ical methods; on the other hand, there is a lot of numerical data on records in sports (national, Olympic, world records), hydrology (for instance, the floods in St. Petersburg have been recorded for 300 years), meteorology (the reader of course has heard on radio or TV something like "the air temperature today in our city was the lowest for the last 95 years"), etc., which motivate the mathematicians to build models compatible with the available record observations and to try to predict the future record values. It is difficult to separate the theory of records from the theory of order statistics. Records are especially closely related to extremal order statistics. A systematic ex-

position of the theory of order statistics and extremes can be found in books by H. A. David "Order Statistics" (1970, 1981) and J. Galambos "The Asymptotic Theory of Extreme Order Statistics" (1978, 1987). Regarding records, a comparatively detailed review of results (without proofs) and related bibliography can be found only in the form of articles (Nevzorov (1987), Nagaraja (1988), Nevzorov and Balakrishnan (1998)). As mentioned above, the number of publications has practically doubled for the last 10 years. Hence it becomes necessary to relate the classical results for records with the latest advances. This book can be viewed as an "Introduction into the Theory of Records." We tried to present the material in the form combining the features of a textbook and a survey of literature. Thus, after learning the basic methods utilized in the theory of records, the reader will find in Appendix 1 bibliographical notes which, together with the list of references comprising about 300 papers on records and related topics, will allow him to acquire a deeper knowledge of the subject. Moreover, this book contains about 50 exercises which will allow the reader to assess the degree of his mastering the material. Hints and solutions are collected in Appendix 2. The book is written on an intermediate level which presumes the knowledge of only standard courses of probability theory and mathematical statistics. In the first part of the book we present briefly the necessary material on order statistics ix

x

PREFACE

which is used in the theory of records. The reader interested in a more detailed knowledge of the theory of order statistics and its applications is referred to the book by David (1970) mentioned above and references therein. The book can be used for preparing courses on order statistics and records, as well as for studying these areas of probability theory and mathematical statistics on one's own. A large part of the material was employed by the author in lecture courses on order statistics and records at the Faculty of Mathematics and Mechanics

at the St. Petersburg State University and the Department of Statistics at Ohio State University. I hope that the methods for analysis of records, numerous record models, and

the various applications of records treated in the book will attract the attention not only of mathematicians, but also of engineers (especially those engaged in reliability of constructions), actuaries, sport statisticians, specialists in hydrology, meteorology, gerontology, and many others dealing with analysis of extremal values of various random variables and processes. Writing the book gives me an opportunity to set out systematically the results of my research on order statistics and records. It is my pleasure to express gratitude to V. V. Petrov from whom I always received attention and support beginning with my first steps in science. I am indebted to my former students A. V. Stepanov and S. V. Malov, now scientists themselves, who have been the most scrupulous readers of my works, and with whom constant communication stimulated my research in this area. Of great importance for my scientific activity was an exchange of ideas with colleagues. I am thankful to M. Ahsanullah, N. Balakrishnan, P. Deheuvels, V. A. Egorov, G. Haiman, and H. N. Nagaraja with whom I obtained a number of joint results included in this book.

Comments useful in improving the text were made by S. M. Ananievskii, D. M. Chibisov, O. V. Rusakov, and V. V. Slavova. Advice from A. M. Trevgoda and assistance from my wife Lyudmila and my son Igor helped to speed up the process of typesetting the manuscript.

V. B. Nevzorov

St. Petersburg, April 2000

Introduction

Lecture 1

One of the most popular books in the world, after the Holy Bible, is the "Guinness Book of Records" first published in September 1955. For almost half a century this book, regularly updated and republished, stimulated the appearance of many similar publications, such as the "St. Petersburg Book of Records" first published in 1995. Why are records so popular? Maybe this is because we often encounter them in

everyday life, single out the records from the multitude of data, fix and memorize the record values. Of particular interest are records in sports and record values related to natural phenomena. The spectators are drawn to stadiums not only by competition excitement, but also by an opportunity to witness new record achievements which broaden the horizons of human abilities. Indeed, for the majority of people who watch sport competitions or participate in them, achievement of a new record is undoubtedly associated with progress. In order to attract potential record-breakers (and, in turn, numerous spectators and advertisers), the organizers of many athletic competitions set special prizes for breaking records, which are, as a rule, of higher value than the prizes for the winners. One can frequently see a show where daredevils risk their necks trying to break a speed record on water, ground, or in the air, or to leap a motorcycle over a record number of cars, or to achieve a record depth without using an aqualung, in order to place their names into the book of records. After Sir Edmund Hillary and Tenzing Norgay reached the summit of Mt. Everest in 1953, having thus achieved the absolute record in mountaineering for our planet, other mountaineers found the "way out" in climbing Everest or other mountains along the routes of record difficulty. People are excited by records related to geography and the various natural or social phenomena. Tourists strive to see the highest waterfall in the world, the oldest tree, or the largest cave. Cities compete with one another by erecting the highest skyscraper or building the largest stadium. Journalists will not miss a chance to report on the birthday of the oldest person in the world, to describe the life of the richest person, or to show the consequences of the most devastating hurricane in a particular region. In weather reports, newspapers publish the lowest and highest temperatures observed in the locality for the last, say, 100 years, and readers are interested to see if these local records would be broken, while old-timers tell stories to envying listeners about the extremely cold winter they survived some sixty years ago. In English the word "record" means not only the unsurpassed performance in some area, but also a report, account, chronicle, diary, or relics of the past. One can see that all these meanings are logically interrelated. I

I. INTRODUCTION

2

Each record "achievement" - such as the oldest record in the Guinness Book of Records which is due to St. Simeon Stylites the Younger who lived for 45 years standing on top of a stone pillar near Antioch, Syria, some 14 centuries ago; the record miracle near the Sea of Galilee when five thousand men, without counting women and children, were fed with five loaves of bread and two fishes; the pandemic

of plague in Europe in XIV century that claimed about 75 million lives; or the fantastic Bob Beamon's long jump of 8.90 m in 1968 - is registered in chronicles and accounts or is preserved in people's memories. Even in mathematics, record achievements are not infrequent. It is well known, for instance, that the sequence of prime numbers is infinite and there is no largest prime number. Hence, attempts to demonstrate the largest particular prime number continue persistently. The enthusiasm of researchers combined with computer equipment progress resulted in obtaining successively in 1996-1999 the following record values of prime numbers: 21257787 _

1,

21398269

-

1,

22976221

-1,

23021377 _ 1, and 26972593 - 1.

The last of them, found by N. Hajratwall, G. Woltman, S. Kurovski, et at. on June 1, 1999, consists of 2,098,960 digits for decimal representation. Maybe by now the reader can write down a much larger prime number.

Another example refers to the probability theory. A longstanding research was devoted to evaluation of the absolute constant in the well-known Berry-Esseen inequality which sets a bound on the difference between the normal distribution and the distribution of a sum of independent identically distributed random variables

with finite third moment. Esseen (1956) has shown that this constant is no less than

3+ 6

10 27r

The attempts to approach this value were made by Berry, Esseen, Bergstrom, Zolotarev, and others. Making substantial use of computer facilities, Shiganov (1982) obtained for this constant the value 0.7655 lowering by 0.032 the previous record 0.7975 which was due to van Beek (1972). Thus it took 10 years to beat the record. The records as such are memorials of their time. The annals of records reflect

the progress in science and technology and enable us to study and forecast the evolution of mankind on the basis of record achievements in various areas of its activity.

This motivates the necessity to construct mathematical models of records and to develop the corresponding mathematical theory. In 1952 there appeared the first paper on the mathematical theory of records by Chandler (1952). Since then there were many mathematicians who devoted their attention to this subject. All of them, working in different areas of probability theory and mathematical statistics, found interesting problems related to records which required diverse mathematical methods. Presently the author's card-file contains more than 300 papers, half of which appeared during the last 10 years. In a number of them, mathematical models for the analysis of growth of Olympic or world records were suggested (with special attention given to the mile run) and, based on this analysis, attempts were made to predict future records. While the forecast of sports achievements serves mostly for comprehension and is important primarily for the authors of record models themselves as a possibility to assess the fit of the models to real processes, a

I. INTRODUCTION

forecast of the scale of the next record earthquake or flood would make it possible to take adequate precautions. Statistics of the various records contains a large number of data which sometimes cover a very long time interval. Hence a statistician processing the data often deals with a censored sample containing only the record values. This happens, for example, in estimating the durability of some devices in the situation where the production process is modified after each failure. A similar situation arises in processing sports data when out of the multitude of data registered in numerous competitions of different level, only the most valuable and informative ones are preserved for history, and, of course, records are this kind of data. It turns out, surprisingly, that the theory of records is connected with cycles of random permutations, with some algorithms of linear programming, with forming platoons (caravans) of vehicles (when moving in a single lane road for a long time, the vehicles form platoons whose leaders' speeds are a sequence of lower record values), with the optimal selection problem (the so-called secretary problem).

The theory of records relies largely on the theory of order statistics, and is especially closely connected to extreme order statistics. Hence we will discuss first the order statistics and their properties.

PART 1

Order Statistics Lecture 2. ORDER STATISTICS AND THEIR DISTRIBUTIONS

Consider n random variables X1,. .. , Xn. Without loss of generality, we will assume that all of them are defined on the same probability space (f2 = {w}, _F, P). Having arranged these variables in increasing order, we obtain order statistics

Xln <... <XnnThus

X1,n = min{XI, ... , Xn}, X2,n is defined so that X2,n(w) for every elementary event w equals the second smallest value among X, (w), ... , Xn (w), and so on, up to Xn,n = max{X1r ..., Xn }.

The order statistics arise in a natural way, for example, in life testing: if we test n items with life times X1, ... , Xn, then the consecutive times of the first, second, ... , through the last failures will be the order statistics X l.n, X2.n, Let us note the following equality:

,

=Xl+X2+...+Xn.

X1,n+X2,n+...+Xn,n

Hence, whenever the corresponding expectations exist,

=EX1+EX2+

+EXn.

In what follows we will consider sequences of independent identically distributed (i.i.d.) random variables X1i X2, ... with common distribution function (d.f.) F and corresponding order statistics X1,n < . . < Xn,n related to the first n terms of the sequence, n = 1, 2, .. . Of course, by ordering independent random variables we lose the independence .

property. Hence the question arises whether it makes sense to pass from independent X's to dependent order statistics. We can present the following arguments in favor of studying and using order statistics: 1) As was pointed out, in many situations we obtain ordered observations directly, and the experiment often does not yield the complete set of order statistics, but only part of it, which does not allow us to apply the methods of estimation using independent observations. For example, when life testing destroys expensive items or it takes too long to complete testing of all items, it is often preferable to stop testing after a certain part of items tested fail and to make inference based on the observations thus obtained, i.e., to use only the available part of order statistics. 5

I.

(RI)I;1i si,vrrS'rI('S

2) In some cases, the best in a certain sense (efficiency. robustness, or simply convenience for application), estimates of unknown parameters have the form of order statistics or some functions of them. For example, the best in various respects estimate for parameter 0 of the Laplace distribution with density

p(x) = 2 exp(-Ix - 01),

-3e < x < oc,

coincides with the sample median, i.e., the mid-point of the set of order statistic Tests based on extreme order statistics are often applied for detection of outliers. 3) If X1i... , Xl()o denote the annual highest levels of the Neva river in the 21st century, then the citizens of St. Petersburg and hydrologists will be primarily interested in the distribution of several extremal order statistics Xloo,100, X99,100. . . . 4) In insurance, there is a well-known rule "20-80" according to which maximal 20 percent of claims comprise at least 80 percent of the total amount of insurance payment. 5) While each observation in the sample X1,. .. , Xn by symmetry bears the same information, the passage to order statistics violates this symmetry and a few order statistics may contain more information than the rest. This property allows one to select the most informative order statistics and thus store much less data. Recall that we consider a sequence of i.i.d. random variables X1, X2, ... with common d.f. F. It is easily seen that n

Fn:n(x) = P{Xn,n < x} = P{X1 <x,. .. , Xn < x} = II P{Xk < x} = (F(x))n k=1

and similarly

Fl:n(x) = P{X1,n < x} = 1 - P{X1,n > x}

=1-P{X1>x,...,Xn>x}=1-(1-F(x))n. The general formula for an arbitrary order statistic is somewhat more complicated:

Fk:n(x) = P{Xk,n < x} = P{at least k variables among X1, ..., Xn lie on the left of x} n

(2.1)

= E P {exactly m variables among X1,. .. , Xn lie on the left of x} m=k n I

m =k

M

I (F(x))m(1 - F(x))n_m,

1 < m < n.

EXERCISE 2.1. Prove the identity

m=k

(1)m1 =

n!

J" (k - 1)!(n - k)!

tk-1(1

- t)n-k dt.

0 < y < 1.

By comparing (2.1) and (2.2) we arrive at the equality (2.3)

Fk:n(x) _

F(s)

1

(k-l)l(n-k )1 tk-1(1 - t)n-k dt, ,

-oo < x < oo.

2. (BI)ER STATISTICS AND THEIR DISTRIBUTIONS

If we introduce the incomplete beta function I,z(a b) =

p`

1

B(a b) J

to-1(1 -t)b-1 dt.

where B (a, b) =

' t°-1(1 fO'Y

r(a)r(b) - t)6-1 dt = r(a+b) ' i

then the d.f. of the order statistic Xk,,, can be written as

Fk:n(x) = IF(.)(k,n - k + 1).

(2.4)

Equality (2.4) enables one to evaluate Fk,n(x) using widely accessible tables of the function Ix(a,b) (see, e.g., Pearson (1934) or Bol'shev and Smirnov (1983)).

EXERCISE 2.2. Show that for 1 < i < j < n and x < y the following equality holds:

Fi,j:n(x, y) _ `P{Xi,n < x, Xj,n < y} (2.5)

= L.

s=j r=i r!(s - r)!(n - s)!

(F(x))r(F'(Y) - F(x)) 3 r(1 - fly) )n-,,.

Assume now that the underlying random variables X1, X2,... have a common density function f. Then for almost all x (i.e., except, possibly, a set of zero Lebesgue measure) F'(x) = f (x), the right-hand side of (2.3) is differentiable, and Fk:n(x) = fk:n(x ,where (2.6)

N!

fk:n = (k - 1)!(n - k)!

(F(x))k-1(1

- F(x))n-kf (x),

-00<X
In particular, (2.7)

fl:n(x) = n(1 - F(x))n-1 f(x) and fn:n(x) = n(F(x))n-If(x)

Since for any absolutely continuous distribution its density is defined up to a set of zero Lebesgue measure, fk;n(x) is a density of the order statistic Xk.n. Let fk(1),k(2),...,k(r):n(x1i x2, ... , xr) denote the joint density function of r order statistics Xk(1),n, Xk(2),n, ... , Xk(r),n, where 0 = k(0) < k(1) < k(2) < . . . < k(r) <

k(r+l)=n+1, 1
fk(1),k(2)....,k(r):n(x1, x2, ... , xr) Ti.

r+1

r+1

H (F(x9) - F(xs-1))k(s)

- k(s

I) -

r

11I f(xs)

(k(s) - k(s - 1) - 1)! 9=I=1

s=1

if -00=xo <x1 <X2 <... <Xr <xr+1 =oo, and fk(1),k(2),...,k(r):n(x1, x2, ... 9 xr) = 0,

1. ORDER S'T'ATISTICS

N

otherwise. In particular, if r = n, (2.8) gives the joint density function of all order statistics Xl,n, X2,,,, , Xn,n.: J

1,2,...,n:n(xl, x2, ... , x,) = n! fl f (x. ) 3=1

if -oo < x1 < x2 <

< x,. < oo, and f1,2,...,n:n(x1, x2, ... , xn) = 0

otherwise.

We will prove (2.8) only for r = 2 assuming for simplicity that f is continuous at the points x1 and x2 under consideration. In the general case the proof is based on the same ideas, but becomes more cumbersome. Consider the probability

P(5, 0) = P{x1 < Xk(1),n < x1 + 5 < x2 < Xk(2),n < x2 + O}.

We will show that as 5 - 0 and 0 -i 0, the limit f (XI, x2) = Tim

P(5, 0)

50

exists.

We see that P(5, 0) = P(A) + P(B), where A = {x1 < Xk(1),n < x1 + 5 < x2 < Xk(2),n < x2 + 0 and each of the intervals [x1, x1 + 5) and (x2i x2 + 0) contains exactly one order statistic} and

B = {x1 < Xk(1),n < x1 + S < x2 < Xk(2),n < X2 + A and the intervals [x1, x1 + 5) and [x2, x2 + 0) contain totally at least three order statistics}.

Obviously, P{B} < P{C} + P{D}, where C = {at least two out of n variables X1,. .. , Xn fall into [x1, x1 + 5)} and

D = {at least two out of it variables X1,---

,

fall into [X1-, x2 + A))-

Since

(n) (F(xi + 5) - F(x1))k(1 - F(x1

P{C} _

+6)+F(x1))n-k

k=2

(n)

< (F(xi + 5) - F(x1))2

< 2n(F(x1 + 5) - F(x1))2

k=2

=0(52),

5-+0, P{D} = O(O2),

we obtain Tim

P(5, Aa0 P(A) = 0

0 - 0,

as 6-0, 0

0.

2. ORDER STATISTICS AND THEIR. ms,rnIRI'TIONS

9

It remains to observe that n!

P{A} _ (k(1) (F(x1)) - 1)!(k(2) - k(1) - 1)!(n - k(2))!

k(1)-1

6))k(2)-k(1)-I

x (F(xi + 6) - F(xl)) (F(x2) - F(xi +

))n-k(2)

x (F(x2 + 0) - F(x2)) (1 - F(x2 +

It is seen from this equality that the limit we are interested in exists and I k(2))!(F(xI))k(I)-I

f(xl,x2) = (k(1) - 1)!(k(2) - k(1) - 1)!(n -

F(xl))k(2)-k(I)-1 (1 -F(x2))n-k(2)f(xl)f(x2),

x (F(x2) which coincides with (2.8) for r = 2.

Note that we considered 8 > 0 and 0 > 0, so that we have found actually the right limit P XI + 0, x2 + 0). But taking into account continuity of f at the points x1 and x2 we can obtain in a similar way the same expression for the limits f (XI + 0,X2 - 0), A XI - 0,X2 + 0), and f (x1 - 0,x2 - 0).

EXAMPLE 2.1. Let F(x) = x, 0 < x < 1, which corresponds to the uniform distribution on the interval [0, 1]. Then (2.10)

fk:n(X) =

k)!xk-1(1

(k - 1)!(n -

-

0 < x < 1,

x)n-k,

i.e., the order statistic Uk,n related to i.i.d. uniform random variables U1, U2,..., Un

has the beta distribution with parameters k and n - k + 1, and (2.11)

f1,2,...,n:n (X1, ... , xn) = n!,

0 < x1 < ... < xn < 1.

EXERCISE 2.3. Let independent random variables XI, X2, ... , Xn have geometric distributions with parameters P1, p2, . , pn (0 < pi < 1, i = 1, 2, ... , n), i.e.,

P{Xi=m}=(1-pa)pi"

m=0,1,....

Find the distribution of the random variable Y, = min{XI, X2, ... , X,, I.

EXERCISE 2.4. Find the joint distribution of the maxima 1lln = max{XI,... , X,,} and Mn+i = max{XI,... , Xn+1 }, where XI, X2, ... are i.i.d. random variables with d.f. F. EXERCISE 2.5. Let i.i.d. random variables X1, X2,. distribution, i.e.,

F(-x) = 1 - F(x + 0),

.., X,, have a symmetric

x > 0.

Show that in this case Xk,n and -Xn_k+l,n are equally distributed and that the sample median Xn+1,2n+1 also has a symmetric distribution.

I.

H)

I)I)DIili s'I'AI'IS'rIC'S

Lecture 3. THREE CLASSICAL REPRESENTATIONS FOR ORDER STATISTICS

We pointed out in Lecture 2 that the convenient independence property is lost when we pass from original independent random variables X1, X...... X to order statistics X1,n < ... < X,,n. Nevertheless, in the particular cases of exponential and uniform distributions, the order statistics can be expressed in terms of sums of independent random variables. One more important result consists of the possibility to pass from order statistics related to some particular distribution (e.g., uniform) to arbitrary order statistics by means of monotone transformations. REPRESENTATION 3.1. Let X j, 5 ... <_ X,,,n, n = 1, 2, ... , be order statistics related to i.i.d. random variables with continuous d. f. F, and let

Uln <...
Then for any n = 1, 2,... the vectors (F(X 1,n), ... , F(Xn,n)) and (U1.n, ... , Un.n ) are equally distributed. COROLLARY 3.2. Define the inverse function (3.1)

G(s) = inf{x: F(x) > s}.

Then Representation 3.1 implies that (3.2)

(G(Ui,n),... , G(Un.n)),

(X1,n, ... , Xn,n)

where X = Y means equality in distribution of random variables or vectors X and Y. REMARK 3.3. The relation (3.3)

(F(X I , n ) , ... , F(Xn,n))

(U1.n, ... , Un.n)

is fulfilled only for continuous d.f.'s F, which is obvious because the uniform random variable U can take any values in [0, 1], hence F(x) cannot have discontinuity points

if (3.3) holds. On the other hand, (3.2) can be shown to hold for arbitrary F(x).

Let us prove (3.3). Under the additional assumption that the underlying X's have a density f this relation is a simple consequence of formulas (2.9) and (2.11). In the general case we can use the following easily verifiable relation: Y k = F(Xk)

d

Uk,

k = 1, 2, ... ,

which relates the distributions of the initial and uniform random variables. Since y = F(x) is a monotone transformation, we obtain for any k < n Yk,n = F(Xkn), where Y1,n < . < Yn,n, n = 1, 2, ... , are the order statistics related to the sequence Y1i Y2, ... , which implies (3.3).

I IONS

3.

REPRESENTATION 3.=1. Consider exponential order statistics

Zi.n. < ... < Zf.n, related to a sequence of i.i.d. random variables Z1, Z2,... unth d.f.

H(x) = max (0, 1 - exp(-x)). Then for any n = 1, 2, ... we have (3.4)

(Zl.n, Z2,n, ... , Zn,n)

d

vl , Ll +

v2

v2 n n n-1 , ... , vln + n-1

+ ... + vn) ,

/

where vi, v2.... is a sequence of i.i.d. random variables with them on d.f. H(x).

Thus any exponential order statistic Zk,n, 1 < k < n, n = 1, 2, ... , is representable as a sum of independent random variables: (3.5)

d Ul +

v2

yk

+ ... +

n n-1 n-k+1 COROLLARY 3.5. It immediately follows from (3.4) that

(3.6)

Zk,n

(Zl,n, Z2,n - Zl,n, ... , Zn,n - Zn_l,n)

d

y1

v2

- (n 'n- 1' ... , vn

and (3.7)

(nZi,n, (n - 1)(Z2,n - Z1,n), ... , Zn,n - Zn_ 1.n)

d

(vl v2, ... , vn).

The relations (3.6) and (3.7) mean that the spacings Rk = Zk.n - Zk-l.n, k = 1,2,.. . , n, where Zo,n = 0, are independent and, moreover, all random variables (n - k + 1)Rk have the standard exponential distribution. Let us note, furthermore, that not only the order statistics Zk,n themselves, but also their linear combinations, which are often used for parameter estimation in the reliability theory, are representable as sums of independent random variables. The equalities (3.6) and (3.7) remain valid for any arbitrary exponential distribution originating at zero, i.e., when the underlying variables Z1, Z2, ... , as well as V1, v2, ... , have common d.f. H,\(x) = H(x/A), A > 0.

The above relations are particularly important for the reliability theory. Let a system consist of n elements of the same kind working simultaneously, whose life times are mutually independent and have the same exponential distribution with parameter A. Then (3.6) implies that the time intervals between consecutive failures of the elements are independent and have exponential distributions (with corresponding parameters A/n, A/(n - 1), ... , A/2, A). PROOF OF (3.4). It suffices to check that the vectors we compare have the same density function. Putting f (x) = exp(-x), x > 0 into (2.9) we obtain that the joint density of order statistics Zl,n, Z2,,

,

, Zn.n has the form n

(3.8)

11.2....... :n(xl, x2, ... , xn) = n! fl exp(-xy) = n! exp ( - L. X

\ 1

s=1

x=1

if 0 < X1 < x2 < < x,, < oo. The joint density of i.i.d. exponential random variables v1, v2, ... , v, in the domain A = {yy > 0, s = 1, 2, .... n} is given by (3.9)

y,).

9(yl,y2,...,yn) = exp s=1

I. ORDER. STATISTICS

12

The linear change of variables

(v1,uz,...,vn) = ( yt

J2

y1

y2

L,

.. +-yn)

n' n +n-1' n +n-1

with Jacobian 1/n!, which corresponds to the passage to random variables

V1 = n,V2= n

+n121,...,Vn=

+nv21+...+un,

n

has the property

Vl +v2+...+Vn

=y1+y2+...+Yn

< vn < oc. Now (3.9)

and maps the domain A into the domain 0 < v1 < v2 < implies that V1, V2, ... , Vn have the joint density n

0
\\

(3.10) s=1

0

The comparison of (3.8) with (3.10) proves (3.5). REPRESENTATION 3.6. Let U1,n < .

.

< Un,n, n = 1,2,..., be uniform order

statistics related to i.i.d. random variables U1, U2, ... with d.f. F(x) = x, 0 < x < 1. Then for any n = 1, 2, .. . (U1,n,...,Un,n) d ( S1

(3.11)

Sn

Sn+l

Sn+1

where

m = 1, 2, ... ,

Sm = 1/1 + v2 + ... + vm,

and v1, v2.... are as in Representation 3.4. REMARK 3.7. Denote by

Tk = Uk,n - Uk-l,n,

with U0,n = 0 and Un+l,n = 1

k = 1, 2, ... , n + 1,

the spacings, i.e., the lengths of the intervals in the random partition of the unit interval b y the points U1, U2, ... , Un. Then the random variables T1, ... , Tn+1 are symmetrically dependent, and the vectors V1

(T1,T2,...,Tn+1) and

Sn+l'

Vn+1

Sn+1

are equally distributed. Let T1,n+1

T2,n+1 < ...

Tn+1.n+1

be the order statistics of the spacings T1, ... , T,,+ 1. Then Tk,n+d

1=

Vk,n+1 Sn+I

and (3.4) implies that

v+v2+...+

vk

n

n-k+2

Tkn+1°( n+1

)ISn+1,

k=1,2,...,n+1.

3. THREE CLASSICAL F1EPR.ISENTATIONS

PROOF OF (3.11). The joint density of the random variables v1, v2 has the form

I3

, Yn+1

n+I (3.12)

S=1

in the domain A = {y, > 0, s = 1, 2, ... , n + 1}. By the linear transformation ( V I

Y 2 ,

,

with unit Jacobian we derive from (3.12) the joint density f (v1, ... , vn+1) of sums Sl, ... , Sn+1, which is equal to f (vl , ... , vn+1) = exp(-vn+1)

in the domain 0 < v1 < v2 < . . . < vn+1 One more transformation Zln

V1

Xn+1)

((X1,

with Jacobian (vn+l)-n = of S1/Sn+1, . (3.13)

.

,

,vn+1

,....

Vn+1

Vn+1

(xn+1)_n results in the assertion that the joint density

Sn/Sn+1, Sn+1 equals

h(xl, ... )xn, xn+l) = (xn+1)n exp(-xn+l )

in the domain 0 < x1 < x2 < < xn, xn+l > 0. Integrating (3.13) with respect to xn+l, we obtain that the first n random variables S1/Sn+1, ... , Sn/Sn+l have the density r(xl,...,xn) = n!

(3.14)

when 0 < x1 < x2 <

< xn. The comparison of (2.11) with (3.14) estab-

lishes (3.11).

REMARK 3.8. One can derive from (3.14) and (3.13) one more useful fact. Since h(x1,...,xn,xn+1) = r(xl,...,xn)S(xn+1),

where

s(x) = xn

exp(-x)

x > 0,

n.

is the density function of the sum Sn+1, the vector ( Si Sn+1

Sn

Sn+1

is stochastically independent of Sn+1. Furthermore, we can see now that for any n = 2, 3, ... the random variables S1/S2, S2/S3, ... , Sn_ 1 /Sn and Sn are mutually independent (see also Exercise 3.1).

I. 0121)K1t sTAuisric's

lI

REMARK 3.9. Combining the above representations. we can express arbitrary order statistics Xk..,,, related to a d.f. F in terms of sums of i.i.d. exponential random variables V1,V2i...:

G( y1 + ... + Uk

Xk.n

(3.15)

V1 +.... + Vnw1

and

+nvl+...+n-k+11J),

Xk,n ` G ``(1-exp{-\n

(3.16)

where the function G is defined by (3.1).

EXERCISE 3.1. Show that for any n = 2, 3, ... the random variables

Un-1 n n-1 66n=(Un.n)n - ,W2= `Udn} ,...,Wn-1=(Un,n 1 2

M

are independent and uniformly distributed on [0, 1]. This implies the following representation of the order statistics Uk, in terms of products of powers of independent uniformly distributed random variables: (3.17)

{Uk.nik=1 = {Wk,kWk+lktl)

... LVn/nJk=1

EXERCISE 3.2. Show that for any n the following relation holds: (3.18)

(U1,n, ... , Un,n)

(S1, ... , S. I Sn+1 = 1),

where S. = V1 + V2 + ... +. vm,

m = 1, 2, ... ,

v1i v2.... being i.i.d. random variables with common d.f. H(x) = max(0,1 - e-=), i.e., the distribution of the vector of uniform order statistics coincides with the conditional distribution of the vector of sums S1, ... , Sn given that Snt1 = 1. EXERCISE 3.3. For each n = 1, 2, ... , let Uk,n and Vk,n, k = 1, ... , n, denote the uniform order statistics related to samples U1,. .. , Un and V1.... , Vn of i.i.d. random variables with uniform distribution on [0, 1]. Show that Uk,n

d

Uk,mVm+l,n

for any

i < k < m < n.

EXERCISE 3.4. Show that for any set of real numbers c1, ... , cn there exists a linear combination n

Ln = E d,nUm,n M=1

with nonnegative coefficients d2i ... , d, which has the same distribution as n

Tn = > cm.Urn,n. m=1

1. J.\RKOV VROPEIRTY AND CONDITIONAL INDEPENDENCE

15

Lecture 4. MARKOV PROPERTY AND CONDITIONAL INDEPENDENCE OF ORDER STATISTICS

As we pointed out in Lecture 2, when we pass from the original random variables

X1, X2, ... to order statistics, the independence property is lost. It can be shown, however, that for continuous underlying distributions the order statistics form a Markov chain.

We will restrict the proof of the Markov property of order statistics to the case of absolutely continuous underlying distributions. Let the random variables X1, ... , Xn have a common density f. In this case, in order to prove that Xl n, ... , Xn_n form a Markov chain it suffices to show that for any 1 < k < n the conditional density (to be denoted by x1, ... , xk)) of Xk+I,n given all previous order statistics, X1.n = x1, ... , f (U Xk,n = xk, coincides with conditional density f (u I Xk) of Xk+l,n given only that Xk,n = xk Using notation introduced in Lecture 2 we can write the conditional densities f(u I x1,...,xk) and /f (u I xk) as I

(4.1)

flu I X1,...,xk) =

f1,...,k+1:n(x1,...,xk,u) f1....,k:n(x1, ... , xk)

and

(4.2)

f(u I Xk) =

fk,k+l:n(xk,u) fk:n(Xk)

It remains to employ formula (2.8) and check that the right-hand sides of (4.1) and (4.2) are equal to F(u))n-k-l

(n - k)(1 -

(1 - F(xk ))

n-k

f(u) ,

1xk.

This means that the order statistics form a Markov chain. The condition that the d.f. F is continuous is essential for the Markov property of order statistics. It turns out that if the underlying distribution is discrete and its support contains at least three points, then the corresponding order statistics do not form a Markov chain. To prove this fact, it suffices to consider three random variables X1, X2, and X3 taking at least three values. Without loss of generality, we will assume that they are integer-valued and

Pk=P{X=k}>0,

k=0,1,2.

To simplify the proof we will also assume that P{X < 0} = 0. One can easily verify the following equalities:

P{X3.3=2,X2.3=1,X1,3=1}=3pip2, P{X2.3 = 1, X1,3 = 1} = Pi + 3p2 i (1 - pn - PI), P{X3,3 = 2, X2,3 = 1} = 3P1P2 + 6poPIP2, and P{X2.3 = 1} =P3 + 3Pi(1

-

P()

- pl) + 3PoP1 + 6Pop1(1 - po - pl)

1. ORDER sTA'rls'rucs

16

This implies that

P{X3.3=21 X2,:s=1,X1,3=1}=

P{X3,3 = 2, X2.3 = 1, X1.:1 = 1 }

P{X2,3=1,X1.3=1} 3p2 pi + 3(1 - po - p1)

and similarly 3p1p2 + 6poP2

P{X3,3 = 2 I X2,3 = 1} =

Pi + 3p1(1 - po - pi) + 3pop1 + 6po(1 - po - pi)

Let us show that P{X3,3 = 2 1 X2.3 = 1,X1,3 = 1} > P{X3.3 = 2 I X2.3 = 1}.

For that it suffices to establish that

(p1+3(1-po-pl))(pl+2po) < (pi+3p1(1-po-pl)+3pop1+6po(1-po-pl)), but this relation is equivalent to the obvious inequality P, (p, + 2po) < pl + 3pop1,

which completes the proof. The following theorem plays an important role in the treatment of order statistics and especially their sums (which is often needed when dealing with truncated samples arising under the various censorship models).

THEOREM 4.1. Let X1,,, < ... < be order statistics corresponding to a continuous d.f. F. Then for any 1 < k < n the random vectors

X") = (X1,n,...,Xk-l,n)

and

X(2)

= (Xk+l,n, ... , Xn.n)

are conditionally independent given any fixed value of the order statistic Xk,n. Fur-

thermore, the conditional distribution of the vector XM given that Xk,n = u coincides with the unconditional distribution of order statistics Y1.k_I, ...,Yk-l.k-1 corresponding to i. i. d. random variables Y1,. .. , Yk_ 1 with common F(u) (x)

=

F(u),

d. f.

x < U.

Similarly, the conditional distribution of the vector X(2) given the same condition coincides with unconditional distribution of order statistics W1.n-k, ... , Wn-k.n-k related to the d. f. F(x) - F(u) F. (x) =

1 - F(u)

x > U.

4. M\R.KOV PItOPEIUY AND CONDITIONAL. INDEPENDENCE

17

PROOF. To simplify the proof, we will assume additionally that the underlying random variables X 1 .... , XU have a density function f. We see from (2.6) and (2.9) that the conditional density function of order statistics X1. ... , Xk_ 1 , Xk+l, ... , Xn given that Xk = u (to be denoted by f (xl, , xk_1, xk+1, - . xn I u)) has the form - -

f (x

1

,

-

,

xk -

1

,

x 1+

I

,

-,

xn I u) =

f1,2,...,n:n (xl, x2, - .. , xk-1, xk+1,.. , xn ) fk:n(u) n k-1

_ (k - 1)!J1

(4.3)

f(xs)(n

s=1

F(u)

- F(

- k)! II

F(u r=k+1) 1

Since f (x)/F(u), x < u, and f (x)/(1 - F(u)), x > u, are density functions corresponding to d.f.'s Fu and F(u) respectively, all assertions of Theorem 4.1 follow from (4.3) and (2.9). 0

The following result similar to Theorem 4.1 can be useful when dealing with two-sided censoring.

THEOREM 4.2. Let Xl,n _< _< Xn,,i be the order statistics related to a continuous d. f. F. Then for any 1 < r < s < n the conditional distribution of order statistics Xr+l,n, , X,-l,n given that Xr,n = y and X3,n = z, y < z, coincides with the unconditional distribution of order statistics Vl,,_r_1i...,V,_r_1,,_r_l corresponding to i.i.d. random variables V1i...,V,_r_1 with common d.f. -

- -

V.,

=(x) = F(x) - F(y)

y < x < z.

F(z) - F(y)'

Theorem 4.2 enables us, for example, to write down the d.f. of

T = Xr+l,n + ... + X,_1.n in the form

P{T < x} = ff

P{T < x I Xr,n = y, Xs.n = z}frs:n(y, z) dydz

o0
If where

Vy',zs-r-1)

Vy,z

oo
s-r-1

)(x)fr,s:n(y, z) dydz,

denotes the (s - r - 1)-fold convolution of the d.f. Vy,z.

REMARK 4.3. Theorem 4.1 implies a number of useful relations, e.g.,

F(x) P{Xk,n < x I Xk+l,n = v}

(F(v))

k

'

x
and

P{Xk+l,n > x I Xk,n = v} -

(11

-

F(F(vx))

y-k'

X

V.

>

EXERCISE 4.1. Show that for exponential order statistics the conditional distribution of the vector (Zk+i,n, - -, Zn,n) given Zk, = x coincides with the unconditional distribution of the vector (x + Z1,n-k, - , x + Zn-k.n-k)-

-

I. OHIIER S VI ISTICS

Is

EXERCISE 4.2. Let the underlying random variables have a two-point distribu-

tion (without loss of generality, assume that 0 < PI X = 0} = 1 - P{ X = 1} < 1). Show that in this case the order statistics Xl,,,, X2.n, .... X,,,, form a Markov chain. Obviously, the Markov property holds also for degenerate underlying distributions. Compare these statements with the above result for discrete random variables taking at least three values.

Lecture 5. ORDER STATISTICS FOR NONSTATIONARY SEQUENCES

In the classical setup the order statistics correspond to a sequence of i.i.d. random variables. In some cases, however, we cannot assume that the underlying sequence is stationary. For example, when testing for the presence of outliers we may need to know the distribution of order statistics in the situation where at least one of the observations X1i . . . , X, comes from another population. A similar problem arises if we have samples from different populations and we test the hypothesis

that the distributions in all populations are the same against the alternative that one of them (unspecified in advance) is shifted to the right or to the left. One more situation of this kind is related to censored observations where instead of original identically distributed X's we observe Y k = min (bk, max(ak, Xk)),

k = 1, 2, ... , n,

where ak < bk are constants specifying the interval on which the values of Xk can be registered.

We state here two representations which will enable us to apply the welldeveloped theory of summation of independent random variables when dealing with order statistics for non-identically distributed random variables.

Consider independent random variables X1..... X with d.f.'s Fl,..., F and the corresponding order statistics X1 < ... < X,,. Define the indicators Ik(x) by letting Ik(x) = 1 if Xk < x and Ik(x) = 0 if Xk _> x, k = 1, ... , n. Note that for any x the random indicators Ik(x) are independent and take the values 0 and 1 with probabilities Fk(x) and 1 - Fk(x), k = 1, 2, .... The following representation is obvious.

REPRESENTATION 5.1. For any x and k = 1, 2, ... , n (5.1)

If F1(x) distributed.

Fk:n(X) = P{Xk,n, < x} = P{II(x) + --- + In(x) > k}. F. (x), the indicators in the right-hand side of (5.1) are identically

Another representation holds f o r exponential random variables Zk, k = 1, 2, ... , with different scale parameters.

REPRESENTATION 5.2. Let Zk,n, 1 < k < n, be order statistics related to independent random variables Z1,.. . , Zn with d. f. 's

Fk(x) = max{0, 1 - exp(-.kx)}.

'.

l)It)FI STATISTICS FOR vONSI:ATIONARY SILQI EN('ES

19

Then for any n the distribution of the vector {Zk_n}A=1 coincides with the distribution of the vector n!

{Yk}k=nt = Emixtpl

U1

Ar(1) +, *, + Ar(n)

1=1

U2

+

+ .. .

Ar(2) + ... + Ar(n)

n

U-

Ar(k) + . . + Ar(n)

k=1'

where vi, v2i ... are (as in (3.4)) i.i.d. random variables with standard exponential distribution, and n!

n

mixtpl{Wt}k=1 l=1

denotes the random vector whose distribution is the mixture of the distributions of n-dimensional vectors W t = (W1,1, ... , W1,n) with components

_

U1

WI'k

k

+ ... +

Ar(1) + ... + Ar(n)

1
Ar(k) + ... + Ar(n)

taken with weights 7),

('\r(1) + ... + Ar(n))(Ar(2) + ... + Ar(n)) ... Ar(n) '

this mixture involves the distributions of n! vectors corresponding to all n! permuta-

tions (r(1), ... , r(n)) of numbers 1, 2,... , n. Let us point out that the mixture with weights p1, ... , pn associates with (univariate or multivariate) d. f . ' s Fl , ... , Fn the d.f. F=p1F1+...+pnFn.

REMARK 5.3. The following compact form of Representation 5.2 using antiranks was suggested by Tikhov (1991). Let D(1), ... , D(n) be the antiranks corresponding to X1,. .. , Xn, which are defined by

{D(r) = m} _ {X, = Xm},

1 < r < n,

1 < m < n.

One can see that in our case

P{D(1) = r(1),... , D(n) = r(n)} = P{Xr(l) < (Ar(l) + ... + Ar(n))(Ar(2) + ... +

< Xr(n)} Ar(n))... Ar(n)

Then Representation 5.2 can be written in the following equivalent form: n

d

{Zk.n}k-1 =

VI

U2

AD(1) +...+AD(n) + AD(,,)+...+AD(n)

+... Uk

+ AD(k) + ... ±'\D(n) where the vectors ( v i ,... , Un) and (D(1),..

.

,

n

} k=1

D(n)) are independent.

The following two corollaries can be easily deduced from Representation 5.2.

1. ORDER STATISTICS

20

COROLLARY 5.4. The following representation holds:

(Z1,n, Z2.n - Z1,n, ... , Zn,n - Zn-1.n) n!

d

mixtpl

I/1

r(1) +

+ Ar(n)

, Ar(2)

I/2

+

+ )1x(11)'

n

... , In Ar(

COROLLARY 5.5. Let c1,. .. , cn be arbitrary real numbers. Then n

n!

E CkZk.n = k=1

mixtp1711,

1=1

where

Cl+...+C C2+...+Cn ]l = ( Ar(1) + ... + Ar(n) vl + \ )1r(2) + ... + Ar(n)) v2 + ... +

cn I/

Ar(n)

)In,

and the probabilities p1 and the random variables v1, v2,. . ., In are defined in Representation 5.2. PROOF OF REPRESENTATION 5.2. The joint density function A X1, x2,

,

xn)

of the random variables Z1, Z2,. . ., Zn has the form

x1 > 0+ ,xn > 0

f(x1,x2, - ,xn) = al 2

,

and the joint density function f1.2....,n:n(x1 i X2, ... , xn) of the order statistics Z1.n, Z2,n, , Zn,n is given by

e-

f1.2....,n:n(x1,x2,-..,xn) = 1\1)12...)ln (n))

0 < X1 < ... < Xn, where the summation extends over all n! permutations (r(1), ... , r(n)) of numbers (1, 2, ... , n). Next we see that the joint density function of the spacings Zl.n, Z2.n Zn-l,n is given by h(y1,y2,...,yn)

cn>)yt+(a1«1+ +a l >)vs+ +a.(,lyn)

_ 1\11\2 ... An (r(1),..,,r(n))

Plfl(y)f2(y)...fn(y),

Y1 > 0,...,Yn > 0,

(r(1)....,r(n))

where

fk(y) = (Ar(k) + ... + are the densities of the random variables

Ar(n))e-(a,(k) +-.-+a,-(,.))Yk

vk k = 1,...,n, Ar(k) + ... + Ar(n)'

and v1i v2.... are i.i.d. random variables having the standard exponential distribution. The equality (5.2) is equivalent to the assertion of Corollary 5.4, which is 0 easily seen to be equivalent to Representation 5.2.

NIONIEN I S OF ORDER S'I'A I'ISI'I('S

r,.

21

EXERCISE 5.1. Let Zk,,,, 1 < k < n, be order statistics corresponding to independent random variables Z1, ... , Z,, having d.f.'s

Fk(x) = max{O,1 - exp(-Akx)}. Show that the random variable Z1,n and the random vector (Z2.n - Z 1 . " , ... , Zn.n are independent.

EXERCISE 5.2. Show that under the conditions of the previous exercise the random variable Z1,n and the linear combination c1Z1,n +''' + CnZn,n + cn = 0.

are independent if c1 +

Lecture 6. MOMENTS OF ORDER STATISTICS

In the second lecture we derived formulas for the d.f.'s of order statistics. Here we will consider a number of problems about their moments. When the random variables X1,. .. , Xn have a common density function f, we immediately obtain from (2.6) that the moments Nk:n = EXk,n

are given by the formula 00

t

xr(F(x))k-1(1

kn = (k - 1) ( n - k)!

- F(x))n-kf(x) dx.

If X's have an arbitrary common V. F, we can use formula (2.3) for the d.f. of the order statistic Xk,n, F(s)

nt

tk-1 (1 - t)n-k dt, -00 < x < 00, (k - 1)!(n - k)! from which the reader familiar with the theory of Stiltjes integral (see, e.g., Rudin Fk:n(X)

(1966)) can easily deduce the equality (6.1)

(r)

xr dFk:n(X)

k1k:n = 00

°O

1

(k - 1)!(n - k)!

X.(F(x))k-1(1 - F(x)) n-k dF(x).

oo

This equality implies that (6.2)

nt µkr)

(k - 1)!(n - k)! nt

f

Ixlr(F(x))k-1(1 - F(x))n-kdF(x)

a

f.

n1

Ixlr dF(x) = (k - 1)!(n - k)! E(IXIT), (k - 1)!(n - k)! which means that any order statistic Xk,n, 1 < k < n, n = 1, 2, ... , has a finite rth moment provided that this moment is finite for the underlying distribution. Since X l,n < X1 < X,,, for any n = 1, 2, ... , we have IX1Ir < maX{IX1,nj', IXn,nIr} :5

IX1.nIr + IXn.nlr

I. 01il)[1,11? STATISTICS

22

and

E(IXI'') < E(IX...,I'). Thus if for some n = 1, 2.... the extreme order statistics have finite rth moments, then the moment of the same order is finite for the underlying distribution. On the other hand, comparing the integrals

I

'o Ixlr(F(x))k-1(1

r

and

-

F(x))"-k

dF(x)

00

I xl r dF(x),

J we see that for 1 < k < n one can find distributions such that 1 (1-F(x)) n_k IxIr/F dF(x)
J

althou gh

(x)\\ Jk

f

0O

Ixlr dF(x) = 00. 00

This means that when the underlying distribution has no finite rth moment, this moment may exist for intermediate (and even for one of the extreme) order statistics.

EXERCISE 6.1. Consider the Cauchy distribution with density function 7r(1 + x2)'

which has no finite expectation. Since the tails of the d.f. 1 - F(x) and F(-x) are equal asymptotically, as x --+ oo, to 1/irx, one can see from (6.1) that in this case any order statistic Xr,n, k < r < n - k + 1, has a finite moment of order k, but X,,,, has no finite kth moment if r < k or r > n - k + 1. REMARK 6.2. In connection with Example 6.1 we mention without proof the following result by Sen (1959): if E(IX1°`) < 00

for some a > 0, then the moment (r)

µk:n = E(Xtv.. s)r

is finite for all k such that

r a

r
Formula (6.1) enables us to evaluate the moments for any distribution, but by far it does not always provide convenient analytic expressions, especially for large n. Even for the widely used normal distribution the moments can be expressed in terms

of elementary functions only for n < 7. For example, for the standard normal distribution we have 5

15 aresin s

+ 2ir3/2 For many distributions the moments of order statistics have been computed by numerical integration and tabulated (see, e.g., "Tables Guide" in David (1979)). X5.5 =

471/2

6. NUMIEN'I'S OF ORDER S'I'A'I'IS'1'I('S

2:1

We will consider here some methods of obtaining convenient formulas for various numerical characteristics of uniform and exponential order statistics. Besides the moments (about the origin) and variances we will consider the product moments 1 < r < s < n.

!Lr,s:n = EXr,nXs:n,

Note that if the underlying distribution has the density function f, then E(Xk n! (6.3) ,(r) n) = (k - 1).(n k:n = - k).Ti

r(.F(x))k-1(1

- F(x))n-kf(x) dx,

and, as follows from (2.8), (6 . 4)

n!

= EX r,n X s:n = (r - 1)!(s - r - 1)!(n - s)! r°° oo xy(F(x))k-1(F(y) - F(x))9-r-1 (1 - F(x2))n-sf(x)f(y)dxdy.

Ar,s:n X

J

oo

x

Exponential Distribution. It suffices to consider only the standard exponential distribution. Let Z1,n < .. < Zn,n be exponential order statistics related to the sequence of i.i.d. random variables Z1, Z2.... with density function

f (x) = exp(-x),

x > 0.

Then, as we know, for any n = 1, 2.... relation (3.4) holds:

,- +

Vl

d (Z1,n,Z2,n,...,Zn,n)= n

Vl

V2

n

n -1

V2 ,..., Vl-+ +...+Vnll/ n n 1

where vl, v2,... are i.i.d. exponential random variables with V.

H(x) = max{0,1 - exp(-x)}. Since EVk = 1 and Var vk = 1, k = 1, 2, ... , (3.4) implies that (6.5)

(6.6)

µk:. = E(Zk,n) = E (VI n + n 1

1

n

n-1

n-k+1 1

Var(Zk,n) = Var (I!1 + n

1

vk

v2

n-k+1' 12

n-1

1

n2 + (n --1)2

+ ... +

+ ... +

VR

n-k+1 I 1

(n

- k + 1)2'

and

(6.7)

cov(Zr.n, Zs,n) = E(Zr,n - EZr,n)(Zs,n - EZs,n) = Var(Xr.n) _ 1 1 1 +...+

- n2 + (n-1)2

(n-r+1)2'

r<s.

I. ORDER STATISTICS

24

Uniform distribution. Now consider the order statistics Uj.n < n = 1, 2, ... , related to the sequence of i.i.d. random variables U1, U2

F(x)=x,

< U,,,n,

.. with d.f.

0<x<1.

Since for any n = 1, 2.... we have the relation (see (3.11)) Uk n d

Sk

,

Sn+1

where Sm = vl + v2 +- + v..., m = 1, 2, ... , with vl, V2.... being i.i.d. exponential random variables, we immediately obtain from symmetry considerations that ESk

(6.8)

EUk,n= Sn+1

_

vi -kE(Sn+1)

=

k

(Sn+l

k

n+lESn+l n+l'

and that the expectations of the subintervals (spacings) into which the interval [0, 1] is divided by the random points U1, U2, ... , Un are equal to 1/(n + 1) each. For evaluation of an arbitrary moment E(Uk,n)" it is more convenient to use the expression (2.10) for densities of the uniform order statistics, lxk+a-1(1 n! (6.9) E(Uk,n)a = J 1xaJk:n(x) dx = (k - 1)!(n - k)! fo

_ B(k+a,n-k+1) _

B(k,n-k+1)

-

x"-k dX

n!r(k+a) r(n+a+1)(k- 1)!

To derive (6.9) we have used the equality

B (a, b) -

r(a)r(b)

r(a + b) relating the gamma function r(s) and the beta function B (r, s), as well as the fact that r(n + 1) = n!, n = 0,1,... For a = 1, (6.9) implies (6.8). Next, 2 EUk,n =

EUk,n =

_ k(k + 1) n! (k + 1)! (n + 2)!(k - 1)! (n + 1)(n + 2)' k(k + 1)(k + 2) (n + 1)(n + 2)(n + 3)'

and

(6.10)

2 Var(Uk,n) = EUk,n -

(EUk,n)2=

k(n-k+l) (n + 1)2(n + 2)

Note that Var(Uk,n) = Var(Un_k+1,n). The product moments can be evaluated with the help of representation (3.17):

ln - 1. W1/k W1/(k+l) ... Wl/n n fUk,nJk=1 n Jk=1+ l k k+1 where W1, W2, ... are i.i.d. random variables uniformly distributed on [0, 11. Let us find first the moment E(Ur,n, Us,n) for r < s. Observe that for the uniform distribution on [0, 11

EW" =

1

a+1

G. MOMENTS OF ORDER STA'I'IS'I'1(.:S

25

Then it follows from (3.17) that (6.11) We+js+l)

E(Ur,nUs,) = E(Wr /r Wr+Ir+l) ... Wn/n Wy /s

... Wn/n)

... Wn/n = E(W, /r Wr+lr+l) -W / is-1) Ws /s e 1))E(W$ /a)E(W +(8 +1)) ... E(W,, = E(Wr Ws+ls+l)

/r)E(Wr+ir+l))

-

l(

l

r

l(

... E(Wn/n)

l

1 ((1+r)(1+r+1)...(1+311)(1+s)(1+s+1/...(I+n)l-

r(s + 1)

r(r + 1) ... (s - 1)s(s + 1)...n

(n + 1)(n + 2)'

(r + 1)(r + 2)...s(s + 2)(s + 3)...(n + 2)

In a similar manner one can find arbitrary moments of the form E ((Ur(1),n)a(l)(Ur(2),n)a(2) ... (Ur(k),n)Q(k)) . For example, (6.12) E ((Ur,n)O(Us,n)a) +( r+l) ... WsZls-1) W;a+Q)/s W;+I 0)/(a+1) ... = E(W,?/r w

Wna+l3)/r+)

1

r(r + 1)...(s - 1)s...n

(r+a)(r+1+a)...(s-1+a)(s+a+a)(s+1+a+a)...(n+a+a) r(n + 1)r(r + a)r(s + a + a)

= r(r)r(s+a)r(n+1+a+ Next, we can derive from (6.8) and (6.11) the expression for the covariance of uniform order statistics:

r(s + 1)

rs

(n + 1)(n + 2)

(n + 1)2

(6.13) cov(Ur,n, U,,n) =

r(n - s + 1) =

r < s,

(n + 1)2(n + 2)'

and (6.10) and (6.13) enable us to evaluate the corresponding correlation coefficient (6.14)

P = P(Ur,n, Us,n) =

cov(Ur,nr Us,n )

_

(Var(Ur,n) Var(Ua,n))1/2

r(n - s + 1) 1/2 s(n - r + 1) /

l

For some d.f.'s F, relation (3.2) combined with expressions for the moments of uniform order statistics enable us to evaluate the necessary characteristics of the corresponding order statistics Xk,n. EXAMPLE 6.3. Consider the standard power distribution having d.f.

F(x)=x',

0<x<1.

In this case (6.15)

Xk,n

(Uk,n)l/a

and EX n = E(Uk,n)Q/Q =

n!r(k +

r(n +

+ 1)(k - 1)!

In particular, (6.15) yields (6.16)

EXk n =

n!r(k +

r(n+ + 1)(k - 1)!

and EXk n =

n!2 (k +

r(n + a + 1)(k - 1)!

I. ORDER STATISTICS

26

hence n!

(6.17)

var(XA..,,) = (k - 1)!

n!(r(k +

r(k + 2-,)

(r(n + Q + 1))2(k -1)!

r(n+ n + 1)

In a similar way, using the equality it

(Xr,n)y(Xs,n)a

and formula (6.12) we can obtain the product moments E(Xr,n)'(Xs,n)°. For example,

EXr.nXs.n = E(Ur.n)1/a(Us,n)1/a =

(6.18)

r(n+ 1)r(i + Q)r(s+

)

r(r)r(s + i-)r(n + 1 + 2

EXERCISE 6.1. Prove the following relations for the moments of exponential order statistics:

1
Alrn=n

and µ'kn-Ak_1:n-1+n rµkrnl)+

n

EXERCISE 6.2. Consider the order statistics Xk,n related to the d.f.

F(x) = 1 - x-ry,

x > 1,

where -y > 0, and find the moment characteristics similar to (6.15)-(6.18). For what -y does the moment E(Xk,n)a exist? EXERCISE 6.3. Let X1 and X2 have jointly the bivariate normal distribution N(a1, a2, vi, U22, p), and let a2 = al + a2 - 2pal a2

and

a = (a2 - al )I a.

Show that

a > 0, if a = 0,

EX2,2 = al + a(ac(a) + sp(a))

if

EX2,2 = max(a1, a2)

and

where 4)(x) and p(x) denote the d.f. and the density of the standard normal law.

Lecture 7. MOMENTS OF ORDER STATISTICS (CONTINUED)

Here we state some relations which may be useful for evaluation of moment characteristics of order statistics. If two distributions are related by the equality

Y=aX+b,

a>0,

it is easily seen that the same relation d=

Yk,n

aXk,n + b

holds for order statistics. When a < 0, the passage from X's to Y's reverses their ordering, which leads to the equality d

Yk,n = aXn-k+1,n +b.

i. MOMENTS OF ORDF.li

27

Hence under these linear transformations of random variables we get the following relations for expectations and variances: EYk,,, = aEXk,,, + b,

Var Yk,,, = a2 Var Xk,,,

for

a > 0,

and

EYk,,, = aEXn-k+l,n + b,

Var Yk, = a2 Var Xn-k+l.n

for

a < 0.

Note that not only variances, but all central moments do not depend on b. If our sample is drawn from a symmetric distribution (i.e., if X = -X), then the above arguments lead to the equalities d

Xk,n d -Xn-k+l,r1, EXk,n = -EXn-k+1.n, Var Xk ,.n = Var X. - k + l . n

For joint distributions of order statistics the following analog of (7.1) holds: (7.4)

(Xs(1),n, ... , X.,(,),,,)

(-Xn-s(1)+1,n) ... , -Xn-s(r)+l.n)-

This equality implies the following relations for the covariance and the correlation coefficient in the case of symmetric distributions: COV(Xr,n, Xs,n) = COV(Xn-r+l.n, Xn-s+l.n),

P(Xr,n, X,,.n) = P(Xn-r+l,n, Xn-s+i.n)

Moreover, it follows from (7.2) that for symmetric distributions the expectations of sample medians Xk+1,2k+1, k = 1, 2, ... , are equal to zero, provided, of course, they exist. A number of relations for moments of order statistics can be deduced from obvious equalities n

[nom

/

\uX )m = j=1

j=1

and 71

n

n

(7.6) i=1 j=1

n

Xi,nX n = E E X1Xj , i=1 j=1

which hold for arbitrary n, k, 1, m = 1, 2.... For example. taking m = 1 in (7.5) we obtain that n

+1

EX n = E > Xk = nEXk,

(7.7)

j=1

j=1

while (7.6) for l = k = 1 yields the equality n

(7.8)

n

n

n

E E EXi,nXj,,, = E E EXiXj = nEX2 + n(n - 1)(EX)2. i=l j=1

i=1 j=1

For some particular underlying distributions one can derive moment relations making use of specific properties of these distributions. This concerns, first of

1. OR.DF.R. STATISTICS

28

all, the normal law. As is well known, if X1,. .. , Xn are independent and have a common normal distribution, then the vector (X1-X, ... , Xn - X) and the sample + Xn)/n are independent. This implies that any random mean X = (X1 + variable of the form f (X I - X_., X, - X) is independent of X. Furthermore, if we arrange the components of the vector (XI - X, ... , Xn - X) in ascending order, and hence any function of the then the resulting vector (Xl,n - X) ... , Xn,n form g(X l,n - X, ... , Xn,n - X), is independent of X. In particular, for the normal distribution the differences between order statistics (including the so-called sample range Xn,n-X1,n) are independent of X. This implies that for the standard normal distribution one has for any r = 1, 2, .. .

E(XT,n - X)X = 0 or

E(Xr,nX) = E(X)2 = 1/n.

(7.9) Since

nX = X1 + ... + Xn = X1,n + ... + Xn.n, we obtain from (7.9) that n

(7.10)

r = 1,2,...,n.

FIE(Xr,nX,,n) = 1,

8=1

Next we consider some asymptotic relations for moments of order statistics. It follows from (3.4) that the expectations and variances of exponential order statistics Z,N,n satisfy equalities (6.5) and (6.6):

E(Zkn)=,1-+ n n

1

1+...+ n-k+1'

and 1

1

Var(Zk,n) = W2- +

+ ... + (n

n - 1)2

1

- k + 1)2

Observe that for n -> no the following asymptotic relations hold:

E(Zn,n)=

n+ n-1+ 1

1

+1=logn+ ry +o(1),

where ry = 0.577215... is the Euler constant, Var(Zf,n) _ 2 + (n 11)2 +...+1 = (7.11) E(Z(Qn)+1.n)

=1 n

1

}

n- 1

+ ... +

2

6

+O\n

n- (anj = - log(1 - a) + O (n1), 1

and (7.12)

where 0
Var(Z(an)+1,n) _ (1

a)n +O(n2),

7. MONMENTS OF ORDER. STATISTICS

0n 211

Let us mention that the order statistic X!nn)+l,n is called the sample a-quantile

(0 < a < 1), and for a = 1/2 it is called the sample median. For the exponential sample median we have from (7.11) and (7.12)

E(Z!n/21+1,n) = µ1/2 + O() and Var(Z!n/21+1.n) _

+

(n )

as n - oo, where µ1/2 = log 2 is the median of the standard exponential distribution. Similar equalities for uniform order statistics are derived from (6.8) and (6.10):

E(U!.n)+l,n) = a + 0(n) ' all a) Var(U)an1+1,n) = +0( n

1

),

0 < a < 1.

In particular, for the median of the sample from the uniform distribution on [0, 1] we get

E(Z(n/2)+1,n) = A1/2 +0(1 n)

and

Var(U(n/2)+1.n) =

1

4n

1 +0 (\7L2)

as n -+ oo, where µ1/2 = 1/2 is the median of the uniform distribution. We know from Representation 3.1 and Corollary 3.2 that the order statistics Xk,n, k = 1, 2, ... , n, related to the d.f. F satisfy the distributional equality ( X 1, , (X1,,,,,

.

.. , X.,.) A (G(U1,n), ... ] G(Un,n)),

where G(s) is the inverse function to F. Therefore a number of relations for moments of arbitrary order statistics Xk,n can be derived from the knowledge of the moments of Uk,n. We will demonstrate the method which allows one to write down the moments of Xk,n in the form of series expansions in powers of n (more precisely, we will obtain expansions in powers of (n + 2)) whose coefficients are expressed in terms of moments of uniform order statistics. We will assume that the function G is continuous and has r continuous deriva-

tives G00, s = 1, 2, ... , in a neighborhood of the point k

a = ak,n = EUk,n = n +

Using the Taylor expansion about the point a we obtain EXk,n = EG(Uk,n)

G(2)(a)(2k.n - a)2 + ... = E(G(a) + GO)(a)(Uk,n - a) + a)r-I)

(7.13)

+ G(r-1) (a)(Vk,n (r - 1)! = G(a) + G(1)(a)E(Uk,n - a) +

+ An

G(2)(a)E(2 k.n - a)2 + ...

+ G(r-1)(a)E(Uk.n - a)r-I + An, (r - 1)!

30

1.

OItDGI3. STATISTICS

where the remainder 0 is bounded by a quantity of order Ir

IG(r)(a)I EIUk,n r!

Without discussing the accuracy of this approximation we will only rewrite the right-hand side of (7.13) in a simpler form: EXk.n = G(a) +

+ ... + G(r-1)(a)E(Uk.n - a)r-1

G(2) (a) Var Uk,n 2

(r - 1)!

+

Using the expression for the variance obtained above, VarFT, ,n

=

k(n-k+1) (n + 1)2(n + 2)'

and the numerical values of the central moments E(Uk. - a)' for s = 3,4 we arrive at the following approximate equality: (714)

EX k n

G(a) +

+

G(2)(a)a(1 - a) 2((G(3)(a)(1 n + 2)

\

((+

2a) + 1 G(4) (a)a(1 - a) I .

In a similar manner we obtain approximate expressions for variances and covariances: (7.15)

all

VarXk,n

2(n(+ 2)(a))2 + a(l

a)

(2(1

- 2a)G(1)(a)G(2)(a)

+ a(i - a) (G(1) (a)G(3) (a) + 1 (G(2) (a) )211

cov(Xk.n, Xm,n) :

abG(n1)(+)

ab

(1)(b)

+ (n +

2

2)2((1 - 2a)G(2)(a)G(1) (b)

+ (1 - 2b)G(1)(a)G(2)(b) + 2a(1 - b)G(3)(a)G(1)(b)

(7.16)

+ 1b(1

- b)G(1)(a)G(3)(b) + 1a(1 - b)G(2)(a)G(2)(b)).

In (7.16), b = m/(n + 1). Formulas (7.14)-(7.16) and their refinements involve derivatives of the function G. For example, if the underlying distribution has a differentiable density f, with f(1) denoting its derivative, then

G

(1)

1

(x)

7(G ((x))

and

G(2)(x) _

f(1)(G(x)) f3(G(x))

In general, formulas for G(') (x) become cumbersome and computationally involved

already for r = 3, but for particular distributions they may have rather simple form. For example, for the logistic distribution with density

f(x) =

exp(-x) (1 + exp(-x))2

OF ORDER srxrls'rl(rs

31

the following equalities hold:

G(x) = logs - log(1 - x), G(r)(x) = (r

- 1)1(-1)r-1 I

- (1

1X)r I,

0 < x < 1.

For the standard normal distribution with density

f(x) =

1

(27r)1/2 exp

(_

1

z/

2x

the inverse function C cannot be expressed analytically in terms of elementary functions, but the identity f l1) (x) = -xf (x) enables us to obtain comparatively simple formulas for G(') (x): 1

f (G(x))' G(x)

f2(G(x))' 1 + 2G2(x)

f3(G(x)) G(x) (7 + 6G2(x))

f4(G(x)) 7 + 46G2(x) + 24G4(x)

f5(G(x)) and so on. EXERCISE 7.1. Show that the relation COV(Xr,n - X)X = 0

valid for the standard normal distribution, implies that for any r = 1, 2, ... , n and n = 1,2.... the following equality holds: n

ECOV(Xr,n,X3,n) = 1. s=1

EXERCISE 7.2. Show that for any continuous distribution, any k, m = 1, 2, ... , and n = 1,2? ... the following equality holds:

E(Xr nX n) = n(n - 1) E(X 2X2 2).

2

1
EXERCISE 7.3. Show that for any continuous distribution, for all n = 1, 2, all in = 1, 2, ... , and 1 < r < n the following equality holds:

rE(Xr+l,n)m + (n - r)E(Xr,n)'n =

nE(Xr,n_1)'n.

...

1. ORDER STATISTICS

32

Lecture 8. ASYMPTOTIC DISTRIBUTIONS OF THE MIDDLE ORDER STATISTICS

When dealing with large samples we have to use asymptotic distributions of various statistics, including order statistics. Consider the order statistics Xk,n

for k = 1,2,... , n. We are interested actually in the limiting distributions as n --r o0 of the random variables Xk(n),n, suitably centered and normalized when necessary. Depending on the behavior of the sequence k(n), the order statistics Xk(n),n are referred to as extreme (when k = k(n) or n - k(n) is fixed), middle

(when 0 < lim infn_,,, k(n)/n < limsupn-. k(n)/n < 1), or intermediate (when k(n) -+ oo, k(n)/n 0 or n - k(n) -+ oo, k(n)/n 1 as n , oo). Sometimes the term extremes (extremal order statistics) is used only for the maximal (Xn,n) and minimal (X1,n) members of the sample, while the order statistics Xk,n (kth minimum) and Xn_k+l,n (kth maximum) for fixed k = 2,3,... are referred to as "kth extremes." We will study here the asymptotic behavior of the middle and intermediate

oo as n - co.

order statistics. We will assume that k = k(n) - 00 and n - k Let, for definiteness, k < n/2, i.e., k < n - k.

Exponential distribution. Consider first the exponential order statistics Representation (3.4) implies that Zk,n can be written as a sum of inde-

Zk,n.

pendent terms, d VI

Zkn

V2

Vk +n-1+...+n-k+l'

n where vi, v2.... are i.i.d. random variables with standard exponential distribution. Since

k=1,2,...,

a=Evk=1 and a2=Varvk=1, we have n

1

EZk,n = E

n

and

Var Zk,n =

m=n-k+l Let us point out also that

1

L, ,m2 . m=n-k+l

k=1,2,...

7=Elvk-Evk13=6-10e-1,

Representation (3.4) enables us to study the asymptotic behavior of order statistics Zk,n using well-known results for sums of independent random variables. In particular, we will need the following Lyapunov inequality (see, e.g., Petrov (1987)): if X1, X2, ... are independent random variables with expectations al, a2, ... , vari-

ances vi, v, ... , and finite third moments

7k=EIXk-ak13,

k

then

P{ >(Xk - ak) < xBn}

-

'(x)I < CLn,

k=1

where

Bn=Ol+...+Un, Ln = Ek=1 yk/Bn is the Lyapunov ratio, C is an absolute constant, and 4 is the standard normal d.f. Before applying the Lyapunov inequality in our case, we will

8. ASYMP'T'OTIC DISTRIBUTIONS

3:3

estimate the Lyapunov ratio for the sums

n +

+

n-k+1'

Using the inequality

n-k --= n 1

1

n

j,n- x- dx> nk m=n-k+1

-;m2-> 1

n+1

1-2,4, -

-k+1

n-k+1 - n+1' 1

1

we obtain k

Var Zk, -

(8.1)

n(n - k)I

<

1

(n - k)2

In a similar way we obtain the inequality k

rn =

E m=1

3

um -E( n-m+1 n-m+1

v-

=ry

m3

m=n-k+1

so that the Lyapunov ratio L,, = rn/(VarZk,n)3/2 for the sum of independent random variables

n

+...+n-k+1 1/k

n-V2

1

is bounded by 8-y

L <

n - min{k1/2, (n - k)1/2} Therefore the Lyapunov inequality implies that (8.2)

sup IP{Zk,n - EZk,n < x(VarZk,n)1/2} - (p(x)I < x

C min{k1/2, (n - k)1/2}'

where C is an absolute constant and (D(x) denotes the standard normal d.f. Taking into account (8.1) and the fact that n

EZk,n= E m=lognk+nek, n m-k+1

where 191 < 1, and using elementary inequalities for -D(x) of the form I

Pk

(x) - '(x - E)I :5 Iel(27r)-1/2, 4) (Ax) - 4) (x) < {

A > 1, if 0 < A < 1, if

(27re)-1/2(A - 1)

(27re) -1/2(1/A - 1)

(see, e.g., Petrov (1987)), we can rewrite (8.2) as (8.3)

sup x

n l {Zk,n-log\n-k, <x( n(n-k)

) 1/2

Cl min{k1/2, (n - k)1/2}'

where C1 is again an absolute constant.

(x)

1. ORDFR STATISTIC'S

31

(slip

If k = k(n) = (an.[ + 1, 0 < a < 1, which corresponds to the sample a-quantile then one can easily deduce from (8.3) the estimate

l1/2

sip P{Z(rtn+lJ.n + log(1 - a) < x(

(1

a)/

C(a)

1 - t(x) < nl/2

where the constant C(a) depends only on a. In a similar way one can prove asymptotic normality of a sum of exponential order statistics n

Tn = E CmZm,n, m=1

which by (3.4) has the same distribution as the sum

, n

Sn =

bmvm,

m=1

where bm =

cm

n-m+1

+...+ 21

+cn,

m= 1,2,...

In this case n

n

ETn=ESn

bm,

VarTn=VarSn=>bm, m=1

m=1 and

n M=1

where -y = 6 - 10e-1. Then the Lyapunov ratio becomes Ln

n

rn

(VarSn)3/2

3

(Em=1 b2m)3/2

and, whenever Ln -+ oo, we have

supIP{Tn - ETn < x(VarTn)112}-4) (x) I-+0

as n -+oo.

X

Uniform distribution. In the case of uniform distribution we will immediately estimate the rate of convergence for linear combinations of order statistics. Let, as usual,

0=Uo,n
n=1,2,...

denote the order statistics for the uniform distribution on [0, 11. Consider a linear combination n Tn = T CmUm,n m=1

with arbitrary real weights c1i c2, ... and denote n

b,n=ECk, 'm =1,...,n, k=m

Then we have the following estimate.

1

bn+1=0,

n+l

b=7l+r bk. k=1

S. ASYMPTOTIC DIST111BUTIONS'

T-1

THEOREM 8.1. For any I < k < n and n = 1, 2.... the following inequality holds:

sup IP{Tn - ETn < x(VarTn)1/2} - (D(x)I < C

(8.5)

x

'm `I Ibm - b1:1 (G.m=1l(b, - 6)2)3/2

where C is an absolute constant.

If we take Ck = 1 and cm = 0 for m # k in Theorem 8.1, then T = Uk.R and In this case for for

ETn =

k

VarTn =

n+

n+1 (b

m- b) 2 }

3/2

m=1

and

n+1

=(

k(n-k+l) (n + 1)2(n + 2)'

k(n - k + 1)

3/2

n+1

k(n - k + 1)(k2 +(n-k+ 1))2

3

m=1

Now we can easily check that

2 n+1 M=1 bm

(n - k +

k3/2

bl3

i

1)

3/2

(n - k + 1)1/2(n + 1)3/2 + k1/2(n + 1)3/2

{Em=1(bm, - b)2}3/2 1

1

1

(n-k+1)1/2+k1/2' which leads to the following important corollary to Theorem 8.1.

COROLLARY 8.2. For any 1 < k < n and n = 1, 2,... the following inequality holds:

(8.6)

sup P{Uk,n x

- n k-1

1

<xI2121 -fi(x)

where

/32=VarTn=

1

< C (n - k + 1)1/2 + k1/2

k(n-k+1) (n + 1)2(n + 2)

and C is an absolute constant.

Note that inequality (8.6) ensures asymptotic normality of suitably centered and normalized order statistics Uk,n if min{k, n - k + 1} -+ oo as n --+ oc. PROOF OF THEOREM 8.1. We obtain from Representation 3.6 for uniform order statistics that n

ETn=

mcm

n+1 =

7 -n+ b' = n+1

bm n

VarTn =

1

r i(n - j + 1)c,c4) m(n - m+ 1)cm + 2 (n + 1)2(n +1 2) r 1<,<j
__ (n + 1)2(n + 2)

E bm

2

(n + 1)2(n + 2) i<:<;
I. ORDER STATISTICS

36

and r

n+1

n+1

PIT, -ET,<x}=P{ >bmv,n<(x+ETn)Evm} l m=1 n+1

m=1

=P{ 1: (bm - x - ETn)vm < 0}, m=1

where v1, v2,... are i.i.d. random variables with standard exponential distribution. Denote n+1

E=>(bm-x-ETn)vm M=1

Then (8.7)

P{Tn - ETn <x} =PIE <01 =P {

(VaE-EE

rE)'/2 < - (Var E) 1/2

where n+1

n+1

(8.8) EE = E (b,n - x - ETn) = E b,n - x(n + 1) - (n + 1)ETn = -x(n + 1) m=1

M=1

and n+1

n+1

(8.9)

VarE= 1: (bmx-ETn)2= 1: (bm-ETn)2+(n+1)x2 m=1

m=1

_

n+1

n+1

b2 _ (Em=1 6m)

m=1

M

2

n+1

+ (n + 1)x2.

The equality EIVm.l3=Ev,3n=6,

m=1,2,...,

and inequality Ia + bl3 < (lal + Ibl)3 < 4(lal3 + lb13)

imply that n+1

n+1

(8.10) 1'n = E EI(bm.-x-ETn)vml3 < 24{ E Ibm-x-ETnl3+(n+1)Ixl3}. m=1

M=1

We will need the following result by Bikelis (1966).

THEOREM 8.3. Let X1, X2, ... be a sequence of independent random variables with expectations a1, a2 .... , variances o , 02, ... , and finite third moments rym =

EIXm -aml3, m = 1, 2, ... Then for any n = 1, 2.... the following inequality holds: (8.11)

(Xm - am) < xBn} - $(x)I < C

P{ m=1

Ln

1 + Ixl3'

rIoNS

4.

37

where Bn = (L.m=1 am)1/2r Ln = j:nt=1 ym/Bn is the Lyapunov ratio, and C is an absolute constant. Applying (8.11) to the sum of independent random variables n+1

E = E (bm - x - ETn)vm m=1

and invoking relations (8.9) and (8.10) we arrive at the bound

E - EE

(8.12)

_

P{(VarE)1/2 <

C

<

In (Var E)3/2

EE

EE

(VarE)1/2} \- (VarE)1/2) rn ( 1 + IEE13 \

(Var E)3/2 J < C (Var E)3/2 + IEEI3

En 11

Ibm - ETn I3 + (n + 1)IxI3 m=1(bm - ETn)2}3/2 + (n + 1)3Ix13 n+l 3 1 +

{En+l

'm=1Ibm-ETnI

{Em=1(bm - ETn)2}3/2 n+1

C

.m=1 Ibm - ETnI

(n + 1)2

3 ETn)2}3I2.

{Lsn 11(bm -

Note that in our formulas C may mean different absolute constants. The last inequality in (8.12) is based on the inequality Em==11 Ibm - ETnI3

(8.13)

{n li(bm - ETn)2}3/2 - (n + 1)1/2

which implies that n+1 3 Zm=1 Ibm - ETnI+

{m=l(bm - ETn)2}3/2

n+1 < m=1 Ibm - ETnI 3 2 (n + 1)1/2 - {>n+ 11(bm - ETn)2}3/2 1

To prove (8.13) in the most transparent way, it suffices to introduce the random variable 17 taking values b1, b2, ... , bn+l with equal probabilities. Then E1r7 - Er7I3

n+1 m_

1

rn+1

13 nbm -ETn

+

E' 177 - E,712 =

1

'

6

- ETn)2

and (8.13) follows directly from the well-known inequality for moments (EI77 - E7713)2 > (EI77 - E1)12)3.

Now we obtain from (8.7)-(8.9) and (8.12) that (8.14)

IP{Tn - ETn < x} - 4i

`

_

=IPIT, -ETn<x}-4i(

EE (Var E)1/2)

x(n+1)

{Em 11(bm - ETn)2 + (n + 1)x2}1/2 J

n+1

C

lI

m=1 Ibm - ETnI

3

{Em=11(bm - ETn)2}3/2

= C.

n+1

Em=1 Ibm -

bI

3

{Em 11(bm - b)2}3/2

1. ORDER STATISrlc5

:3114

Observe that n+I

n+I

1: (b,n -ETn)2 = > (bm -b)2

m=I

m=1

n+1

2

n+11
Ebnt n+1 m=I and denote

x(n + 1)1/2 U

(n + 2)1/2(VarTn)1/2

and

B2 = (n + 2) Var Tn n

_

n + 1

n+1

n (n + 1)

g

b2 m=1 m

-

2 (n + 1)

btb

3

1<1<j
Then (8.14) can be rewritten as (8.15)

sup

P{Tn-ETn
Y

2

y

(1+y /

n+1

3

By standard manipulations with the normal d.f. we obtain (8.16)

sup ID(y) -,D( (1 v

y +y2/(n+ 1))1/2

C ) < - n+1

Now the conclusion of the theorem follows from (8.13), (8.15), and (8.16).

Lecture 9. ASYMPTOTIC DISTRIBUTIONS OF THE MIDDLE ORDER STATISTICS (CONTINUED)

Having obtained the asymptotic distribution of uniform order statistics and using Representation 3.1 we can derive the limiting laws for arbitrary order statistics. First we give heuristic arguments which will enable us to guess the possible form of limiting distributions.

Let X1.n < X2,, < ... < Xn,n, n = 1, 2, ... , be order statistics of a sample from d.f. F (sufficiently smooth to justify the arguments to follow). Since for any 1 < k < n we have the representations F(Xk,n) = Uk,n and

Xk.n.

G(Uk.n)r

where G is the inverse function to F, we can use the fact that the ratios (Uk.n

k

n3/2

- n+1 ) k'/2(n - k)1/2

,1SYMI'I'O'I'I(' DISTRIBUTIONS (CON IINI EDT

!).

:11)

are asymptotically normal whenever min(k, n - k) -> oc as n this fact in the following form: (F(Xk,,,)

(9.1)

-

k

n3

l

n+1/

O. We can rewrite

d

k(n - k)

where 17 has the standard normal distribution. Since F(Xk.,,) can be represented as

F(Xk,n)

(9.2)

F(G(n+

k

111 + (Xk'n

k

n+ 1 + (Xk'n - G(n+

G(n+ 1) /F'(G\n+ 1)/F'(G(n+

111'

(9.1) and (9.2) imply that 3 (Xk'n-G(n+1F'(G\n+1JI

d

k(n - k) z

77

and, in particular, we should expect that the sample quantile X(Q) = X1Qnl+t,n, 0 < a < 1, will fulfill the relation

sup I P((X(a) - G(a)) f (G(a))n1/2a-1/2(1 - a)-1/2 < y} - p(Y)I v

0,

where f (x) = F'(x) (the existence of a density of the underlying distribution is unnecessary here, it suffices that the M. F be differentiable in a neighborhood of the point G(a) with F'(G(a)) > 0). The situation becomes more complicated if f (G(a)) = 0. In this case G(a) is a minimum point of the function f and f'(G(a)) = F(2) (G(a)) = 0 if F is twice differentiable in a neighborhood of the point G(a). Then F(Xk'n)

n+ 1 +F'(G(n+ 1Jl (Xk'n +

2F(2)(G(n+1)1(Xk,n

+1F(3)(G(nk

- G(n+ 111 -G(ri+1)/2

(Xk,n-G(ri+1)13

and

Ulan)+1,n

d

F(X(a)) PL- a + F'(G(a)) (X(,,) - G(a))

+

1

(X(0) - G(a))`

1 F(2)

pa))

+ 1 F(3) (G(a)) (X(") - G(a))3 a + 1 F(3) (G(a)) (X(Q) - G(

In this case it is the random variable 1(X(,7)

- (G(a))3F'(3) (G (a)) a(ln a)

Q))3.

I. ORDER

to

which has the limiting standard normal distribution. Now we will prove the following theorem. THEOREM 9.1. Let the underlying random variables X1, X2, ... have a differentiable density function f such that

sup If'(x)I < M < oo.

where the supremum is taken over the set A = {x: f (x) > 0}. Then for any k = 1, ... , n and n = 1, 2.... the following inequality holds:

sup x

Pl (Xk,n - G(k/(n + 1))) f (G(k/(n + 1)))

<

x11

- t(x)l

Qn/2

M 3,V 2

<

C(k-1/2

+ (n - k + 1)-1/2 + f2 (G(k/(n

l

+ 1)))

where

R. =

k(n - k + 1) (n + 1)2(n + 2)

and C is an absolute constant.

PROOF. Since

P{Xk,n <x + G(n

+llJ=

P{F(Xk,n) < F(x

+G( k n + 111 J

=P{Uk,n
(l/+ 2 J'l xzB

where 181 < M, by using inequality (8.6) we obtain (9.5)

P

(Xk.n - G (n + 1 / I f (G ( n + 1 ,n/2

)1

l <x+2f

P{_/z(Uk'n

n+11

j

x < z 4120

Gn))

1

)))+O(k-''2+(n-k+1)-1/2) n

fi(x) + O (k-1/2 + (n - k + 1)-1/2 + 2f`2 (G(n )) 2

)'

which already implies (9.4). Let us point out that in the proof of (9.5) we used the estimates for closeness of the values of the normal d.f. at two close points which were given in Lecture 8.

9.

,ASYiNIP'I'O'I'I(' DISTRIBUTIONS (CON'rINVED)

41

COROLLARY 9.2. Under the conditions of Theorem 9.1, if f (G(a)) > 0, the following bound holds for the sample quantile X(,,) = X[af]+I.n:

P{ (X(,) -

G(a)) f

(G(a))nl12

<

a1/2(1 - a)1/2

x} -

(x) < C(a, M)n-1/2,

where C(a, M) is a constant depending only on a and M. Thus the sample a-quantile is asymptotically normal

a 21 - a)

N G(a), of (G(a)) It was shown already by Mosteller (1946) that if we consider r sample quantiles X(a(1)), X(a(2)), .. , , X(a(r)),

where

0 < a(1) < a(2) < . . . < a(r) < 1,

then their joint distribution converges to the r-variate normal distribution with expectations

G(a(l)), G(a(2)), ... , G(a(r)) and covariances COV(X(a(i)), X(a M) =

a(i)(1

a(7))

of (G(a(i)))f (G(a(j)))'

1
provided that the density f is differentiable and positive in neighborhoods of points G(a(k)), k = 1, 2, ... , r.

EXERCISE 9.1. As is well known, the sample mean X = (X1 + - - + is asymptotically normal N(a, a2/n), where a and o2 are the mean value and the variance of the underlying distribution. For distributions symmetric with respect to the unknown parameter a, X and the sample median X(12) are often considered as competitors in estimation of a. Calculate the asymptotic efficiency of X(12) with respect to X, i.e., the asymptotic ratio of the variances of these estimates for the underlying densities -

f, (x) = (2 11/2 exp{-(x - a)2/2} and

f2(x) = 2 exp{-Ix - all. We have presented a number of results related to asymptotic normality of middle and intermediate order statistics. In the next lecture we will consider the asymptotics of extremal order statistics.

I. Ol 1)I-,R SIAl'ISI'I( S

12

Leci,ure 10. ASYMPTOTIC DISTRIBUTIONS OF MAXIMA

There are numerous applied problems related to extremal (maximal or minimal)

order statistics. For example, an architect who is going to design a skyscraper in California will be interested in the possible magnitude of the maximal earthquake in this region during, say, the next 100 years. The constructors of dams in the Netherlands are supplied by statisticians with forecasts of the maximal levels of the North Sea. The viability of many systems is determined by the weakest element. Hence one has to know the distribution of the minimal life time of the various elements of the system. In agriculture, growing some plants (e.g., rice) requires the minimal precipitation for each ten-day period to exceed some fixed level. Many similar examples could be given by meteorologists, hydrologists, ecologists. As far back as in early 18th century N. Bernoulli was interested in the distribution of extremal insurance premiums. The fundamentals of the theory of extremes were laid in the first half of the 20th century by Frechet, von Mises, Fisher and Tippett. This period of development of the theory culminated in appearance of the outstanding paper by Gnedenko (1943) that summarized the studies of the limiting distributions of extremes in the classical setup dealing with i.i.d. random variables. It should be mentioned that de Haan (1976) substantially simplified the exposition of the basic results of the theory of extremes. The further development of the theory was related to the study of kth extremes (note here the results by Smirnov (1949)), multivariate extremes, extremal values for dependent observations. In the recent period the studies of extremes became more oriented to applications (specific mathematical models intended for various applied problems, estimation of parameters of the asymptotic distributions of extremes, assessment of the accuracy of approximation of finite sample distributions by asymptotic distributions). In what follows we restrict ourselves to a brief account of the basic ideas of the classical theory of extremes. A detailed exposition of this theory can be found in the books by Galambos (1978, 1987) and Leadbetter, Lindgren, and Rootzen (1983).

Let X1, X2, ... be a sequence of i.i.d. random variables with the common V. F. Consider the maxima

M,, = X,,,n = max{XI,... , X,,,},

n = 1, 2, ... .

We know already that

F,,(x) = P{M,, < x} = F'(x). As is well known, the classical theory of summation of random variables establishes

all possible asymptotic distributions for suitably centered and normalized sums S, = Xl + + X,, (which are stable distributions) and describes their domains of attraction, i.e., it associates with an underlying distribution the corresponding limiting distribution. What asymptotic distributions can be obtained if we consider the maxima M,, instead of sums We begin with several simple examples.

EXAMPLE 10.1. Let F(x) = max{0,1 - exp(-x)}. Then for any fixed x and sufficiently large n we have the equality

P{Mn - logn < x} = (F(x+logn))" = (1

- exp(-x) n

J

'

10. AsY\IF''Io'rIC I)ISI'Rlilt ''VIO;NS O1 MAXI\I:\

I:1

therefore

(10.1)

P{M,, -logn<x}-+A(x)=exp(-exp(-x))

as

n

EXAMPLE 10.2. Let F(x) be equal to the d.f. A(x) defined in (10.1). Then

P{M,, - logn < x} = (A(x + log n))' = (exp ( - exp(-x - logn)))" = exp ( - exp(-x)) = A(x). i.e., Xl and the centered maximum (Mn - log n) have the same distribution for

each n = 1,2,.... EXAMPLE 10.3. Consider the Pareto distribution with

F(x) = 1 - x-Q,

x>1, o>0.

Then

P{Mn/7tt/a < x} =

(1 -

(xnl/°)-°)n

x>0, x<0.

- exp(-z-°),

0,

Hence in this case the limiting d.f. has the form (10.2)

(DQ(x) =

(0

for x < 0,

S` exp(-x-a)

for x > 0.

EXAMPLE 10.4. Let

F(x)=1-(1-x)Q,

0<x<1, a>0

(in particular, for a = 1 this is the uniform distribution on [0,1]). Then

P{(Mn - 1) n l/Ot < x } = ( xn -1/° + 1 ) " = (1

-

_an )

)

exp

(- (- x )°)

if x < 0 and

P{(MM - 1)nl/° < x} = 1 The limiting d.f. in this case has the form

for

x > I.

exp (- (-x)a)

for

x < 0,

1

for

x > 0.

(10.3)

'PQ

(x)

In Examples 10.1-10.4 we obtained three types of possible limiting distributions for maxima. Of course, for any d.f. F one can find a sequence of normalizing

constants {an} such that the sequence of random variables llln/an will have a degenerate limiting distribution. Therefore we are interested in finding all possible nondegenerate limiting distributions for centered and normalized maxima Al,,. They are described in the following theorem by Gnedenko, which summarized the numerous studies in this area. Before formulating it, let us point out that the type of the distribution F, along with the d.f. F itself, comprises all d.f.'s F(ax+b), where a > 0 and b are arbitrary

constants. Two d.f.'s Fl and F2 belong to the same type if there are constants a > 0 and b such that F2 (x) = Fi(ax + b).

1. ORDER STATISTIC'S

44

THEOREM 10.5. The set of all nondegenerate distributions for suitably centered and normalized maxima consists only of distributions that belong to the types A(x), 4),,(x), and `yn(x), a > 0.

REMARK 10.6. For different positive values of the shape parameter a the d.f.'s (%, (and, similarly,',) determine different types of distributions. For the proof of the theorem we need several lemmas. The following lemma is due to Khinchine. LEMMA 10.7. Let a sequence of d.f 's Fn converge weakly to a nondegenerate d. f. G, i.e.,

n - oc, Fn(x) - G(x), for any continuity point of G. So that the sequence of d.f.'s Hn(x) = Fn(anx+bn) converges, for some constants an > 0 and bn, to a nondegenerate d.f. H it is necessary and sufficient that an --+ a, bn --# b as n -+ oc, where a > 0 and b are some constants, and H(x) = G(ax + b). PROOF. Let us prove necessity. Since H is a nondegenerate d.f., one can find

its continuity points y and z (-oo < y < z < oo) such that 0 < H(y) < H(z) < 1. Without loss of generality, we will assume that y = 0 and z = 1 (which can be achieved by a linear transformation). Moreover, one can find continuity points u and v of function G such that

-00
0 < G(u) < H(0) < H(1) < G(v) < 1. From the last inequality, the definition of functions G and H, and monotonicity of d.f.'s Fn we conclude that

u
-oo 0 and b such

that as k -+ oo. ar(k) -+ a and bn(k) -+ b Therefore for any fixed e > 0 and k large enough the following inequalities hold:

a-e 0 and arbitrary e > 0 the inequalities Fn(k) (ax + b - e( x + 1)) < Hn(k) (x) : F,.(k) (ax + b + e(x + 1))

10. ASYMIF''I'O'I'IC DISTRIBUTIONS OF AIAXI%IA

a5

hold for k large enough. Letting n(k) tend to infinity and then e tend to zero, we obtain H(x) = lim Hn(k) (x) = G(ax + b). Similar arguments can be applied for negative x. Thus we have shown that for almost all x (except for at most countably many points)

H(x) = G(ax + b), i.e., H and G belong to the same type of distributions. Now we show that for different sequences n(k), k = 1, 2, ... , tending to infinity the sequences an(k) and bn(k) cannot have limits a1 and b1 different from a and b. Indeed, if there were such a1 and b1, then for almost all x the equality G(ax + b) = G(ai x + b1)

would hold, implying that for any growth point x" of H

ax*+b=alx*+ b1. Since H is nondegenerate, there are at least two such points, hence a1 = a and b1 = b.

The proof of sufficiency is obvious.

We are interested in possible limiting (as n -+ co) d.f.'s H(x) for sequences

Hn(x) = (F(ax + bn))n, where F is the underlying V. and an and bn, n = 1, 2, ... , are some centering and normalizing constants. LEMMA 10.8. In order for a nondegenerate d.f. H to appear as the limit of a sequence (F(anx+bn))n for some d. f. F and constants an > 0 and bn, n = 1, 2, ... , it is necessary and sufficient that for any s > 0 and x (10.4)

H6 (A(s)x + B(s)) = H(x),

where A(s) > 0 and B(s) are some functions defined for s > 0. PROOF. Sufficiency is obvious since if (10.4) holds, one can take F = H as the

underlying V. and an = A(n) and bn = B(n) as the centering and normalizing constants. Let us prove necessity. Of course, along with the relation (10.5)

Hn(x) = (F(anx + bn))n -. H(x),

the relation (10.6)

(F(ain3ix+b(n31))in"l -+H(x),

oc,

n

holds for any s > 0, where [x] denotes the integral part of x. This implies that (10.7)

(F(ain3ix + b(,,3)) n -+ (H(x))1/',

n -+ oo.

Observe that (10.7) can be rewritten as (10.8)

Hn (an(s)x + /3 (s)) -+ (H(x))113,

I. omwit s'l'A'I'IS'I'I('S

16

where

an.(s) = a(,,.I/a > 0 and Nn(s) = b(n.a) - b,,. Comparing (10.5) with (10.8) we obtain by Lemma 10.7 that for any s > 0 there are constants A(s) and B(s) such that an,(s)

A(s) > 0,

fn(s) -+ B(s) > 0,

and

H(x)'I' = H(A(s)x + B(s)).

This completes the proof of Lemma 10.8.

Thus we reduced the problem of finding limiting distributions of maxima to finding all solutions of the functional equation (10.4). This problem can be solved using the following well-known result, which we formulate without proof. The first part of this statement can be found, e.g., in Galambos (1978, 1987), the second part follows easily from the first. LEMMA 10.9. If a monotone function v(t) for all t and s fulfills the equality

v(t + s) = v(t)v(s),

(10.9)

then either v(t) = 0 or v(t) = exp At, where A is a real constant. If v(t) fulfills the equality

v(t + s) = v(t) + v(s),

(10.10)

then v(t) = At, where A is a real constant. Now we can proceed to the proof of Theorem 10.5.

It is seen from Examples 10.1-10.4 that all d.f.'s of types A(x), ' (x), and WQ(x), where a > 0, satisfy (10.4). We will prove that there are no other nondegenerate d.f.'s satisfying (10.4). Taking logarithms of both sides of (10.4) twice, we obtain the equality (10.11)

-log(-log(H(A(s)x+B(s)))) -logs=-log(-logH(x)),

which holds for all x such that 0 < H(x) < 1. Denoting by T(x) the inverse function to V(x) log (- log H(x)) we can rewrite (10.11) in the form

A(s)x + B(s) = T(log s + V(x)). This equality is equivalent to the relation

A(s)T(x) + B(s) = T(log s + x), which implies that

A(s) (T (x) - T(0)) = T(log s + x) - T(log s). Substituting y = logs and denoting AI(y) = A(el)

and

Tj(x) = T(x) - T(0),

we arrive at the equality (10.12)

Ti(x + y) - Tj(y) = A1(y)Ti(x)

Relation (10.12) and the equality Ti(x + y) - Ti(x) = A,(x)TI(y)

1

1. DOMAINS OF ATTRACTION

17

symmetric to (10.12) imply that (10.13)

T1(y)(1 - A, (x)) = 71(x)(1 - A1(y)).

Consider two cases.

(i) Let A1(x) - 1. Then (10.12) implies T1 (x + y) = Ti (x) + Ti (y).

By Lemma 10.9, T! (x) = .\x, which implies that

H(x) = exp{- exp(-,\x + a)},

where \ and a are some constants. Since H(x) is a d.f., we observe that '\ > 0 and H belongs to the same type as A. (ii) Assume now that there is a point x with Al (x) 54 1. We will show that in 1 for any x # 0. Assume that A! (y) = 1 for some y # 0. Then this case A, (x) it follows from (10.13) that Tl (y) = 0, and (10.12) implies that T, (x + y) = T! (x)

for any x, which contradicts the fact that T! (x) has at least one growth point. Consequently, A1(x) 0 1 for all x 34 0. Fix some y = yo

0 and denote

T1(yo) c

1 - A1(yo)'

Then we see from (10.13) that for all x (10.14)

T1(x) = c(1 - A1(x)).

Using (10.14) we deduce from (10.2) the relation (10.15)

A1(x+y) = A1(x)A1(y)

whose solution by Lemma 10.9 is T1(x) = exp Ax. It is obvious that A 0 0. If A > 0, it can be easily shown that H(x) belongs to the same type as the d.f. 4Q(x) for a = 1/A, while for A < 0 the function H(x) belongs to the same type as the d.f. !PQ(x) for a = -1/A. The proof is completed. 0

EXERCISE 10.1. Show that if F(x) = cI (x), then the limiting d.f. for the random variable Mn/n1/Q coincides with (PQ.

EXERCISE 10.2. Show that if F(x) = WQ(x), then the limiting d.f. for the

random variable Mn"' ncoincides with 41Q. Lecture 11. DOMAINS OF ATTRACTION FOR THE LIMITING DISTRIBUTIONS OF EXTREMES

After finding all possible limiting distributions, the second important issue in the theory of extremes is the description of the corresponding domains of attraction D(A), and D(',,) of the three limiting types. That is, for each underlying d.f. F we wish to indicate the corresponding limiting distribution A. 4iQ, or 41Q. It should be pointed out, first of all, that there are d.f.'s F for which the only possible limiting distribution of maxima is the degenerate one. For example, if X1, X2,... are independent and take two values 0 and 1 with probabilities p and 1 - p, then

P{A,IM = 0} = pn and P{Mn = 1} = 1 - pn. Therefore loin can have only the

1. ORDER STATISTICS

48

degenerate limiting distribution at point 1. Less trivial examples of this kind are given by geometric and Poisson distributions. A continuous d.f. which possesses this property is, e.g., F(x) = 1 - I/ log x, x > e. There are necessary and sufficient conditions for an underlying d.f. F to belong to each of the domains of attraction D(A), D(4),,), and D(WY0), but their form is cumbersome and the proofs are tedious (see, e.g., Calambos (1978, 1987)), so that we present only some sufficient conditions and some necessary conditions close to them.

THEOREM 11.1. Let d. f. F have positive derivative for all x > xo. If for some

a>0

lim

(11.1)

xF'(x) = a,

xoo 1 - F(x)

-

then F E D((Da). The centering, bn, and normalizing, an, constants can be taken to be

n=1,2,..., bn=0 and an=G(1- 1), n

(11.2)

where G is the inverse function to F. PROOF. Denote

a(x) =

xF'(x)

1 - F(x)

Since

a(t)

f.x o

t

dt = f x d( - log(1 - F(u)))

log(1- F(x)) + log(1 - F(xo)),

o

we obtain that

1 - F(x) _ (1 - F(xo)) expI( - f

a(t) dt y, t

11

for x > xo, and for an = G(1 - 1/n) we have

1-F(an)=1, n

n=1,2,...,

and

n(1 - F(x)) = exp { - J any a(t) dt1 = exp { - r a(ant) dt y. l t l 1 t Condition (11.1) and the definition of an imply that an -+ oo as n -+ oo and for any fixed t the integrand in the right-hand side of (11.3) tends to a/t, hence n(1 - F(anx)) for any fixed x > 0 tends to x-° as n -+ oo. It remains to observe that

f

(11.3)

(11.4)

Hn(x) = P{ = (

Mn an

< x} _

(F(aux))"

) exp(-x- ) 1 -r-nx-+ n

We have shown that Ha(x) converges to

H(x) = exp(-x-')

as

n-+oc.

It. DOMAINS OF ATTRACTION

49

for any x > 0. Since H(0+) = limxlo H(x) = 0, we have that H,,(x) for x < 0 also tends to zero. Thus we have shown that F belongs to the domain of attraction of the limiting d.f. REMARK 11.2. The following necessary condition which we formulate without proof is close to the sufficient condition in Theorem 11.1: if the d.f. F has a density

which is monotone in the domain {x: 0 < F(x) < 1} and F E D(I ), then (11.1) holds. Note also that if F E D(4),,), then F(x) < 1 for all x < oo. Now we give the corresponding conditions for the limiting d.f. 41Q. THEOREM 11.3. Let d. f. F have positive derivative F'(x) for x in some interval

(x1,xo) and F'(x) = 0 for x > xo. If for some a > 0 we have (11.5)

lim

x-xo

(xo - x)F'(x) = a 1 - F(x)

then F E D(WYa). The centering, bn, and normalizing, an, constants can be taken to be (11.6)

bn=xo

and an=xo-G(1- 1), n

n=1,2,...,

where G is the inverse function to F. The proof is similar to that of Theorem 1(1.1. For x E (x1,lxo) (11.7)

1 - F(x) = (1 - F(xl)) expI( -

where

Jx1

a(t) dt xot

JJJ

a(t) _ (xo - t)F'(t) 1 - F(t)

Since G(1) = xo, we have that an - 0 as n -* oo. It follows from (11.7) that for

x<0

(11.8)

rn(x) = n(1 - F(xo + anx)) = exp { -

ll fxo-a.

= exp { - fl-x

a(xo t

a(t) xo - It

l

dt } JJ

ant) dt JJJ

We see from (11.5) and (11.8) that

rn(x) -+ (-x)°i

as

n -. oo

for any fixed x < 0, and then

Hn(x) = p f Mn - bn

l = (1

an

< x} = (F(anx + bn))n

)n -+ H(x) = exp (- rn(x) (-x)-Q) n

Since H(0-) = 1, we obtain that Hn(x) --p 1 as n -+ oo for x > 0. Thus F E D(qla).

I. ORDER STATISTICS

50

REMARK 11.4. It can be shown that if F has a monotone density in the domain {x: 0 < F(x) < 1} and F E D('Y,.r), then condition (11.5) holds, i.e., the sufficient condition given in Theorem 11.3 is close to the necessary and sufficient one. It can

also be shown that if F E D(',,), then sup{x: F(x) < 1} < oc. Now we formulate a theorem on the domain of attraction of A.

THEOREM 11.5. Let d.f. F have negative second derivative F(2)(x) for x in some interval (x1, x11), and let F'(x) = 0 for x > x0. If (11.9)

limo F(2)(x)(1

X

-

F(x))

then F E D(A). The centering, bn, and normalizing, an, constants can be taken to be

(11.10)

bn,=G(1- -') and an=f(bn), n

where

f(t) =

1 - F(t) F'(t)

The proof is similar to the proofs of Theorems 11.1 and 11.3.

REMARK 11.6. It can be shown that if F has a monotone increasing second derivative F(2) (x) and F E D(A), then condition (11.9) holds.

EXERCISE 11.1. Show that if F is the standard normal d.f., then F E D(A), and the centering and normalizing constants can be taken to be bn = (2logn - loglogn - log47r)112

and

an = 1/bn.

EXERCISE 11.2. Show that the Cauchy distribution with d.f.

F(x) = 2 + - arctanx belongs to the domain of attraction D(4?1), and one can take

a=n and b=0. Lecture 12. SOME TOPICS RELATED TO THE DISTRIBUTION OF EXTREMES

In two preceding lectures we studied asymptotic behavior of maximal order statistics. These results can be easily carried over to minima,

Inn = min{X1,...,Xn} = X1_.. This is done by using the following obvious relation between maxima and minima: (12.1)

lnin{X1,...,X} = -max{Y1i...,Y},

Yk = -Xk,

k = 1,2,... .

For positive random variables X1,X2,... the following equality may also be useful: (12.2)

min{X1,... , X,} =

1

max{V1,..., Vn}

,

Vk = 1 . k = 1, 2,... . Xk

12. SOME Rr-'.[.AI'Er) TOPICS

51

It is seen from equality (12.1) that its suitably centered and normalized righthand side can converge in distribution to a limiting d.f. L(x) satisfying the equality L(x) = 1 - H(-x), where H(x) is the limiting d.f. for maxima which corresponds to the underlying d.f.

F(x) = 1 - F(-x + 0) with F(x) = P{X < x}. Accordingly, L(x) must belong to one of three types of limiting distributions for minima, which we will write as (12.3)

(12.4) (12.5)

A(x) = 1 - A(-x) = 1 - exp ( - exp(x)), -oc < x < oc, 1 - exp (- (-x)-a) for x < 0. fa(x) = 1 - 4ia(-x) = S for x > 0, l 1, 1 - exp(-xa) for x > 0,

Wa(x) = 1 - IF,, (-x) =

f

for

0,

x < 0.

where a > 0. _ Denote by D(A), D( $a), and D(Wa) the corresponding domains of attraction of the limiting distributions for minima. It is easily seen that the d.f.'s from the domains of attraction of the limiting laws for maxima and minima are related in the following way: (a)

F E D(A)

F E D(A);

(b)

F E D(4ia)

F E D(4i0);

(c)

F E D(Wa) b F E D('Pa).

Since the asymptotic behavior of maxima and minima is determined by the right and left tails of the underlying d.f. F respectively, one can easily give examples where

_ F E D(A)

and simultaneously F E D(4ia)

or

F E D('Pa) and F E D(Wp) or, say,

F E D(4ia)

but F belongs to none of the domains D(A), D(4ia), D('ia).

Let us point out that if F is the d.f. of a symmetric law, then simultaneously

F E D(H) and F E D(H), where H denotes one of the symbols 4ia, 'Pa, or A. EXERCISE 12.1. Using (12.1) and the material of the previous lecture, find the form of centering and normalizing constants in the limit theorems for minima. EXERCISE 12.2. Show that if

F(x) = 1 - exp(-x),

x > 0,

then F E D(f1). What are centering and normalizing constants in this case? Let us return to the three types, A, 4ia, and %Pa, of limiting distributions for maxima. It turns out that all these distributions can be presented in a unified form

1. ORDER. S 1'A'l'IS'rI( S

52

using the so-called generalized distribution of extreme values. For this it suffices to introduce the one-parameter family of d.f.'s

H(x, a) = exp ( - (1 +

xa)_1"0'),

-00 < a < oo,

which are defined by this formula in the domain 1 + xa > 0 and are equal to zero or one (depending on the sign of a) for 1 + xa < 0. If a > 0, then H(x, a) has

the same type as ' j/Q(x). For a < 0 the d.f. H(x,a) coincides, up to a shift and scale transformation, with W_1/,(x). For a = 0, by H(x,0) is meant the limit of H(x,a) as a -+ 0, and we obtain that H(x,0) is equal to A(x). With all limiting d.f.'s written in this unified form, the problem of determination of the type of limiting distribution based on a sample of maxima reduces to estimation of the unknown parameter a or to testing certain hypotheses about this parameter. After we complete the study of the asymptotic behavior of maxima, let us see what will change if instead of Xn,n or Xl,n we consider the kth maxima Xn-k+1.n or the kth minima Xk,n as n -+ oo with k fixed. We will formulate some results for the sequence of the kth maxima Xn-k+l,n The corresponding results for the kth minima can be immediately obtained from the theorems for Xn_k+l,n if we use the following obvious extension of (12.1):

Xk,n = -Yn-k+l,n, where Yl,n <

< Yn,n are the order statistics related to the random variables

Y1 = -XI, Y2 = -X2, ..., Y. = -Xn. It is not difficult to deduce the following theorem from comparison between the formulas

Fn,n(anx + bn) = (F(anx + bn))n and k-1

Fn-k+l,n(anx + bn) _

\

(n I (F(anx + bn))n-m (1 - F(anx + bn))m. m m=0

THEOREM 12.1. For any fixed x the relation

Fn,n(anx + bn) -+ H(x),

n -+ oo,

holds if and only if for any fixed k = 1, 2,... we have Fn-k+l,n(anx + bn) -+ Hlkl (x) =

k-i H(x)(- log H(x))m

E

M=0

m!

This theorem immediately implies that there is a one-to-one correspondence between limiting d.f.'s

H(x) = A(x), fa(x), 'PQ(x) for maxima and H(k)lx) = A(k)(x), (pnk)(x), jp(k)(5)

12. SOME RELATED TOPICS

5:3

for kth maxima, where k-1 (12.6)

A(k)(x) = exp ( - exp( -x))

eX mx) E p(

m=0

> (- 100. (x))'

(D(k)(x) = pa(x) (12.7)

m!

m=

=0

= exp(-x-a)

k-t -am m=0

M!

'

x>0,

and

(-logT.(x))m

Tak)(x) = Va(x) (12.8)

m!

m=0

= exp(-(-X)a)

(-x) 1am

L m=0

'

x<0.

REMARK 12.2. The centering and normalizing constants and the structure of the corresponding domains of attraction do not change when we pass from maxima to kth maxima.

We conclude with the following useful fact: for any fixed r and s the rth maximum and the sth minimum are asymptotically independent as n -+ oo. EXERCISE 12.3. Show that the d.f.'s A(k), yak), and yak) defined by (12.6)(12.8) can be written in the following equivalent form: A(k)(x) = (p (ak)(x)

=

lp ak)(x)

=

1

(k - 1)! 1

(k - 1)!

f

xp(-x)

f e-tt-1 dt, e-ttk-1 dt,

1

(k - 1)!

a-ttk-1 dt,

(

x)o

- oo < x < oc, x > 0, x < 0.

PART 2

Record Times and Record Values Lecture 13. MAXIMA AND RECORDS Consider a random sample X1,. .. , Xn of size n from a population with continuous d.f. F. Wilks (1959) found the distribution of the minimal number of additional observations N(n) which are needed in order to get an observed value exceeding Mn = max{X1, ..., X. 1. It is easily seen that (13.1)

P{N(n) > m} = P{ max{Xn+1i...,Xn+m} < lbtn)

_ x Fm(u) d(Fn(u)) = n rm Fm+n-1 (u) dF(u) x f x =n

,,,m+n-1

Jo

dv =

n

n+m'

and then 00

(13.2)

00

x

EN(n) =O r P{N(n) > m}On+m = r n = n E k1 = 00.

The form of the right-hand side of (13.1) prompts one more (quite elementary) proof of this equality. Indeed, by continuity of F the independent random variables Xl,... , Xn+m take different values almost sure, and since all of them have the same distribution, the events {Mn+,,,, = Xk}, 1 < k < n + m, are equiprobable, i.e.,

P{ Mn+m =Xk}-n+m 1 It remains to note that

P{ max{Xn+l,... , Xn+m} < Mn} = P{Mn+m = Mn} n

n

= I: P{Mn+m = Xk} = n+m k=1 By symmetry, we will obtain the same distribution as in (13.1) if we consider the additional number of observations needed to get an observation falling into the random interval (-oo,mn), where mn = min{X1....,Xn}. Now, let N(1,n) 55

2. RECORD TIMES AND RECORD VALUES

56

denote the minimal size of the additional sample containing at least one observation falling outside the interval [mn, MnJ. Then

P{N(1,, n) >r m} = P{mn < Xn+k < Mn, k = 1, 2, ... , m} (13.3)

= J x00 Ju (F(v) - F(u))mn(n - 1)(F(v) - F(u))n-2 dF(u) dF(v) = n(n - 1) J

_

0o Ju

F(u))-+n-2

(F(v) -

dF(u) dF(v)

n(n - 1)

(n+m)(n+m- 1) 00

oo

x

I 0o its

(n + m) (n + m - 1) (F(v) - F(u))

n+m-z

dF(u) dF(v)

n(n - 1)

(n+m)(n+m-1) PI-00 < mn+m < Mn+m < oo} _

n(n - 1)

(n+m)(n+m-1) and 00

00

(13.4) E N(1, n) _ > P{N(n) > m} = n(n - 1) m=0

M=0

1 (n+m-1)(n+m)

00

= n(n - 1) E

m=n m(`

1

)

=

n(n - 1) E (m l l

1)=n.

m=n

Like (13.1), the relation (13.3) also admits a simple proof. In this case the problem can be reduced to the following one: out of n + m places, two places are randomly chosen for allocating the maximum and minimum on them, and we have to find the probability that the numbers in both places do not exceed n. An elementary calculation gives for this probability the value

n(n - 1) (n + m)(n + m - 1) It is of interest to compare (13.2) with (13.4). We see that E N(n) = oo already for n = 1, whereas E N(1, n) < oo for any n = 1, 2, .. . Consider the simplest case of the Wilks' problem: we observe the random variable X1; denote L(1) = 1 and find in the sequence X2, X3, ... the first observation (to be denoted by XL(2)) exceeding X1. It follows from (13.1) that

1,2,...,

P{L(2) > (13.5)

P{L(2) = j} = P{L(2) > j - 1} - P{L(2) > j} 1

j = 2,3,...,

and (13.6)

E L(2) = oo.

13. MAXIMA AND RECORDS

57

Along with the random index L(2) define the index

L(2)=min{j> 1: Xj <X1}. Obviously (13.5) and (13.6) remain valid with L(2) replaced by Z(2). It is curious

that P{ min (L(2), L(2)) = 2} = 1 and E (min (L(2), L(2))) = 2, although

E L(2) = E L(2) = oo. The variables L(2) and Z(2) are the upper and lower record times respectively, and

X(2) = XL(2)

and

X(2) = XL(2)

are called the upper and lower record values. In view of (12.1) the results for minima mn and various functionals thereof can be easily obtained (restated) from the corresponding results for maxima Mn. For this reason we will restrict ourselves

to the upper record times and upper record values and refer to them simply as record times and record values. We have already introduced the record time L(2). Define the subsequent record times L(n) by the recursive relation (13.7)

L(n + 1) = min{ j > L(n) : Xi > XL(n) },

n = 2,3,...,

i.e., having observed the sample X1i ... , XL(n) of size L(n) and the maximum ML(n) = XL(n),L(n) = max{X1, ... , XL(n)}, we find in the sequence XL(n)+1, XL(n)+2,

the first random variable (with index L(n + 1)) that exceeds ML(n). The random variables (13.8)

X (n) = XL(n) = XL(n),L(n) = ML(n),

n = 1, 2, ... ,

are called record values. If we replace the strict inequality in (13.7) by ">" we will obtain weak records.

This corresponds to the situation when the repetition of a record mark is also counted as a record. For continuous underlying distributions the passage from ordinary records to weak records does not make any difference. For this reason we will invoke the weak records in Lecture 16 when dealing with record times and record values for discrete distributions. There is another way to define the record times and record values. Consider the nondecreasing sequence

-oo<M1 <M2:5 ... <Mn-1 <Mn <... and find those indices n (to be denoted by 1 = L(1) < L(2) < ...) for which the strict inequalities

Mn-1 <Mn hold. These indices coincide with record times, and the maxima ML(n),

n=1,2,...,

form the sequence of record values.

Now we define the record indicators e1 = 1

and

n=

n = 2, 3, ... ,

iR

2

PEC0111) 'I'INIK'S AND Id- ;COW) VALI ES

which are equal to one if M, > Mr_ 1 and to zero if M < M _ 1. Thus 51 = 1 if n is a record time, i.e., if n coincides with one of the variables L(1). L(2),.... The following lemmas establish two important properties of record indicators. LEMMA 13.1 (Renyi (1962)). Let X1, X2.... be independent random variables with common continuous d.f. F. Then the indicators 1;1,f2.... are mutually independent and 1 P{Sn=1}=n, C

n=1,2,....

LEMMA 13.2. Under the conditions of Lemma 13.1 for any n = 1, 2.... the random variables 1;1;1;2; ... , l; and Mn are mutually independent. PROOF OF THE LEMMAS. Take an arbitrary n > 2. We see by symmetry that

k = 1.2,... , n.

P{en = 1} = P{Xn = Mn} = P{Xk = By continuity of the d.f. of X's,

P{Xa=X3}=0 for any i # j. Then 1 = P{Mn = X1} +

+ P{Mn = Xn} = nP{Mn = Xn}

and

Pfjn=1}=P{Xn=Mn}=n.

(13.9)

The indicators 1;2,1.3, ... take only two values. Therefore, for the proof of the lemmas it suffices to show that for any x, r = 2, 3, ... , s = 1, ... , r, and 1 < a(1) < a(2) <

< a(s) < r

the following equalities hold: 1,...,1;«(2) = 1, 111,. <x}

(13.10)

=1} ... P{I.(2) =1}P{Mr < x}.

=

Indeed, it is not difficult to deduce from (13.10) that this equality will continue to hold with some ones replaced by zeroes. In order not to overburden the proof we will consider only the cases s = 1 and s = 2. In the former case, taking into account (13.9). we obtain

P{I.(1) = 1, Al" < x} = P{max(X1,... , X,,(1)_1) < X,,(1) < x;

=

(F(x))r-a(i)

x _

(F(u))Q(1)-1

cc

= P{l;Q(1) = 1}P{Mr < x}.

dF(u) =

X,.) < x} FT(x)

a(1)

I

I.

1)IS'I'RIl1l"1'IONS OF HECOHD TIMES

.7

In the latter case 1, M,. <x}

= P{max(Xt,...,Xa(1)-1) < I ) ), ... , x

_ (F(x))r-a(2) f

x;

X, (2)- I) < X,t(2) < x: maX(XQ(2)+1.... , Xr) < x} r

f

(F(u))a(1)-1(F(v))a(2)-a(1)-1

xu

dF(v)dF(u)

Fr(x) = 1}P{A1r < x}. a(1)a(2) The equality (13.10) for an arbitrary s can be proved in a similar way.

0

EXERCISE 13.1. Let X1, ... , Xn be i.i.d. random variables with a continuous d.f. and let X1,n < < Xn,n be the corresponding order statistics. Define the random variable N,(n) as the minimal number of additional observations Xn+l, Xn+2, ... appearing until the first value exceeding Xn-m+i,n occurs. Find the distribution and the expectation of the random index N,n(n) for m > 1 (for m = 1 see (13.1) and (13.2)). EXERCISE 13.2. Find the conditional expectations P{N(n) > m I Al" = Xk}, k = 1, 2, ... , n, and P{N(n) > m I Mn = x} for the random index N(n) defined in the beginning of the lecture. Lecture 14. DISTRIBUTIONS OF RECORD TIMES We already know from (13.5) that

P{L(2)=j}=j(jl

j=2,3,...

1),

Using Lemma 13.1 one can show that P{L(1) = 1, L(2) = j(2),...,L(n) = j(n)}

(14.1)

1

(j(2) - 1)(j(3) - 1) ...(j(n) - 1)j(n)'

if 1 = j(1) < j(2) < ... < j(n)

Indeed, P{S1 =

=

Mj(2) = 1,£j(2)+1 =

j(3) =

1,

j(n-1)+1 =

1}

0.

0} ...

01 ...

PVj(:1)-1 =

0}

1}P{ej(n-1)+1 = 0}...PV3(,1)-1 = 0}P{S1(n) = 1} = P{b2 = 0} ... P{£j(n) = 0}

0}

1

(j(2) - 1)(j(3) - 1)...(j(n) - 1)j(n)

...

P{£,(2) = 0}

2. RECORD TIMES AND RECORD VALUES

60

WVe see from (14.1) that (14.2)

P{L(n) = k} =

E 1
1

(k(2) - 1) (k(3) - 1)

... (k(n) - 1)k(n)

Also, let us point out the following equalities: (14.3)

P{L(n + 1) > j I L(n) = i} P{6+1 =0, 6+2

i

=0} = i+1 ...

j-1

and

(14.4)

P{L(n + 1) = j I L(n) = i} `

= P {Ci+1 = 0, Si+2 = 0, ... , i -1 = Mi = 11

(14.5)

s

P{L(n+ 1) = j I L(n) = i,L(n - 1) = in_1,...,L(2) = i2,L(1) = 1} i

j(j - 1) for any 1 < i 2 < ... < in-1
L(1) = 1 ,

L(n + 1) = [L(n) exp(W,,)] + 1,

n = 1, 2, ... ,

where W1, W2, ... are i.i.d. random variables with standard exponential distribution and [x] denotes the integral part of x. Let us prove (14.6). Since

P{exp(-Wn)<x}=P{Wn>-log x}=x,

0<x<1,

the recurrence relation (14.6) is equivalent to (14.7)

L(1) = 1 ,

L(n+ 1)

(n)] + 1,

n = 1,2,...,

where U1, U2,... are i.i.d. random variables uniformly distributed on [0, 11. The

relations (14.6) and (14.7) are to be understood in the sense that the random variables which they determine have the same distribution as the record times L(n).

Since (14.6) and (14.7) are equivalent, we will prove the latter. For the proof it suffices to show that the random variables defined by (14.7) form a Markov chain with transition probabilities given by (14.4). By construction, L(n) in (14.7)

W. DISTRIBUTIONS OF R.ECOR.D TIMES

depends only on U1, U-2,. . ., U. we obtain

and does not depend on U.. Hence for any j > i

P{L(n+1)=jIL(n)=i}=P{[

Un

)]+1=jIL(n)=i}

=P{[Un.]+1=jI L(n)=i}=P{[ i

=P {j
61

j-1

]+1=j}

3(j-1).

Since L(1), ... , L(n - 1) are also independent of Un, we can show by similar ar-

guments that for any 1 = it < i2 < . . < in-1 < i < j the following equality .

holds:

P{L(n + 1) = j I L(n) = i, L(n - 1) = in-1, ... , L(2) = i2, L(1) = ij } =

.

1)

Thus the random variables defined by (14.7), as well as the variables given by (14.6), form a Markov chain with the same transition probabilities (14.4) as the record values. The record times L(n) are closely related to the random variables N(n) defined

as the number of records among X1, X2, ... , Xn. Indeed, for any n, m = 1, 2, .. . the following equalities hold:

P{L(n) > m} = P{N(m) < n}

(14.9)

and (14.10)

P{L(n) = m} = P{N(m - 1) = n - 1, N(m) = n}.

This equalities allow us to express the distributions of the record times in terms of the distributions of N(m) = 1 + +£m, which are sums of independent indicators, so that one can apply the well-developed theory of summation of independent random variables to the study of record times. The relations (14.9) and (14.10) can be rewritten as follows:

P{L(n) = m} = P{N(m - 1) = n - 1, Sm = n} P{S1+ m Since

N(m) = yl + ... + Sm, where G, k = 1 , 2, ... , are mutually independent, and

P{G=1}=1-P{G=0}=k we obtain that the generating function Pm(s) of N(m) has the following form: (14.13)

P,n(s) = E SN(m) =

s(1 + s)(2 + s)... (m - 1 + s) M!

2. HE('Oi l TIMES AND RECORD VAI.I ES

(i2

or

,5(s - 1) ... (s - m + 1). (14.14)

MI

Recall that the Stirling numbers of the first kind Sn are defined by the equalities (14.15)

x(x - 1)...(x - n + 1) _ E Snxk k>O

(see, e.g., Abramovitz and Stegun (1964) or Balakrishnan and Nevzorov (1997)). Then we obtain from (14.14) and (14.15) that k

(-1)kP{N(m) = k} = mm and k

P{N(m) = k} _ (-1)k m

(14.16)

k

(QkI

It follows from (14.16) and (14.12) that

P{L(n) = m} =

(14.17)

P{N(m - 1) = n - 1} _ ISm-'II m

m!

REMARK 14.1. Using the asymptotics of the Stirling numbers, Renyi (1962) and Westkott (1977a) showed that rz

P{N(m) = n} -

(

(n

)

n 2

1

1)I

and P{L(n) = m} - (mo(n)

2)!

asm -+oo. EXERCISE 14.1. Using formulas (14.11)-(14.13) show that the generating func-

tion of the random variable L(n) has the form (14.18)

(- log(' - s))k

Q ,,(s) = Es L(n) = 1 - (1 - s) k=0

=j-log(l-a)

k!

y1 exp(-v) dv

(n-1)! In Lecture 20 we will find the form of the generating function for the so-called kth record times. The equality (14.18) is its particular case for k = 1. Now we consider some asymptotic relations for the random variables N(n) and L(n). Consider again the record indicators 6,6.... They are independent..

Ek

=

k

and

Var l;k =

1

k

- 1, ,

k = 1, 2, .. .

Therefore

(14.19)

Ek =logn+7+O(nn-oo, k=1

I

I

DISTRU 1lI °I'IONS OR RECORD TIMES

63

where ^y = 0.5772... is the Euler constant, and n

VarN(n)

(14.20)

1

k=1

n- x.

loge+ -y - 6 +O(n-1),

n

6

k=1

Using uniform and nonuniform bounds in the central limit theorem for indepen-

dent bounded (0 < n, < 1) random variables (see, e.g., Petrov (1987)) we obtain the following inequalities: (14.21)

CI

sup IP{N(n) - EN(n) < x(VarN(n))1/2} -t(x) I

(VarN(n))1/2

X

and (14.22)

sup IP{N(n) -EN(n) < x(VarN(n))1/2} - -D(x) X

C2

(Var N(n))1/2(1 + Jx13)

where C1 and C2 (as well as C3, C4,... in the sequel) are absolute constants. Using (14.19) and (14.20), by standard arguments from (14.21) and (14.22) one can obtain the following more convenient relations: (14.23)

sup IP{N(n) - logn < x(logn)112} - 4i(x)I < X

C3

(logn)1/2

and

(14.24)

C4

sup IP{N(n) - logn < x(logn)1/2} - p(x)I < (log n)1/1(1 + jx13) x

Thus the sequence of random variables

N(n) - logn (log n) 1/2

converges in distribution to the standard normal distribution. This fact implies (using (14.11)) asymptotic normality of the random variables

log L(n) - n n1/2

Indeed, let us fix x and denote

no = n(x) = exp(xn1/2 + n). Assume for simplicity that no is an integer. Then

P{log L(n) < xn1/2 + n} = P{L(n) < exp(xn1/2 + n)} = P{N(no) > n} )112} = P N(no) - log no > -x( n I.

n+xn1/2

(logno)1/2

-+1-(D(-x)=$(x)

as

n

A more detailed derivation using (14.24) enables us to obtain the following inequality: (14.25)

sup IP{IogL(n) - n < xnl/2} - -D(x)I

n'/z'

2. RECORD TIMES AND RECORD VALUES

64

To conclude this lecture, consider the limiting behavior of the ratios of the record times,

L(n+ 1) Tn = We see that for any x > 1,

n = 1,2,...

L(n)

P{Tn > x} = P{L(n + 1) > xL(n)}

(14.26)

00

_

P{L(n + 1) > xL(n) I L(n) = i}P{L(n) = i} i=n 00

_ E P{L(n + 1) > [xi] I L(n) = i}P{L(n) = i} i=n 00

_ i=n

[i] P{L(n) = i}.

For an integer x, 00

CO

i=n (xi]

P{L(n) = i} _

a

ion xi

00

P{L(n) = i} _

P{L(n) i=n

x

1

x

For an arbitrary x we have 1

< i <

1

x

[xi]

x

+

1

x[xi]'

hence 00

x

i=n (xi]

00

P{L(n) = i} <

which means that

x + i=n x(xi]

P{L(n) = i} < z +

P1 L(n+'1)>x.

1 x L(n) for any x > 1 as n --+ oo. Thus we have proved the following theorem.

THEOREM 14.2. For any x > 1, -+L

lim P{ noo L

()1)

> x} =

-

.

If x > 1 is an integer, then

P{L(n+ 1) L(n)

> x} =

I X

EXERCISE 14.2. Prove the following result by Shorrock (1972b): for a fixed

r = 2,3,... the ratios L(n + k)

'n TA :,n

L(n+k-1)'

k = 1,2,...,r,

are asymptotically independent as n -+ oo. The following exercise is closely related to Exercise 14.2.

15. DISTRIBUTIONS OF RECORD VALUES

65

EXERCISE 14.3. Galambos and Seneta (1975) considered the ratios L(n) and defined integer-valued random variables T(n) by

n=2,3,....

T(n)-1 < L(n(n)1)
Prove their result that T(2),T(3),... are independent and identically distributed with

P{T(n) = j} = P{L(2) = j} =

,

.1

7(J

-

j

1),

>

2.

Lecture 15. DISTRIBUTIONS OF RECORD VALUES

From record times L(n) we turn now to the record values

n = 1, 2, ... .

X (n) = XL(n) = XL(n),L(n) = ML(n),

It follows from Lemma 13.2 that for any n, m = 1, 2,... the event

=n-1,tm=1} and the random variable Mm are independent. Since

P{Mn < x} = Fn(x), we obtain that (15.1)

P{X(n) < x} = P{ML(n) < x} 00

> P{ML(n) < x I L(n) = m}P{L(n) = m} M=I 00

_ E P{M,n < x I L(n) = m}P{L(n) = m} M=1 00

_ E P{Mm < x}P{L(n) = m} M=1 00

=

r F-(x)P{L(n) = m} = E (FL(n)(x)) = Qn(F(x)), M=1

where

Qn(s) = EsL(n)

is the generating function of the random variable L(n). Now (14.18) implies that (15.2)

(- log(1 - F(x)))k

P{X(n) < x} = 1 - (1 - F(x))

k!

k=O

=J

log(i-F'(x)) vn-I exp(-v) dv

(n - 1)!

,

If

F(x) = 1 - exp(-x),

x > 0,

n=1,2,...

2. IIEC'C)RI) I'INIES AND RECC)R.D \AI.1'1.:S

(i(i

then

(15.3)

P{X (n.) < x}

-J

(n - 1)!

i.e., X (n) has the gamma-distribution with parameter n. An analog of Representation 3.1 for order statistics holds also for record values.

REPRESENTATION 15.1. Let X(1) < X(2) < ... be the record values in a sequence of i.i.d. random variables with continuous d.f. F, and let U(1) < U(2) < ... be the record values related to the uniform distribution on 10, 11. Then for any n = 1, 2, ... the random vector (F(X (1)), ... , F(X(n))) has the same distribution as (U(1), ... , U(n)). The proof of this result is based on the ideas which were used in the proof of Representation 3.1. The basic idea is that the probability integral transformation F(X), which transforms a random variable X with continuous d.f. F into a random variable uniformly distributed on [0, 11, does not affect the ordering of the random variables.

Now we obtain some useful corollaries to Representation 15.1.

COROLLARY 15.2. If F is a continuous d. f. and G is its inverse, then for any n = 1, 2, ... we have (15.4)

(X(1),...,X(n)) `I (G(U(1)),...,G(U(n))).

REMARK 15.3. In contrast to relation (3.2), the equality (15.4) may fail if F

is not continuous. For example, let P{X = 0} = P{X = 1} = 1/2. Then G(s) = inf{x: F(x) > s} = 0 for 0 < s < 1/2 and G(s) = 1 for 1/2 < s < 1. Therefore the vector in the right-hand side of (15.4) has at most two different components, which for n > 2 contradicts the property that X(1) < . . . < X (n). This example relies on the fact that record values may not exist with probability one if the rightmost point of the support of the distribution, x" = sup{x: F(x) < 1}, is an atom, i.e., if P{X = x'} > 0. However, taking a somewhat more complicated distribution than in this example, having, say, a single atom xo such that

P{X<xo}>O, P{X=xo}>0, and P{X>xo}>0, we also see that (15.4) fails.

COROLLARY 15.4. For any n = 1, 2,... the record values X (l) < X(2) < ... and Y(1) < Y(2) < ... corresponding to continuous d.f.'s FI and F2 are related as follows: (15.5)

(X ( 1 ) ,

... , X (n))

d

(H(Y(1)), ... , H(Y(n))),

where H(x) = GI (F2(x)) and GI is the inverse function to FI. Let us mention the following important particular case of (15.5): if X(1) < X(2) < ... are record values related to a continuous d.f. F and Z(1) < Z(2) < ... are exponential record values (related to the standard exponential distribution). then (15.6)

(X(1),...,X(n))

d

(H(Z(1)),...,H(Z(n))).

67

iS. fFS'ITil13l;'I IONS OF RECORD VALI'ES

where H(x) = C(1 - exp(-x)) and C is the inverse function to F. Now we find the conditional distribution

x

cp(x I x1,. .. , x,,) = P(X(n + 1) > x I X(1) = x1, X(2) =

First we show that for any n = 1, 2.... the random variables Yl = XL(n)+1, Y2 = XL(n)+2, ...

are i.i.d. with common d.f. F, and are independent of X (1), X (2), ... , X (n). To this end, observe that for any m the events Cn,m = {L(n) = m}

are determined only by the random variables X1, X2, ... , X,n and do not depend

on Xm+l, Xm+2,... For an arbitrary event B generated by the record values X(1),X(2),...,X(n) and A = {Y1 < xl,...,Yk < xk}, the probability P{AB} can be written as 00

00

P{AB} = E P{ABCn,m} _> P{A I BCn,m}P{BCn,,n} m=0

M=0 00

_ E P{Xm+l <x1,...,Xm+k <xk I BCn.m}P{BCn.m}. M=0

Since the event BC,,,,,, is determined by the random variables X1, X2.... , X,n, which are independent of Xm+l, ... , X,n+k, we see that 00

P{AB} =

, Xm+k < xk}P{BCn,m}

P{X,n+1 < XI, M=0 00

= F(x1) ... F(xk)

P{BCn,m} = F(xl) ... F(xk)P{B}, m=0

which proves the above assertion. Next, the random variable X(n + 1) can be written as XL(n)+T(x(n)) or Yr(X(n)), where -r(u) denotes the index of the first random variable in the sequence Y1, Y2, ... exceeding u. Taking into account the above assertion we obtain that co(xI

x I X(1)

= =

=xn}

x I X(1) = xl,..., X(n) = xn} x} = P{Yr(x..) > x}.

Finding the distribution of the first random variable in the sequence Y1, Y2.... exceeding xn we obtain

Ax I xl,...,xn) = P{Yr(x.,) > x}

=P{Yl > x}+P{Yl <xn,Y2 > x}+... +P{Yl <xn,...,Yk_1 <xn,Yk>XI + _ (1 - F(x)) +F(xn)(1 - F(x)) +... +Fk-1(xn)(1 - F(x)) +..

_

1 - F(x) 1 -

F(x)

,

x>xn.

2. RECORD TIMES AND RECORD VALUES

68

Therefore (15.7)

P{X(n + 1) > x I X (1) = xl, X (2) = X2, ... , X (n) = xn}

_ 1 - F(x) 1-F(xn)'

x>xn,

and the conditional density function fn+1(x I x1, ... , xn) of X (n + 1) given that X(1) = x1, X(2) = x2i ... , X (n) = xn has the form (15.8)

n),

F()

fn+1(x I x1,...,xn) = 1

x > xn,

provided the underlying distribution has density f. It follows from (15.7) that X(1), X(2).... form a Markov chain, and (15.8) implies that the joint density function fn(x1, ... , xn) of the record values X(1), X(2), ..., X (n) is (15.9)

fn(xl,.,xn) = fn(xn Ix1,...,xn-1)fn-1(x1,...,xn-1) f(xl)f(x2)...f(xn)

(1 - F(x1)) (1 - F(x2)) ... (1 - F(xn-1)) = R(x1)R(x2) ... R(xn) (1 - F(xn)) for x1 < x2 <

< xn, where

R(x) =

f (x)

1 - F(x)'

In the general case, similar, but more tedious arguments prove the following result.

THEOREM 15.5. For any continuous d.f. F the joint distribution of the record values X (1), X (2),. .., X (n) is given by the formula (15.10)

P{X(1) < xi, X(2) < x2, ... , X(n) < xn}

r n-1l

_J

dF(u3) dF(un)

1 - F(uj)

where the integration is over the domain

B = {uj < xj, j = 1, 2, ... , n, -oo < u1 < ... < un < 00}. We see from (15.10) that

P{X(n) < xn} = f...

JH 11 j_1

dF(uj)

1 - F(uj)

dF(un),

with integration over the domain

B = {uj < xn,

j = 1, 2, ... , n, -oo < u1 < ... < un < 00}.

16. SHORROCK'S REPRESENTATION

69

By symmetry

(

F(u)))n-I

(- log(1 -

dF(u) (n - 1)!

!

00 1

1)I

(n -

1 - F(v)

lyn

fxn

dF(v) n-I dF(u)

fu

P{X(n) < xn} =

log(1-F(xn))

Un-te-° dv,

i.e., we have proved (15.2) by another method. For the exponential distribution with density function f (x) = exp(-x) and d.f. F(x) = 1 - exp(-x) the joint density of the exponential record values

Z(1) < Z(2) < . . . < Z(n) is given by (15.11)

fn(xt,

.

,

xn)

- { exp(-xn) 0'

If 0 < x1 < ... < xn < 00, otherwise.

When proving Representation 3.6 we have shown that for i.i.d. random variables .... with common d.f.

F(x) = 1 - exp(-x),

x > 0,

+ lk, k = 1, 2, ... , n, is equal to the rightthe joint density of sums Sk = e1 + hand side of (15.11). Hence we obtain the following useful result for exponential record values Z(1) < Z(2) < .... REPRESENTATION 15.6. For any n = 1, 2,... the following equality holds: (15.12)

(Z(1), Z(2), ... , Z(n)) d (Si, S2, ... , Sn),

where Sk = 6 +

+ Sk, k = 1, 2, ... , and e1, C2,... are i.i.d. random variables

with standard exponential distribution. The following statement follows from (15.12).

COROLLARY 15.7. The random variables Z(1), Z(2) - Z(1), Z(3) - Z(2), .. . are mutually independent and have the standard exponential distribution. EXERCISE 15.1. Show that if F is a continuous d.f., then for any n = 2, 3, .. . and m = 2,3.... the following equality holds:

P{X(m) > x I X(m - 1) = u} = P{Xn,n > x I Xn_1,n = u}.

Lecture 16. SHORROCK'S REPRESENTATION FOR DISCRETE RECORDS

Now we consider the record values for discrete distributions. For all variables X (n), n = 1, 2, ... , to be well-defined almost sure it is necessary that the underlying distribution have no largest growth point, i.e., a point a such that

P{X < a} < P{X < a} = I.

2. RECORD TIMES AND RECORD VALVES

711

Without loss of generality, we will treat; only random variables that assume values 0, 1,... with positive probabilities. Define the random indicators ii,,, n = 0, 1, ... , as follows: let q, = 1 if n is a record value, i.e., if X (m) = n for some m = 1, 2, ... , and On = 0 if the sequence X (1), X (2), ... does not contain the value n. Shorrock (1972) proved the following result. THEOREM 16.1. The random indicators 77ei 711, ... are mutually independent and

(16.1)

n=0,1,....

P{X>n}

PROOF. The relation (16.1) follows from the equalities

P177 1} = P{X1 = n} + P{X1 < n, X2 = n} + P{X1 < n, X2 < n, X3 = n} + .. . = P{X = n} (1 + P{X < n} + P2{X < n} + ... )

P{X=n}

P{X=n}

1 - P{X < n}

P{X > n}

Since the indicators ?70, 77, take only two values, for the proof of their independence

it suffices to show that for any r = 2, 3, ... and for all 0 < a(1) < a(2) < ...

= P{ria(1) = 1,17x(2)

(16.2)

= 1, ... 77.(r) = 1}

H

k=1

P{X -> a(k)j'

Let M(r - 1) denote the time when the record value equal to a(r - 1) occurs. Then (16.3)

P{%(1) = 1,rla(2) = 1,...,7 (r) = 1} = E P{7la(1) = 1 , 77«(2) = 1 ,--- ,7 7 . ( r ) = 1, M(r - 1) = m} m 00

= E E P{ 7r7.(1) = 1, 77a(2) = 1, ... , 77a(r-1) = 1, M(r - 1) = in, in s=1

Xm+1 < a(r - 1), ... , Xm+s-1 < a(r - 1), Xm+s = a(r) } 00

_ E j:P{17a(1) = 1777a(2) = 1, ... ,77 (r_1) = 1, M(r - 1) =m) m s=1

x P{X,,,+1 < a(r - 1)} ... P{X,,,,+s-1 < a(r - 1)}P{X,,,+s = a(r)} _ E P{77x(1) = 1, rla(2) = 1, ... 7 77a(r-1) = 1, M(r - 1) = m} in cc

x E P{X,+1 < a(r - 1)} ... P{Xm+s-1 < a(r - 1)}P{Xm+s = a(r)} s=1

_

1,77x(2) = 1,...,17«(r-1) = 1, M(r -- 1) = m} in

=

P{ 71x(1)

P{X = a(r)} > a(r- 1)}

= 1,77a(2) = 1,...,r7a(r-t) = 1}P{X

P{X = a(r)} P{X > a(r - 1)}

16. SHORR.OCK'S REPRESENTATION

71

Using (16.3) successively we obtain (16.2). Theorem 16.1 implies the following useful result. REPRESENTATION 16.2. Let X1, X2,... be i.i.d. random variables taking non-

negative integer values, and let X(1) < X(2) < ... be the corresponding record values. Then for any m = 0,1, ... and n = 1, 2, .. . (16.4)

P{X(n) > m} = P{rro +'r71 + ...

+77m <

n}

and (16.5)

P{X(n) =m}

=n-1,l7m = 1}

=m} =P{r7o+77,+ +rlm-1 =n-1}P{X P{X > m} The relations (16.4), (16.5) can be used for obtaining exact and limiting distributions of suitably centered and normalized record values X(n). We will not formulate general theorems and restrict ourselves to the following example.

EXAMPLE 16.3. Consider the geometric distribution supported on the set 11,2.... }. Thus, let

P{X = n} _ (1 - p)pn-1,

n = 1, 2, .. .

Then

P{X>n}=pn-1 and P{77,,=1}=1-p,

n=1,2,...,

i.e., the random indicators 771,772.... are not only independent, but also identically distributed. In this case the sum

Sm=771+ +llm has the binomial distribution with parameters m and (1 - p), and min(m,n-1)

P{X(n) > m} _ >

(rn)(1

- p)rpm-r.

r=o Since

ETlr = 1 -p and Var?7r = p(1 - p),

r = 1, 2, ... ,

we see that the random variable

Sm,-m(1-p) (mp(1 -p))"2 has asymptotically the standard normal distribution. Then the random variable

(1 - p)X(n) - n (np) 1/2

will be also asymptotically normal. Indeed, using (16.4) and denoting x(np)1/2 m(x)= n + 1 -p

72

2

C)RI) C!\IE5 AND RECORD VALI ES

(assume for simplicity that m(:r,) is an integer) we obtain that for any fixed x

P{

(1-p)X(n)-n >x }=P( X(n)> n+x(np)1/2 _ 1-p (np)1/l m(x)(1 - p) n - m(x)(1 - p) f P (m(x)p(1 - p))

1 /2

PJS,n(x)-m(x)(1-p) I

< (m(x)p(1 - p)) <

(m(x)p(1 - p))t/2

l

I/2

x (1 + x(p/n))I/2

Since m(x) -b o as n - oo, we see that

P

{ Szn(x) - m(x)(1 - p) (m(x)p(1

-

p))1/2

<

x (1

+ x(p/n))

1/2

converges to 0(-x)(for any fixed x, where (D is the standard normal d.f. Then

P{

(1

(nXi/2 - n

< X J -y 1 -,D(-x) = fi(x).

One can derive a number of useful corollaries from Theorem 16.1.

COROLLARY 16.4. For any j > k > n - 1

(16.6) P{X(n+ 1) = j I X(n) = k} = P{?7k+I = 0,'qk+2 = 0, ... , 77j_1 = 0, rlj = 1}

_P{X>k+1}...P{X>j-1}P{X=j} _ P{X=j} P{X > k + 1} ... P{X > j - 1}P{X > j}

P{X > k + 1}

and

(16.7)

P{X(n+ 1) > j I X(n) = k} =

P{X > j}

P{X>k+1}'

EXERCISE 16.1. Show that for i.i.d. random variables X1i X2, ... taking nonnegative integer values the joint distribution of the record values X(1), X(2).... is given by the equalities (16.8)

P{X(1) = i1iX(2) = i2,...,X(n) = in I P{X = ir} = P{X = in} n_ 11 P{X > jr}' r=1

0 < i1 < i2 < ... < in.

EXERCISE 16.2. Show that if X1, X2r ... are random variables as in Exercise 16.1, then the record values X(1) < X(2) < ... form a Markov chain with transition probabilities given by (16.6).

Now we state one more interesting relation for discrete records. Let 00

pk=P{X=k}, qk=P{X>k}=Ep,, j=k

k=0,1,...,

Iti. SIIORIIO('K'S REPRESENTATION

-:1

and Ti

n=0,1,....

A(n) =1:Lr, r=0 9r

Consider the conditional expectations

E (A(X(n + 1)) 1 X(1),...,X(n - 1), X(n)). Taking into account Corollary 16.4 and Exercise 16.1 we obtain (16.9)

E (A(X(n + 1)) 1 X(1),. .. , X (n - 1), X(n) = m) = E (A(X(n + 1)) 1 X(n) = m) 00

A(k)P{X(n+ 1) = k I X(n) = m} k=m+1

P{X = k} _ E0"A(k)P{X > m} k=m+1

E

k

00

1: Pk

k=m+1

m _r--±

00

Pk Pr r=0 4r k=max(m+l,r) qm+1

r=0 9'R'+1 00

00

Pk

+-0 Qr k=m+1 4m+1

m

+

00

Pk

r=m+1 4r k=r 9m+1

m

00

rr=O Qr

Prr

Pr r=m+1 9m+1

Er+1=A(m)+1.

r=0 4r

Putting

T(n) = A(X(n)) - n, we can rewrite (16.9) as

E(T(n+1) 1 X(1),X(2),...,X(n)) =A(X(n)) - n = T(n) or (16.10)

E (T(n + 1) 1 T(1),T(2),...,T(n)) = T(n),

n = 1,2....

The reader familiar with the theory of martingales, say, on the level of Shiryaev (1980), can observe from (16.10) that the random variables

T(n) = A(X (n)) - n,

n = 1, 2, ... ,

form a martingale with respect to the sequence of Q-algebras

.In = a(X(1),X(2),...,X(n)).

2. RECORD TIMES AND RECORD VALUES

7-1

As a consequence, we obtain that

ET(n) = ET(n - 1) _ = ET(1) = EA(X(1)) - 1 = EA(X1) - 1 00

k

00

= EA(k)P{X

=k}-1=Epk>Pr

k=0

k=0

00

00 =E-r

-1

r=0 qr

00

1: pk-1=1: pr-1=0

r=0 qr k=r

r=0

and

EA(X(n)) = n,

(16.11)

n = 1,2,... .

Turning again to Example 16.3 we see that A(n) = n(1 -p) and hence (16.11) for the geometric distribution has the form

(1-p)EX(n)=n and

E X(n) =

(16.12)

n p,

n = 1, 2,... .

1

Now we state one more important result for the geometric distribution as in Example 16.3. For this distribution (16.8) becomes

P{X(1)=itiX(2)=i2,...,X(n)=tin}

(16.13)

=(1-p)np`n-n

and

(16.14) P{X (1) _ il, X (2) - X (1) = i2, ... , X (n) - X (n - 1) = in} p)npil+...+in-n = = (1 P{X = i1}P{X = i2} .. -P{X = in}, which implies the following theorem. THEOREM 16.5. Let X1, X2, 1, 2, ... with probabilities

pk = P {X j = k} = (1 -

... be i.i.d. random variables assuming values p)pk-1

j=1,2_.,

k = 1, 2, ... .

Then the random variables X(1), X(2) - X(1), X(3) - X(2).... are mutually independent and have the same geometric distribution as the original random variables X1, X2, ... . Let us point out that (16.13) can be easily derived without recourse to formula (16.8). Indeed,

P{X(1) = i1i X(2) = i2, ... , X (n) = in} 00

00

1: ... = P{Xl =i1,X25it,...,Xk2
kn=1 Xk2+...+kn-i+2

in-1) ... 7 Xk2+....}kn < in-1, Xk,+...+kn+l = in }.

.

16. SHORROCK'S REPRESENTATION

Using that P{X < i} = 1 - pi, i. = 1, 2, ... we obtain

P{X(1) = i1i...,X(n) = in}

x

0o

_ (1

-pii)k2-1

... E (1

k22=1

(1 COROLLARY 16.6. Under the conditions of Theorem 16.5, for any n = 1, 2, .. . the following equality holds:

X(n)=XI+X2+ +Xn.

(16.15)

Since the random variables X1,X2,... in the right-hand side of (16.15) are independent, identically distributed, and have expectations a = 1/(1 - p) and variances a2 = p/(1 - p)2, we immediately obtain that

X(n) - na _ (1 - p)X(n) - n (np) 1/2

on1/2

asymptotically has the standard normal distribution. This assertion was proved in Example 16.3 by a more complicated method. We have already pointed out that it makes sense to consider the so-called weak records only when the underlying d.f. is discontinuous. DEFINITION 16.7. Weak record times L,,(n) and weak record values X(n) are given by

Lw(1) = 1,

Lw(n+1)=min{j>L,,,(n):Xj >max{X1,X2,...,X,_I}}, Xw(n) = XL,,,(n),

n = 1, 2, .. .

Weak records may arise, for example, in some sports (shooting, athletics) where the athlete who repeats the record achievement is also declared a record-holder. We will consider the weak records for random variables taking only nonnegative integer values. Let us point out the property that distinguishes the weak records from the ordinary ones, namely, for any distribution with probability one there exists any record value Xw(n). W e introduce the random indicators r i n , n = 0, 1, ... , as follows: 17 = 1 if n

is a weak record value, i.e., if Xw(m) = n for some m = 1, 2, ... , and nn = 0 if n does not appear in the sequence X,(1),X.(2)..... Similarly to the indicators nn defined in the beginning of this lecture, the indicators 77n , n = 0, 1, ... , are mutually independent and P{?Jn = 1} = P{7). = 1} =

P{X = n}

P{X>n}'

n=0,1,...

Thus Theorem 16.1 continues to hold with indicators nn replaced by >7n . Obviously. Representation 16.2 fails if 77n are replaced by 77n and X(n) by weak record values.

In order to express the distributions of the X,(n) in terms of distributions of sums of independent random variables we will need other random variables than indicators 771 n.

2. REC'ORU I'IAIF.S AND RECORD VALUES

76

Let µ,,, n = 0, 1,. .., be the number of those weak records in the sequence X1, X2, ... that are equal to n. The proof of the following result can be found in Stepanov (1992).

THEOREM 16.8. The random variables µo, Al, A2.... are mutually independent and

P{µn=m}=(1-rn)r;l,

(16.17) where

rn

n=0,1,..., m=0,1,...,

_P{X=n} P{X > n}

Note that

P{Tjn = 1} = P{µn > 0}. The independent random variables 1U0,µ1,µ2, ... will be used in the following representation. REPRESENTATION 16.9. For any n = 1, 2.... and m = 0, 1, .. .

P{X,,,(n) > m} = P{µo + µl +

+ µn < n}.

EXERCISE 16.3. Find the distribution of the weak record value X.(n) for the geometric distribution as in Example 16.3.

Lecture 17. JOINT DISTRIBUTIONS OF RECORD TIMES AND RECORD VALUES

There are a number of useful relations between the distributions of the record

times L(n), record values X(n), and inter-record times 0(1) = L(1) = 1, 0(n) _ L(n) - L(n - 1), n = 2, 3, .. . Consider the joint distributions of the random variables L(n) and X(n). It is easily seen that for any xi,...,xn and 1 = k(1) < k(2) < --- < k(n)

(17.1) P{X(1) <x1,X(2) <x2i...,X(n) <xn, L(1) = 1,L(2) = k(2),...,L(n) = k(n)}

= P{X1 < xl, max(X2.... , Xk(2)-1) X1 < Xk(2) < X2, - . maX(Xk(n-1)+l.... , Xk(n)-1) Xk(n-1) < Xk(n) < xn} -

.c

...

e

x

h(ul,... , un) dF(ul)... dF(un),

00

where

n-1

h(ul,. . ., un) =

F k(r+l)-k(r)-1(ur)

r=1

if -oo < ul <

< un < oo, and h(ul, ... , un) = 0, otherwise. Let the underlying random variables X 1i X2, ... have a density function f . Consider the function f(n) (k(1), ... , k(n), xl,... , xn) which is a density function

17, .101N'1' DISTRIBUTIONS

77

with respect to the continuous random variables X (1), X (2), ... , X (n) and a prob-

ability distribution with respect to the discrete record times L(1), L(2),.... L(n). Differentiating (17.1) with respect to s1,.. , xn,, we obtain (17.2)

(F(x1 ))k(2)-k(1)-i(F(x2))k(3)-k(2)-1 x ...

x (F(xn_1))k(n)-k(n-1)-If(t1)f('2)...f(xn) if

-00<x1 <x2<...<xn<00,

1
and f(n) (k(1), ... , k(n), xl,... , xn) = 0, otherwise.

If we make the substitution m(r) = k(r) - k(r - 1) in (17.2), which corresponds to the passage from the random variables L(r) to 0(r), then the densitydistribution hn (m(1), ... , m(n), x1,.. . , xn) for the inter-record times A(1), i.(2),

.... A(n) and the record values X (1), X (2),. . ., X (n) are given by (17.3)

hn(m(1),...,m(n),x1,...,xn) =

(F(x1))m(2)- l

(F(x2))

m(3)-l

(F(xn-1))m,n)-lf

...

(xl )f (x2) . . .

f(x)

if

-00 < X1 < x2 < ... < xn < 00,

m(1) = 1,

m(r) > 0,

r = 2, ... , n,

and hn (m(1), ..., m.(n), x1, ... , xn) = 0, otherwise. Now (17.3), together with the equality (15.9) giving an expression for the joint density f n(xl, ... , xn) of the record values, yields (17.4)

P{o(2)=m(2),...,A(n)=m(n) I X(1)=xi,...,X(n)=xn} hn (m(1), ... , m(n), x1, ... , xn)

fn(xl,...,X.) _

(F(x1))m(2)-l...

(F(xn-1))m(n)-lf(xl)...

f(xn)

R(xi) ... R(xn)(1 - F(xn)) where

R(x) _ 1 - F(x) The right-hand side of (17.4) can be rewritten as (F(x1))m(2)-i(1 - F(xl))(F(x2))m(3)-1(1 - F(x2)) x ...

x (F(xn_1))

m(n)

1

(1 - F(xn-1)).

Thus we have actually proved the following result. THEOREM 17.1. Let the underlying random variables X 1,X2, ... be i. i. d. with a common density function f. Then the inter-record times A(1), 0(2).... are conditionally independent given the record values X (l), X(2),..., and (17.5)

P{0(n) = m 1X(1), X(2), ...

(1 - F(X(n - 1))) (F(X(n - 1))) m= 1,2,..., n=2,3,...

.

2. RECORD 'TIMES AND RECORD VALUES

78

REMARK 17.2. Note that the probabilities in the right-hand side of (17.5) cor(for fixed X (n-1) _ respond to the geometric distribution with parameter xn_ 1 ) supported on the points 1, 2, ... . It follows from (17.5) that P{0(n) = m I X ( 1 ), X(2),...

m = 1, 2, ... ,

(F (X (n - 1)))'n,

and P{0(n) (1 - F(X (n))) > y I X (1), X (2), ...

(17.6)

where

N(y,n) = r11 -

Fy(X(n))I

(F(X (n - 1)))

N(y.n)

-

For any y and any fixed values of X (1), X (2).... the right-hand side of (17.6) behaves for n -* oo as exp {

- 1))) } - y(1 1- -F(X(n F(X(n)) J

and it can be easily shown that

limoP{Z(n)(1 - F(X(n))) > y} = E exp

n

- y(1

1

F(X(n

1)))

F(X(n))

}.

It follows from (15.6) that the ratio

1 - F(X(n - 1)) 1 - F(X(n)) has the same distribution as the random variable

exp{Z(n) - Z(n - 1)}, where Z(1), Z(2),... are the record values corresponding to the standard exponential distribution. Corollary 15.7 implies that the difference Z(n) - Z(n - 1) is distributed as the random variable Z with exponential E(1) distribution. Since the random variable U = exp(-Z) has the uniform U(0,1) distribution, we obtain

1 - F(X(n - 1)) d 1 - F(X(n))

1

U

and

G(y) = lim0P{0(n)(1 - F(X(n))) < y} = 1 1

= 1 - f e-y/= dx = 1 0

r z-2e-Zy dz,

J

Ee-y/u

y> 0.

1

EXERCISE 17.1. Using formula (17.3), show that under the conditions of The-

orem 17.1 the sequence of vectors (X (n), 0(n)), n = 1, 2, ... , is a Markov chain with (17.7)

P{X(n) > x, 0(n) = m l X(1), 0(1), ... , X(n - 1), 0(n - 1)} = (F(X(n - 1))),n-1 (1 - F(x)).

18. ASYMPTOTIC DISTRIBUTIONS

79

Lecture 18. ASYMPTOTIC DISTRIBUTIONS OF RECORD VALUES

In Lectures 10 and I1 we studied limiting distributions of suitably centered and

normalized maxima (M(n) - b(n))/a(n). It is also of interest to find all possible asymptotic distributions for the record values X(n). Since X(n) = M(L(n)), the set of limiting laws for the random variables (X(n) - b(L(n)))/a(L(n)) (involving random centering and normalization) coincides with the set of limiting laws for the variables (M(n) - b(n))/a(n) which, as we know from Theorem 10.5, consists of the following types of distributions:

A(x) = exp (exp(-x)),

fa (x)

- (0 t

for x < 0,

exp(-x-11)

for x > 0,

exp(-(x)°)

for

and mi(x)

Il

x < 0, x > 0.

for

How will this result be affected by nonrandom censoring and normalization of record values X(n)?

Consider exponential record values Z(1), Z(2),... Representation 15.6 implies that for any n = 1, 2, ... , d

t

=bl+...+C.., Z(n) where b1, 2, ... are i.i.d. random variables with standard exponential distribution.

Since

Ebn=Varen=1,

n=1,2,...,

by the central limit theorem applied to the sums Sl + (18.1)

P{Z(n) - n < xn1/2} -. fi(x)

as

+ Sn we obtain that

n - oo,

where ' (x) is the standard normal d.f. At the same time for the exponential distribution we have

P{M(n) - log n < x} -. A(x) = exp (- exp(-x))

as

n

oo

(see Example 10.1), so that (18.2)

P{Z(n) - log L(n) < x} -+ A(x).

We see from (18.1) and (18.2) that the asymptotic distributions of record values with random and nonrandom normalization need not be the same. Using relation (18.1), which holds for the exponential distribution, we will try to find all possible limiting distributions of random variables

X (n) - B(n) A(n)

obtainable for a suitable choice of the centering constant B(n) and normalizing constant A(n).

80

IU (O L) TIMES AND RECORD VALUES

2

Using (15.6) we obtain

(18.3) P{

X(n) - B(n)

< x} = P{Z(n) < T(xA(n) + B(n)) }

-n - p{ Z(n) n1/2

<

T(xA(n) + B(n)) nl/2

n

},

where

T(x) = - log (1 - F(x)) and F is the d.f. of the underlying X's which we assume to be continuous. Obviously, (18.1) and (18.3) imply the following result. LEMMA 18.1. In order that for some choice of constants A(n) and B(n), there exist a nondegenerate limiting d. f.

G(x) = limo P{X(n) - 49(n) < xA(n)},

(18.4)

it is necessary and sufficient that there exists a limiting function

g(x) = lim

(18.5)

T (xA(n) + B(n)) - n

n-»oo

nl/2

having at least two growth points. The limits (18.4) and (18.5) are related by

G(x) = 4i(g(x)).

(18.6)

It turns out that there are only three possibilities for the function g(x) (up to linear transformations). THEOREM 18.2. The limiting function g can only belong to one of the following three types: (1)

gi(x) = x;

(2)

92

-y > 0,

ry log x, (18 7) .

(

x) _

for x<0;

-00 ry log(-x),

(3)

93(x)

for x > 0,

-y > 0,

{ oo

for x < 0, for x > 0.

PROOF. In view of monotonicity of g(x) it suffices to consider only those x where g(x) assumes finite values. For such values of x one has as n oo,

T(xA(n) + B(n)) - n and then

lye T1/2 (xA(n)n B(n)) + n1/2

(18.8)

1

2

= 2.

Since

T(xA(n) + B(n)) - n nl/2 (T1 /2 (xA(n)

+ B(n)) - nl/2J

Tl/2 (xA(n) + B(n)) + n1/2 nl/2

14. ASYMPTOTIC DISTRIBUTIONS

'j!

(18.8) implies that (18.5) is equivalent to

nlim (T1/2 (xA(n) + B(n)) - n'12) = 9(9)

(18.9)

00

The main idea of the proof is to relate the limiting distributions for the record values corresponding to the d.f. F to limiting distributions for maxima corresponding to a certain d.f. F*. It turns out that we should take for F* the function

F*(x) = I - exp ( - T' 12(X)) = 1 - exp (-( - log(1 - F(x)))

1/2)

One can easily check that it is a distribution function. The relation (18.9) can be rewritten as limoexp(n'/2)(1 - F*(xA(n) + B(n))) = h(x),

(18.10)

where h(x) = exp (-

9(1)). 2

Take the subsequence n = n(m) = [log 2 m] and denote A*(m) = A([log2m]),

B*(m) = B([log2m[),

m = 1,2,... .

Then we obtain from (18.10) a more convenient relation,

nlim0m(1 - F*(xA*(m) + B* (m))) = h(x),

(18.11)

which implies that (18.12)

lim (F*(xA*(m)+B*(m)))m

M-00

_ Mlinn (1 - (1 - F*(xA*(m) + B*(m))))m = H(x), where

H(x) = exp ( - h(x)). Let Y1, Y2, ... be a sequence of i.i.d. random variables with V.

F*(x) = 1 - exp (-( - log(1 - F(x)))1/2) , and let M*(m) = max{Yl,... , Ym}, m = 1, 2,... . The relation (18.12) means that the sequence of d.f.'s of the centered and normalized maxima

M*(m) - B*(m) A* (m)

converges as m

oo to a limiting d.f. H which is related to the function g of

interest by

H(x) =exp(-exp(_ 9(2x))) As we know, H must belong to one of the types A, & , or tPQ stated above, hence the function g(x) = -2 log ( - log H(x)) must belong to one of the three types listed in the theorem.

2. RECORD TIMES AND RECORD VALUES

82

Theorems 18.1 and 18.2 imply the following result. COROLLARY 18.3. The set of all nondegenerate limiting d.f 's for centered and normalized record values X(n) consists (up to linear transformations) of the functions -:P(g(x)) with g(x) defined in (18.7). REMARK 18.4. It is of interest to compare the limiting distributions for records and maxima, which have the form

and

(D(9(x))

exp (- exp(-9(x))),

respectively, with g defined in Theorem 18.2.

EXERCISE 18.1. Examine the proof of Theorem 18.2 to compare the domains of attraction of the limiting distributions for record values and maxima, as well as the form of the centering and normalizing constants.

EXERCISE 18.2. The relation (18.1) shows that the standard exponential distribution belongs to the domain of attraction of the limiting d.f. of records 4b(g1(x)),

where 91(x) = x. Give examples of distributions which belong to the domains of attraction of the distributions OD(g2(x)) and Z(g3(x)), where 92 (x)

_

y log x,

y > 0,

-0o

for for

x > 0, x < 0;

and

93(x) _

_1y log(-x),

ry > 0,

00

x < 0, for x > 0. for

Lecture 19. THE kTH RECORDS The so-called kth records are a natural extension of records. Dziubdziela and Kopocinsky (1976) introduced the kth record times L(n, k) and the kth record values X (n, k) as follows: (19.1)

L(1,k)=k, L(n+1,k)=min{j>L(n,k):Xj >Xj_k,j_1}, n>1,

and (19.2)

X(n, k) = XL(n,k)-k+1,L(n,k),

n > 1.

An intuitively clearer way to define the kth records is related to the nonde-

creasing sequence of the kth maxima

-00 < X1.k C X2,k+1 < ... < Xn-k,n-1 < Xn-k-1.n < -

- -

-

Select the elements of this sequence that are preceded by the strict inequality sign: X1,k < Xn(2)-k+1,n(2) < Xn(3)-k+l.n(a) <

-

One can easily check that this subsequence coincides with the sequence of the kth record values X (1, k) < X (2, k) < .... and the random indices n(1) = k, n(2).... coincide with the sequence of the kth record times L(1, k) < L(2. k) < ... .

Note that for k = 1 the definitions (19.1) and (19.2) amount to the ordinary record times and record values.

19. THE k'rH RECORDS

83

To deal with the kth records it will be convenient to introduce the so-called sequential ranks R(1), R(2),... of the random variables X1, X2, ... as follows: R(n) denotes the rank of X.n among X1, X2, ... , Xn, i.e., Xn = XR(n),n,

n=1,2,....

We will formulate without proof an important property of the sequential ranks due to Renyi (1962).

THEOREM 19.1. For i.i.d. random variables X1, X2,... with continuous d.f. F the sequential ranks R1, R2, ... are mutually independent, and

P{Rn=k}=n,

k=1,2,...,n, n=1,2,....

REMARK 19.2. Independence of the sequential ranks continues to hold if we consider the more general than in R6nyi's theorem class of symmetrically dependent

random variables X1, X2, ... satisfying the condition P (X, = X2 } = 0 for any i of j. Actually Theorem 19.1 claims that if we consider the order statistics X1,n_1 < X2,n_1 < ... < Xn-1 n_1

(by assumption all of them are different almost sure), then the new random variable Xn has equal chances to fall into each of n random intervals (-oo, X1,n-1), (X1,n-1, X2,n-1), ... , (Xn-1,n-1, oc)

In Lecture 13 we defined the record indicators

f1=1 and bn=1{Mn>Mn-1})

n=2,3,....

Note that Sn = 1{R(n)=n},

n = 1, 2, ... ,

and then Lemma 13.1 on independence of the indicators n, n = 1, 2, ... , becomes just a particular case of Theorem 19.1. By analogy with indicators G we introduce the indicators of the kth records: (19.3)

n(k) = 1{Xn>Xn-k,n-1},

n = k, k + 1,... .

Obviously, (19.3) is equivalent to the equalities (19.4)

l;n(k) = 1{ Xn-k+l,n>Xn_k,n_J } = 1{R(n)>n-k+1},

n = k, k + 1, ... .

The following result is also a consequence of Theorem 19.1.

COROLLARY 19.3. For any k = 1, 2.... the indicators t:k(k), .k+1(k), independent and (19.5)

n=k,k+1,....

... are

81

2.

TIMES AND RECORD VALI FS

Independence of the random variables n enables us to derive the distributions of the kth record times. Indeed,

(19.6) P{L(1, k) = k, L(2, k) = m(2), ... , L(n, k) = m(n) } = P {G (k) = l,G+1(k) =

0.

1}

Gn(2)(k) = 1,...,Gm(n)-1(k) =

= P{G(k) = 11P'(4+1(k) = 0} ... P{bm(2)_ 1(k) = 0} 1} ...P{&m(n)-1(k) =0}P{fm(n)(k) =1} X k1(m(n) - k)! kn-1 m(n)! (m(2) - k) (m(3) - k)... (m(n) - k)

< m(n). for k < m(2) < Proceeding as in the case k = 1 we can infer from (19.6), for instance, that the sequence L(1, k), L(2, k).... is a Markov chain with (19.7)

P{L(n + 1, k) > r I L(n, k) = m} = P{L(n + 1, k) > r L(n, k) = m, L(n - 1, k) = L(2, k) = m2, L(1, k) = k} 0,...,£r(k) = 0} =

- (m+l-k)...(r-k-1)(r-k), r>m>n+k-1, (m+1)...(r-1)r

and

(19.8)

P{L(n + 1, k) = r I L(n, k) = m} 0,...,er-1(k) =

= 0,er(k) = 1}

- (m+1-k)...(r-k-1)k r>m>n+k-1, (m + 1)...(r - 1)r

which corresponds to formulas (14.3) and (14.4). With the record times L(n, k) one can naturally associate the random variables N(n, k) defined as the number of the kth records in the sequence X1, X2, ... , Xn. We have for any k, n, m = 1, 2, .. . (19.9)

P{L(n, k) > m} = P{N(m, k) < n}

and

(19.10)

P{L(n, k) = m} = P{N(m - 1, k) = n - 1, N(m, k) = n}.

The equalities (19.9) and (19.10), like (14.9) and (14.10) related to k = 1, enable us to express the distributions of the kth record times in terms of the distributions of the sums (19.11)

of independent indicators and to use the theory of summation of independent random variables. The relations (19.9) and (19.10) can be rewritten as (19.12)

P{L(n, k) > m} = P{G(k) +

+ ,n(k) < n}

11). THE, kT11 ItEC OHI)S

S. I

and

(19.13)

P{L(n, k) = m} = P{N(m - 1, k) =,n - 1, ,n(k) = 1} =

P{G(k)+

Now we state some asymptotic relations for N(n, k) and L(n. k). To this end remember that the record indicators G (k), G+1(k), ... are independent and observe

that k

and

k k2 Varfn(k)=--z,

n=k,k+1,....

Hence

=k

(19.14)

m=k,k+1,....

m, m=k

and

EN(n,k)=k(logn-

(19.15)

k-i

E

1

-m

n1-),

+y)+O(\

n - oo,

m-1

where -y = 0.5772... is the Euler constant. Accordingly, (19.16)

Var N(n, k) = E N(n, k) - kz

E zl m=k

k-1 1

k6 2

M=1 1

+ D(-), D, since

00

lz 6z = m 6 m=1

m

mz

m=1

n --+ oo,

00

and

k-1

Tr2

0<E

lmz <

1

n

m=n+1

Using the central limit theorem for independent bounded (0 < £,,(k) < 1)

random variables we obtain the following bound: (19.17)

sup

P{N(n,k)-EN(n,k)<x(VarN(n,k))11z}-'D(x) <

Cl

(Var N(n.

As it was done in the case of k = 1 (when deriving (14.23) from (14.21)), by standard methods we deduce from (19.15)-(19.17) the following bound: (19.18)

suplP{N(n,k)-klogn<x klogn} -Cx)I S Cog

.

where fi(x) is the standard normal d.f. and the constant C1 (k) depends only on k.

2. RECORD TIMES AND RECORD VALUES

86

Using (19.9) we can deduce from (19.18) the asymptotic normality of log L(n, k) and to obtain the following extension of (14.23): (19.19)

syplP{k logL(n,k)-n<x/ }-(D(x)I
where C2(k) again depends only on k. We omit the proof of (19.19) because it employs no substantially new methodology as compared to the case k = 1.

Lecture 20. GENERATING FUNCTION OF THE kTH RECORD TIMES

Formula (14.7) gives an expression for the generating function Qn(s) of the random variable L(n):

(- log(' - s))k

QT(S) = E sL(n) k_o

k!

-log(l-')

- fo

vn-1 exp(-v) dv

(n- 1)!

Now we will find the generating function Qn,k(s) of the kth record time L(n, k). Note that this lecture will require of the reader knowledge of the theory of martingales on the level, say, of the monograph by Shiryaev (1980). THEOREM 20.1. For any k = 1, 2.... and n = 1, 2, .. .

Qn,k(s) = EsL(n,k) =r ( - log(1 - s)), where (20.1)

r(v) =

kn

(n - 1)!

f v xn-le-kx (1 - e)' dx. o

For the proof of Theorem 20.1 we will need one more assertion. Let us introduce the sequence of random variables

U(n, k) = ylB (L(n, k) + 1, ky - k),

n = 1, 2, ... ,

where y > 1 and B (a, b) = r(a)r(b) = fo 1 x°-1(1- x)6-1 dx. r(a+b)

Moreover, let Fnk) denote the v-algebra generated by the random variables L(1, k), .... L(n, k), or equivalently, by the random variables U(1, k), ... , U(n, k). Note

that .F(k) C

.k) C .. .

THEOREM 20.2. For any fixed k = 1, 2.... and -y > 1 the sequence of random variables U(n, k), n > 1, is a martingale with respect to the sequence of o-algebras F(k) Furthermore, (20.2)

E (U(n, k) I U(m, k)) = U(m, k),

n > m > 1,

and (20.3)

E U(n, k) = E U(1, k) = yB (k + 1, k(y - 1)),

n > 1.

20. GENERATING FUNCTION OF THE kTH RECORD TIMES

87

PROOF. Let b = k-y - k. Under the conditions of the theorem, 6 > 0. For the proof it suffices to show that (20.4)

E (B (L(n, k), 6) L(n - 1, k) = m, L(n - 2,k),. .. , L(1, k))

- B(m+1,6), y

m> k,

because (20.4) implies that

E (U(n, k) I '-(k!1) = E (U(n, k) I U(1, k), ... , U(n - 1, k)) = U(n - 1, k),

so that the sequence U(n, k), n > 1, is a martingale. Denote Mk(m,) =

r(m + 1)

r(m-k+1)'

It follows from (19.8) that

P{L(n, k) = r I L(n - 1, k) = m, L(n - 2, k), ... , L(1, k) } = P{L(n, k) = r I L(n - 1, k) = m}

-k(r-k-1)!m! _ r!(m - k)!

kMk(m) Mk+1(r)'

n+k-1<m
Hence (20.5)

E (B (L(n, k) + 1, 6) 1 L(n - 1, k) = m, L(n - 2,k),. .. , L(1, k)) 00

B(,+ 1, 6) r=m+1

kMk (m)

Mk+1(r) 00

kr(m + 1)r(6)

r(m-k+1)r(k+1+6) r

B (r - k, k + l + 6). 1

Observe that 00

B (r - k, k + 1 + 6) = f1(1 - x)k+6

(20.6)

r=m+1

0

f(i

E 00

xr-k-1 dx

r=m+1

_-x)k+6-lxm-kdx=B(k+6,m-k+1)

r(k+6)r(m- k+ 1) r(m+6+1) Now we obtain from (20.5) and (20.6) that

E (B (L(n, k) + 1, 6) I L(n - 1, k) = m, L(n - 2,k),. .. , L(1, k)) k r(m + 1)r(5) = 1

k+6 r(m+6+1) = y B (m + 1, 6),

which proves (20.4). Thus the sequence

U(n, k) = ynB (L(n, k) + 1, k-y - k) = yn

r(L(n, k) + 1)r(ky - k) r(L(n, k) + ky - k + 1)

n > 1,

-

2. RECORD TIMES AND RECORD VALUES

88

is a martingale for any fixed k = 1, 2.... and arbitrary y > 1. The equality

n > m > 1,

E (U(n, k) I U(m, k)) = U(m, k),

holds because L(n, k), n = 1, 2, ... , is a Markov chain, hence the sequence U(n, 1), U(n, 2),... is a Markov chain as well. REMARK 20.3. The restriction -y > 1 in Theorem 20.2 is only due to the factor

r(ky - k) in the definition of U(n, k). This factor does not depend on n and one can easily check that the sequence n r(L(n, k) + 1) n> y r(L(n, k) + ky - k + 1)' is a martingale for any positive y. In particular, taking

k-/3

)3
k

in this sequence we obtain the following useful result.

COROLLARY 20.4. For any fixed k = 1, 2.... and /3 < k the sequence Tn(/3) _

(20.7)

(k - p)nr(L(n, k) + 1) knr(L(n, k) - ,Q + 1) '

n>

is a martingale with respect to the sequence of a-algebras '14nk) . Thus we have (20.8)

n > m > 1,

E (TT()3) I Tm(/3)) = Tm(f),

and

(k - /3)r(k + 1)

ETn(/3) = ET1(Q) _

(20.9)

kr(k-/3+1)

Now we proceed to the proof of the main result of this lecture. PROOF OF THEOREM 20.1. We obtain from (20.3) that

EUn=ynEB(L(n,k)+1,ky-k)=yB(k+1,k-y-k)=B (k,ky-k), or, putting a = ky - k, (

(20.10)

E B (L(n, k) + 1, a) _ k + 1) \

n

f xa-1(1 1

Relation (20.10) implies the equality (20.11)

J

x)k-1 dx,

1 xQ-1E ((1 - x)L(n'k)) dx = ( k

+

1) -n

f xa-1(1 - x)k-1 dx,

which can be rewritten as e-avE ((1

(20.12)

- e-v)L(n,k)) dv

0

0o

a-«v

knvn-le-kv

o0

dv

(n - 1)! Jo By properties of the Laplace transform, the function o

r(v) = E ((1 -

a > 0.

0

e-)L(n,k))

e-v)k-1 dv.

89

21. NMONIENT CHARACTERISTICS

is the convolution of the functions

ri(v) = kn(n -ll)!kv

r2(V) = (1 - e-' )k-1,

and

v > 0.

Hence we finally obtain that

Qn.k(3) = EsL(n'k) = r(- log(1 - 3)), where

r(v) =

kn

v

(n-1)!Jo xn-1e-kx(1 -

e-(V-s))k-I dx.

0

Thus the proof of Theorem 20.1 is completed.

Lecture 21. MOMENT CHARACTERISTICS OF THE kTH RECORD TIMES

We have shown in Lecture 13 that E L(2) = oo. Therefore, EL(n) = oo for n > 2. This property of the classical record times substantially restricts their use in various statistical procedures. It turns out that E L(n, k) < oo for any n = 2, 3, ... already for k > 1, while for k > 2 all variances Var L(n, k) are finite. In this lecture we will derive a number of formulas for moments of the random variables L(n, k). Let us turn to relation (20.9), which holds for p < k:

ETn()3)=ET1()3)= (k (k k

r(L(n,k)-0+1)

Recall that the gamma-function r(s) satisfies the relation

r(3 + 1) = sr(s).

Therefore, if we assume that Q in (20.9) is an integer r, r = k - 1, k - 2, ... ,1, w get the relation (21.1)

(k - r)n-lE (L(n, k)(L(n, k) - 1) ... (L(n, k) - r + 1)) = k

r(k + 1)

r(k-r+1)'

Using the notation

Mk(m) -

r(m+1) r(m - k + 1)

=m(m- 1)...(m-k+1),

we can rewrite (21.1) as (21.2)

m(r, k, n) = E M,. (L(n, k))

k

_ k - r)

n

1

r(k+1) r(k - r + 1)

k- (k - 1)! (k - r)n(k - r - 1)!'

This equality provides an expression for factorial moments m(r, k, n) of order r = k - 1, k - 2, ... ,1 of the random variables L(n, k) fork > 1 and any n = 1, 2, ... . Let us state some particular relations which follow from (21.2).

2. RECORD TIMES AND RECORD VALUES

90

COROLLARY 21.1. The following equalities hold

m(1, k, n) = E L(n, k) =

(21.3)

kn

(k -

k >- 2,

n >- 1,

1)n-1I

m(2, k, n) = E L(n, k) (L(n, k) - 1) _ (kn (k 2)nl i ,

EL 2 (n, k) = m(2, k, n) + m(1, k, n) = (k (2)nl)) + (k k1)n-1

'

(21.4)

Var L(n, k) = m(2, k, n) + m(1, k, n) - m2(1, k, n)

_

kn

2n

n

n > 1, k > 3.

kl)n-1 - (k = (k (2)nl 1 + (k As is well known, the factorial moment of order r = 1, 2.... of a random k1)2n-2'

variable l; is defined to be

r(c-r+l)/ For a nonnegative random variable l; we will define factorial moments of negative order by means of the same relation:

ur(n) = E

r

E ((£ + 1) ... ( - r)

1

r +)1)

Now we can consider the factorial moments

m(r, k, n) = E Mr.(L(n, k)) = E ( E

r(L(n, k) + 1) l r(L(n, k) - r + 1)

1 1 ((L(n,k)+1)...(L(n,k)-r)l

of negative order r for the kth record times L(n, k). Since (20.9) continues to hold for negative 0, we can employ (21.1) for r = -1,-2.... to derive for these r the following formula for the moments m(r, k, n): k! k In-1 k )n-1 r(k + 1) _ m(r, k, n) _ (k --r)! r(k-r+1) =

- (k-r/

(k-r

In particular, for k > 1 and n > 1 we obtain (21.5)

(k + 1)n'

_

1

(21.6)

kn-1

1

E (L(n, k) + 1 / E ((L(n,k) + 1)(L(n, k) + 2)I

kn-' (k + 2)n (k + 1)'

Using the property (20.8) of the random variables Tn()3) _

(kn 8)nr(L(n, k) + 1) k r(L(n, k) -,3 + 1) '

we can find the product moments of the random variables L(n, k). For example, if n > m and k > 1 we can argue as follows.

21. MOMENT CHARACTERISTICS

91

Obviously, the kth record times are related to Tn(;3) by the equalities L(n, k) =

k" 1)nT,l(1)

(k -

km

and

L(m, k) _ (k - 1)-

Using the martingale property (20.8) we obtain that (21.7)

kn+m

E L(n, k)L(m, k) = (k-1)n+mETn(1)Tm(1) kn+m

(k - 1)n+mETm(1)E (Tn(1) I Tm(1)) kn+m

kn+m

(1)

(k - 1)2m

"_

(k-1)n+mETm= (k-1)n+m k2EL (m'k) kn-m

(k - 1)n-m

km km(k - 1) (k - 2)m-1 + (k - 1)m-1

Now (21.3) and (21.7) imply the following expression for the covariance between the kth record times: kn m km (k _ (21.8) cov (L(n, k), L(m, k)) _ (k- 1)n-m (k (2) m-1 + (k k1)m-1) kn+m

n>m>1.

(k-1)n+m-2'

EXERCISE 21.1. Using the martingale property (20.8) as it was done in the proof of (21.8), show that for k > 1 and any n1 > n2 > n3 > n4 the following equality holds for the covariances between ratios of the kth record times: L(nlik) L(n3,k)

cov

( L(n2i k)' L(n4, k)) =

0.

EXERCISE 21.2. Let the sequence f (n), n = 0,1, ... , be defined as

f(0)=0 and f (21.9)

n = 1, 2, ... and k = 1, 2.... the following equality holds: nk 1 E f (L(n, k)) = f (k) +

REMARK 21.2. Since

logn=f(n)-7-2n+O\n 1

= f(n) - 7 - 2(n+ 1) +O((n+ 1)(n+2)),

n

---+ 00,

where 7 is the Euler's constant, formulas (21.4), (21.5), and (21.8) can be used for finding logarithmic moments E log L(n, k). For example, we have (21.10)

E log L(n, k) = f (k) + n

1

-7

kn-1

2(k+ 1)n

+O((k+2/ )'

n

oo.

2. RECORD TIMES AND RECORD VALUES

92

Relations of the form (21.10) are important because it is the logarithm of L(n, k), rather than L(n, k) itself, which is asymptotically normal.

Lecture 22. TATA'S REPRESENTATION AND ITS GENERALIZATIONS

Now we pass from the kth record times to the kth record values X(n, k) related to a sequence of i.i.d. random variables X1, X2, ... with continuous d.f. F. Denote by Z(n, k) the kth exponential record values related to a sequence of i.i.d. random variables Z1, Z2, ... with standard exponential distribution. One can easily check that using the probability integral transformation (as it was done for ordinary record values, see Corollary 15.4) we obtain the following important assertion, which will allow us to restrict ourselves to the exponential distribution when dealing with the kth record values.

REPRESENTATION 22.1. For any n = 1, 2,... and k = 1, 2,... we have (22.1)

(X(1,k),...,X(n,k))

d

(H(Z(l,k)),...,H(Z(n,k))),

where H(x) = Q(1 - exp(-x)) and Q is the inverse function to F. In Lecture 15 we stated Tata's representation (see Tata (1969)) for exponential record values Z(n) (see Representation 15.6) which implies that the random variables Z(1), Z(2) - Z(1), Z(3) - Z(2) are independent and have the standard exponential distribution. As an important consequence of probability integral transformation and Tata's representation we get a possibility to express the distributions of arbitrary record values in terms of distributions of sums of i.i.d. exponential random variables. This is given by the following theorem. THEOREM 22.2. We have (22.2)

{X(n)J°°

1

=

where H(x) = Q(1 - exp(-x)), Q(x) is the inverse function to F(x), and W1, W2.... is a sequence of i.i.d. random variables with standard exponential distribution. Dziubdziela and Kopocinsky (1976) obtained the following generalization of Tata's result. THEOREM 22.3. Let Z1, Z2, ... be i.i.d. random variables with exponential d.f.

G(x) = 1 - exp(-x),

x > 0,

and let Z(n, k), n = 1, 2, ... , be the corresponding kth record values. Then for any

k=1,2,... (22.3)

{Z(n,k)}0o

n=1

d { w1 + ...+wn}°o k

n=1'

where W1iW2,... are i.i.d. random variables with standard exponential distribution. Relation (22.3) implies the following generalization of Theorem 22.2.

22. TNI'A'S iEPRESENTATION

THEOREM 22.4. For any k = 1, 2,... {H(w1+...+wn/Jx

{X(n)j0'n=

(22.4)

ll 11

k

n=1

We can derive from (22.4) a number of useful relations.

THEOREM 22.5. The sequence X (1, k), X(2, k),... is a Markov chain, and for

anyk>1, n> 1, andx>u (22.5)

P{X (n + 1, k) > x I X (n, k) = u} =

(1 - F(x) k 1 - F(u))/

PROOF. Theorem 22.4 implies that (22.6)

P{X(n+1,k)>xI X(n,k)=u,X(n-1,k),...,X(l,k)} _us

x I H(w1+

H(w1+

k+wn_1l/)...,H(k

-klog(1-F(x)) -k log(1 - F(u)), w1 + - + wn_ 1, ... , w1 }. The sequence w1, w1 + w2, ... is a Markov chain and

+wn=v}=P{wn+1>u-VI = exp{-(u - v)},

u > v.

Hence the right-hand side of (22.6) equals the conditional probability P{w1 +

+wn+1 > -klog(1 - F(x)) I w1 +

+wn = -klog(1 - F(u))}

and has the form

exp {k(log(1 - F(x)) - log(1 - F(u)))

1-F(x)lk

(1 - F(u)

This proves the theorem.

Consider now two sequences of i.i.d. random variables: X1, X2i ... with continuous d.f. F and

Y1 = min{X1i... , Xk}, Y2 = min{Xk+1,... , X}, .. .

with d.f. G(x) = 1 - (1 - F(x))k. Let X(n,k) denote, as before, the kth record values related to the sequence X1i X2,..., and let Y(n, 1) denote the ordinary record values (k = 1) in the sequence Y1, Y2, .... The following theorem relates the distributions of X (n, k) and Y(n, 1). THEOREM 22.6. For any k = 1, 2, ... , 0o

3{X(n,k)}1 n= = {Y(n,1)}n=1.

2. RECORD TIMES AND RECORD VALVES

94

PROOF. It suffices to observe that the function G- (x), inverse to G, has the form G`(x)=Q(1-(1-x)Ilk),

where Q(x) is the inverse function to F. Then the application of Theorem 22.4 with k = 1 and

H(x) =G'-(1-exp(-x)) =Q(1-exp(- -k yields the expression Q(1 -exp (

- Wl + . k . +Wn ))

in the right-hand side of (22.4), which means that Y(n,1) and X(n, k) have the same distribution.

REMARK 22.7. The relationship between the distributions of the kth and ordinary record values stated in Theorem 22.6 enables us to write down immediately the d.f. of X(n,k). Since we know that

- log(1-F(z)) P{X(n,1) < x} = (n

1

1)I fo

un-1e-,. du,

we have to substitute G(x) = 1 - (1 - F(x))k for F(x) in the right-hand side of the equality. Hence we obtain that for any k > 1 and n > 1 k log(1-F(z))

(22.7)

P{X (n, k) < x} =

un-1e-° du.

1

(n - 1)!

o

Consider now the sequences of random vectors X (n)

= (XL(n,k)-k+1,L(n,k),

,

XL(n,k).L(n,k))

constructed for the sequence XI, X2,..., and Z(n) = (ZL(n k)-k+1,L(n,k),

...

TL(n,k),L(n,k)),

n

related to i.i.d. standard exponential random variables ZI, Z2, .... Using the probability integral transformation one can easily check that the vectors X(n) and (H(ZL(n,k)-k+1,L(n,k)),

, H(ZL(n,k).L(n,k)))

have the same distribution. Note that Dziubdziela and Kopocinsky proved that X (1), X (2).... form a Markov chain. Now we will formulate without proof results by Ahsanullah and Nevzorov (1996) describing the structure of order statistics generated by the kth records. THEOREM 22.8. Let

WI,k-I :5 ... < Wk-,,k-1

and WI k < ... < Wk.k

be the order statistics related respectively to the sets

W1,W2,...,wk_1

and W1,W2,...,Wk

23, CORRELATION COEFFICIENTS FOR RECORDS

95

of i.i.d. random variables having the standard exponential distribution. Then the (2k - 1)-dimensional vector (ZL(n,k)-k+2.L(n,k) -ZL(n,k)-k+1,L(n,k), .. , ZL(n.k).L(n,k) -ZL(n,k)-k+1.L(n.k), ZL(n+1,k) -k+1,L(n+1,k) - ZL(n,k)-k+l,L(n,k),

,

ZL(n+l,k),L(n+l.k) -ZL(n,k)-k+1,L(n.k))

is independent of ZL(n,k)-k+1,L(n,k) and has the same distribution as (w1 ,k-1,

,

wk-1,k-1, w1,k, ... , wk,k).

One can deduce from Theorems 22.3 and 22.8 the following assertions for exponential order statistics: ZL(n,k)-k+1,L(n,k),

,

ZL(n.k),L(n,k)-

COROLLARY 22.9. For any k > 1 and n > 1

T(n, k) - ZL(n,k)-k+1,L(n,k) +"'+ZL(n,k),L(n,k)

d

wl ++wn+k-1,

where w 1 , . . . , Wn+k_ 1 are i. i. d. with standard exponential distribution, i.e., the sum T (n, k) has the gamma-distribution with parameter n + k - 1.

COROLLARY 22.10. Let

S(n, k) = T (n, k) - kZL(n,k)-k+1,L(n,k),

where T(n, k) are defined in Corollary 22.9, and V(n) = S(n + 1,k) - S(n,k).

Then for any k = 1, 2.... the random variables V (n), n = 1,2,..., are i.i.d. with standard exponential distribution, while the random variable S(1, k) is independent of V(1), V(2),... and for k > 1 has the gamma distribution with parameter k - 1.

Lecture 23. CORRELATION COEFFICIENTS FOR RECORDS

We know from (21.7) and (21.4) that for k > 3, cov (L(n, k), L(m, k)) =

k'

(k - 1)n-m

km km(k - 1) (k - 2)m-1 + (k - 1)m-1

kn+m

(k -

1)n+m-2'

+

>m>

and

kn(k- 1) + kn - k2n , n (k - 2)n-1 (k - 1)n-1 (k - 1)2n-2' These equalities enable us to evaluate the correlation coefficients between the kth VarL(n,k) =

record times.

Now we discuss how to obtain moments of the kth record values X(n, k) and, in particular, of the classical record values X(n). The simplest calculations are needed in the case of the exponential distribution, where

F(x) = 1 - exp(-x),

x > 0.

2. RECORD TIMES AND RECORD VALUES

96

We can use the representation (22.3): oc

{Z(n,k)}n=1

d Iwl l

k

J)n=1'

where w1,w2i ... are i.i.d. random variables with standard exponential distribution. Since

E wn = 1

and

Var wn = 1,

we obtain that

E Z(n, k) = k ,

n = 1, 2, ... ,

Var Z(n, k) = T2 ,

m < n.

cov (Z(m, k), Z(n, k)) = Var Z(m, k) = k2

Hence the correlation coefficient p between Z(m, k) and Z(n, k) does not depend on k and is given by

p = p(Z(m, k), Z(n, k)) =(ml (n)

(23.1)

1/2

m < n.

For other d.f.'s we can use representation (22.4):

{X(n,k)}°° n=1

11

n=1

k

where H(x) = Q(1 - exp(-x)) and Q(x) is the inverse function to F(x). Then

EX(n,k) = EH(Z(n,k)) = EH(k ) and

VarX(n,k) = Var H(Z(n, k)) =VarH(k ), where 77, = w1 + (23.2)

E X (n, k) =

+wn has the gamma distribution with parameter n. Therefore

I'

f'H (x)

exp(-x)xn-1 dx

(n-1)!

k

00

kn

(n - 1)! J0

H(v)

exp(-kv)vn-1

dv

kn

(n - 1)!

Jo

- (nkn- 1)! (nkn - 1)!f

Q(1 - exp(-v)) exp(-kv)vn-1 dv

Q( z )( 1-z )k-1 (- l0g( 1-z))n-I dz

u(1 - F(u))k-I ( - log(1 - F(u)))n- dF(u),

in particular, (23.3)

EX(n) = f

00 u( - log(1 - F(u)))n-1 dF(u) (n - 1)! 00

23. COI?.RELATION COEFFICIENTS FOR RECORDS

07

and similarly, (23.4)

H2 (x) exp(-x)xr-1 dx

E X2 (n, k)

(n-1)!

k

fo

00 F(u)))n-1

u2(1 - F(u))k-1( - log(1 -

(nkn1)I

dF(u)

and

u2(- log(1 - F(u)))i-1 dF(u)

EX2(n) _

(n - 1)!

Note also that for m < n E X (m, k)X (n, k) = E H (Z(m, k)) H(Z(n, k)).

Since kZ(m, k) and k(Z(n, k) - Z(m, k)) are independent and have gamma distributions with parameters m and n - m respectively, we obtain (23.5)

f

EX(m,k)X(n,k)

-

H(U)H

Jo

(+V exp(-(u +

kH\

k

/

V))um-lvn-m-1 dudv

(m - 1)!(n - m - 1)!

Formulas (23.2), (23.4), and (23.5) enable us to evaluate expectations, variances, covariances, and correlation coefficients of the kth record values for specific distributions. Here we will show that the largest possible correlation coefficient p(X (m, k), X(n, k)) between two kth records is attained for the exponential distribution and is equal to (m/n)1/2 if m < n. So, we fix m and n assuming without loss of generality that m < n. We will only consider the d.f.'s F with finite second moments E X2(n, k) and E X2(m, k), i.e., we assume that (23.6)

I(r) =

J

u2(1 - F(u))k-1( - log(1 - F(u)))'-1 dF(u) < cc

for r=mandr=n. EXERCISE 23.1. Show that finiteness of either quantity I (m) or 1(n) does not imply finiteness of the other.

THEOREM 23.1. For any k = 1, 2, ... , m < n, and any continuous d.1 F such that

f0,00 u2(1 - F(u))k-1 ( - log(1 - F(u)))r-1 dF(u) < oo, the following inequality holds: (M) 1/2

(23.7)

p(X (m, k), X (n, k)) <

n/

which becomes an equality only if F is exponential.

r = m, n,

2. RECORD TIMES AND RECORD VALUES

98

PROOF. We will present the proof only for k = 1. In the general case the arguments remain practically the same. In the proof we use the Laguerre orthogonal polynomials 1

Ln (x) = 1 exp(x)x-"

(23.8)

dn(e-xxn+a ) dxry

and their properties

L' (x) = a + 1 - x, r(a + n + 1)r(µ) L+,.(a)

Lo (x) = 1,

(23.9)

I

x"(1 - x)µ-1Ln(ax) dx =

(23.10)

/°O (23.11)

J

r(a+µ+n+1)

n

r(a + n + 1)

e-yx"L"n (x)LQ (x) dx = 1 {m= nl r(n+ 1)

(see, e.g., Bateman and Erdelyi (1953), Vol. 2). We prove (for k = 1) a stronger relation than (23.7). Namely, we show that

p(hl(Z(m)), h2(Z(n))) < (m (n

(23.12)

) 1/2

for any functions hl and h2 such that

E (hl(Z(m)))2 < oo and E (h2(Z(n) ))2 < oo.

(23.13)

Then taking hl(x) = h2(x) = H(x) = Q(1 - exp(-x)) we obtain p(hl(Z(m)), h2(Z(n))) = p(X(m), X (n)) <

(mn ) 1/2

Expand the functions hl and h2 into series in Laguerre polynomials: 00

hi(x) _ EarLm-1(x) c* and h2(x) _ Eb,L;-1(x). r=0

a=0

Then

hl (x)x'n-1 exp(-x) dx (m - 1)! Jo °° Lm-1(x)xm-1 exp(-x) dx = Ea r 00 (m - 1)! r=0 r o

E hl (Z(m)) =

(23.14)

Using (23.9) and (23.11) we can simplify the right-hand side of (23.14): 00

00 Lm-l(x)xm-1 exp(-x) dx

E ar r=O 00

-

°° Lo -1(x)Lm-1(x)xm-l exp(-x) dx = ao. (m - 1)! r =o ar Jo

Therefore, (23.15)

Ehl(Z(m)) = ao

and similarly, (23.16)

Eh2(Z(n)) = bo.

23, CORRELATION COEFFICIENTS FOR RECORDS

99

The property (23.11) shows that

E (hl(Z(m)))2

(23.17)

00 (hl(x))2xm-fexp(-x)dx (m - 1)! f00 (Lr- -I(x))2xm-Iexp(-x)dx

-

E 00

rJ0

a2

(m-1)!

r=u0

_

0r0

a2r(r + m)

- O r(m)r(r + 1) Now (23.15) and (23.17) imply m)

00

Var (hl(Z(rn))) = E

(23.18)

r(m)r(r+ 1)

In a similar way we obtain

var (h2(z(n))) _

(23.19)

r(n)r(s + 1)

Now, in order to express the correlation coefficient p(h1(Z(m)), h2(Z(n))) in terms of the coefficients ar and b, it remains to obtain a formula for the product-moment E (h1(Z(m,)) h2(Z(n))). Using (23.5), (23.10), and (23.11) we see that (23.20)

E (h1(Z(m)) h2(Z(n))) 00 h1(x)h2(x+y)xm-lyn-m-1eXp(-(x+y))dxdy 0

-

=

(m - 1)!(n - m - 1)!

J0

h2(y)yn-Ie-" dy

J0

J0

00

00

1

hl(yv)ym_l(1 - y)n_m_1 dy (m - 1)!(n - lml - 1)!

=Ear>b,J

J

o

8=0

r=O

f o

1L"`-1(yv)ym-1(1-y)n_"-Idy1 L;-I(v)vn-Ie-"dv J (m - 1)!(n - m - 1)!

LT_1(v)(n-m- 1)!(r+m- 1)!)

= r=O ar>b, fooo 00 (m+r-1)! _ V arbr

(r + n - 1)!

3==.0

L;-1(v)vn-Ie-"dv (m - 1)!(n - m - 1)!

(m - 1)!r!

r=O

Collecting (23.15), (23.16), (23.18)-(23.20) we arrive at the expression (23.21)

p(hi(Z(m)), h2(Z(n)))

_

Note that

00

r=1

(m+r-1)! arbr (m - 1)!r!

(r+m-1)!

°°

{ r=1

(m + r - 1)!

m!

(n+r-1)!

n!'

(m - 1)!r!

oc

2(r+n-1)! }_h/2

br r=1

r=2,3,...,

(n - 1)!r!

2. RECORD TIMES AND RECORD VALUES

100

for m < n, so that

(m+r- 1)! < ((m+r-1)!)1/2((n+r-1)!)1/2

m 1/2 .

n!

Hence (23.22)

jp(hi(Z(m))h2(Z(n)))I r=1 F_00

(

)1)1/2 n+r-I ! 11/2 ( ! \ 1/2I m. n - 1 an { ,n+r-1 I ) r! br ( r! I

I

I

( ,-r

a2

r

r+rn-1 ! Eoo b2 (r+n-l)! 11/2 ((m r=1 r r! J r!

- 1)!n!)1/2

where the equality holds if and only if arbr = 0 for any r = 2, 3, .... Invoking the Cauchy inequality we arrive at the desirable relation (nm) 1/2

p(hi (Z(m)), h2(Z(n))) I < As can be easily shown, the equality (p(h1(Z(m)), h2(Z(n))) l = rm `n )

1/2

holds if and only if ar = br = 0,

and

r = 2,3,...,

albs > 0.

This condition means that =bo+bjLi-1(x)-

h1(x) =ao+aiLi -1(x) and h2(x)

Substituting the expressions for the Laguerre polynomials from (23.9) we obtain that (nm) 1/2

Ip(h1(Z(m)), h2(Z(n)))I = if and only if h1(x) = ao + a1(m - x)

and

h2(x) = bo + bl(n - x).

We see that equality in (23.7) holds when

hi(x) = h2(x) = H(x) = Q(1 - exp(-x)), where Q is the inverse function to F. Since

Ip(H(Z(m)), H(Z(n))) I can be equal to (m/n) 1/2 only if H(x) is a linear function, we conclude by solving the equation Q(1 - exp(-x)) = Cl + C2X

that

F(x)=1-exp Thus the proof is completed.

- x- c1 C2

I,

x>cl. 0

21. RECORDS FOR NONSTA'I'IONARY SEQUENCES

101

REMARK 23.2. Note that for the correlation coefficient between order statistics the following inequality holds: (23.23)

/

P(Xi,n, X3,n)

ri(n+1-j)\1/2 \ j(n + 1 - i))

where the equality attains only for the uniform distribution. It may be of interest to note that Szekely and Mori (1985), who proved (23.23), used the Jacobi orthogonal polynomials.

Lecture 24. RECORDS FOR NONSTATIONARY SEQUENCES OF RANDOM VARIABLES

There were a number of attempts to apply the classical scheme to records in

various sports. The most attention in this respect was paid to track events in athletics. In some track and field sports (for example, a mile run) records are available for a period longer than a century and there is a sufficient amount of data for statistical processing. It was shown that in almost all situations the stationary model does not fit the available tables of records. This is by no means surprising

because the methods of training, the sport facilities, the conditions of life and, the competition rules change even within a short period of time. The time of the postman who won the Marathon race in Athens in 1896 is as incomparable with the times shown by the professionals in Atlanta in 1996, each of whom was accompanied by a team of trainers and doctors, as cars at the beginning of the 20th century with contemporary Formula-1 racing cars.

In the literature a number of new record models were proposed to take into account the progress of human abilities. We will describe some of them.

Record models incorporating trend. The simplest (for description) model consists in replacing the sequence of i.i.d. random variables X1, X2.... by Y n = Xn + c(n),

n = 1, 2, ... ,

where c(n) are some constants depending only on n. The most conveniently treated is the case c(n) = cn, where c > 0 when the upper records are considered, and c < 0 for the lower records. This model was thoroughly studied by Ballerini and Resnick (1987). Although the model is comparatively simple, even the proof of asymptotic normality of the number of records N(n) meets certain technical difficulties. The problem is that for c 54 0 the record indicators G = 1{M(n)>M(n-1)},

n = 2, 3, ... ,

are no longer independent, so that the classical limit theorems for independent summands become inapplicable. Furthermore, the probabilities pn = P{G = 1} in this case depend on the underlying d.f. F. Hence one has to restrict oneself to certain classes of d.f.'s F, to show that dependence between t;,, and {,,, decays at a due rate with growing difference n - m, and to apply limit theorems for weakly dependent random variables. It is worth mentioning that while in the classical case E N(n) grows at a logarithmic rate, the best explored cases of the model with trend exhibit a linear growth of the expected number of records.

2. RECORD TIMES AND RECORD VALUES

102

Pfeifer's scheme. The next model of records can also be connected with sports. The German athlete Uwe Hohn on July 20, 1984, was the first to exceed the 100 meter mark in javelin throwing. His javelin landed at 104.8 m, which made this sport dangerous for spectators. Then the design of the javelin was changed by shifting its center of gravity forward, which reduced the length of throws. The records for the new javelin have not yet surpassed Hohn's. But the best throws nowadays come close to 100 m, and undoubtedly the rules will change again as soon as this mark is exceeded. The idea of varying distributions of the random variables in the sequence X1, X2, ... is basic for the scheme by Pfeifer (1982, 1984).

Let {Xnk, n > 1, k > 1} be a double array of independent random variables having in each row Xnl, Xn2i ... a common d.f. Fn. It will be convenient to define first the inter-record times 0(n) = L(n) -L(n-1) rather than the record times L(n). They are defined as

0(1) = 1,

A(n + 1) = min{k: Xn+I,k > Xn.A(n) },

n = 1, 2, ...

.

Then the record times L(n) and the record values X(n) in Pfeifer's scheme are given by

L(n) = 0(1) +

+ A(n)

and

n = 1,2,... . If Fl = F2 = .... then these definitions coincide with the definitions of the classical X (n) = Xn,o(n),

records statistics. Pfeifer obtained a number of results for records in his scheme. In particular, he showed that the vectors (i(n), X (n)), n = 1, 2, ... , form a Markov chain with transition probabilities

P{0(n) = k, X (n) > x 10(n - 1) = m, X (n - 1) = y}

(24.1)

= (1 - Fn(x)) (Fn(y))k-1,

x > y.

It was proved that the sequences (L(n), X (n))

and X (n),

n = 1, 2, ... ,

also form Markov chains, though the random variables L(1), L(2), ... need not possess the Markov property. Note that in Pfeifer's scheme the inter-record times

D(1), ... , 0(n) are conditionally independent given the values of X(1), X(2),..., X (n - 1) and

P{0(1) = 1,i (2) = k(2),..., 0(n) = k(n) I X(1),X(2),...,X(n- 1)} n

_

(1 - Ft(X(i - 1)))(F=(X(i -

k(i) = 1,2,...,

i = 2,...,n.

i=2

EXERCISE 24.1. Using (24.1) and the Markov property of the vectors

(0(n), X (n)),

n = 1, 2, ... ,

show that if

Fn(x)=1-exp{-fin},

x>0,

24. RECORDS FOR NONSTATIONARY SEQUENCES

103

with an > 0, n = 1, 2, ... , in Pfeifer's scheme, then the random variables

T(1) = X(1), T(2) = X(2) - X(1),... are independent and

P{T(n) < x} = Fn(x),

n = 1,2,... .

The scheme of Balabekyan-Nevzorov. One more scheme was proposed by Balabekyan and Nevzorov (1986) and developed by Rannen (1991). Consider again sports competitions, for example, the long jump. Let m athletes of different skill jump in turn, making n attempts each. We can assume that their jump lengths can be described by a sequence of independent random variables

X1,...,Xm,...,Xm(n-1)+1,...,Xmn with d.f.'s

F1, ... , Fm, ... , Fm(n-1)+1, ... , Fmn

such that

k = 1,2) .... n, r = 1, 2, ... , m, Fm(k-1)+r = Fr, i.e., this sequence consists of n repetitions of the group of m d.f.'s Fl,... , Fm. This scheme combines features of a nonstationary model (m different distribution functions) and of the classical records model, since the longest jumps in each attempt Yk = max{Xm(k-1)+l, ..

,

k=1,2,...,n,

Xmk}+

form a sequence of i.i.d. random variables with common d.f. M

G = 11 Fr. r=1

Thus we consider a sequence of independent random variables X1, X2, ... with

continuous d.f.'s Fl, F2.... such that Fm(k-1)+r = Fr,

k = 1, 2, ... .

r = 1 , 2, ... , m,

Let N(nm) denote the number of records in the sequence XI, ... , Xm, ... , Xm(n-1)+1, ... , Xmn, i.e., the number of records counted after n runs each of which consists of m random variables.

THEOREM 24.1. For any n > 1 and m > 1, (24.2)

( m2 l sup IP{N(nm) - logn < x logn} - 4'(x)I : c9 \lognl' X

where c is an absolute constant and

l

! g(x) = max lx1/2, x1/2log r

l / }. 1

2. RECORD TIMES AND RECORD VALUES

104

PROOF. Let Yk = max{Xm(k-1)+1,

,

k = 1,2,... .

Xmk},

The random variables Y1, Y2, ... are independent and have the common d.f. M

G=fFr. r=1

Let 1, e2.... be the record indicators in the sequence Y1, Y2,. .., i.e., {k = 1 if max{Xm(k-1)+1, ... , Xm,k} > max{X1,.

, Xm(k-1) }

Define the variables n = 1,2,...,

NI(n) = b1 + ... + en,

which count the number of "record" runs, so that in each run only one record

achievement is counted. Denote by vk the true number of records in the run Xm(k-1)+1,

, Xmk,

k=1,2,....

Now let 11k = max{0, vk - 1}

and let N2 (n)

=r11+...+77n,

n= 1,2,... .

Then

N(nm) = (6 + ... +

.

) + (771 + ... +'in) = N1(n) + N2(n),

n = 1, 2,... .

Since F1 i . . . , Fm are continuous d.f.'s, G is also continuous. Therefore the indicators

1, £2.... are independent, P{en = 1) = 1/n, and N, (n) - log n (log n) 1/2

has asymptotically the standard normal distribution. Invoking (14.22) we can write (24.3)

sup IP{N1(n) - log n < x log n} -t(x) I <

c (log n) 1/2

The next step is to show that the second term N2(n) is negligibly small as compared to N1 (n) and does not affect the asymptotics of N(nm) = N1 (n)+N2(n). For that we will need a number of auxiliary results.

LEMMA 24.2. For any m > 2, n > 1, and r > 1 we have (24.4)

E (771 + ....{- ]n)r < (m -

1)rrr+1

24. RECORDS FOR

SEQUENCES

PROOF. We can write Er = E (771 +

LL

+ 77n )r as

r

Er =

(24.5)

T.

1 k T

T1

k=1

REDk

105

1

... +

F (rla

ii

)

,

where R E Dk means that the summation extends over all vectors

R = (Ti,...,rk) Since O < i, < (m-1),

+ rk = r.

such that Ti > 0,...,rk > O and r1 + i = 1, 2,..., we have

r

(24.6) Er < (m - 1)' k=1 REDk

P{77,, > 0,

TI

r1I

k

77,, > 01-

Let

m=1,2,...,k,

ao=0, /3m=am-am-1-m, G(m1,m2,z) =

fl

G(1,0,x) = 1.

F. (X),

M1 5,95M2

Note that for any 1 < m1 < m2 < n we have 1 - G(m1i m2, x) < 1 - G(X)-

(24.7)

For simplicity of notation, denote v1 = z1 and vl = max{y1-1, z1}. Obviously, (24.8) P{77., > 01 ... , 77ak > 01 m-1 00

00

= f d(Gp1(z1)) E G(1,j1 - 1,v1) 00

j1=1

00

G(1,j2 - 1,v2) f

d(GQ'(z2))

m-

dG(71 + l,m,y1) l 00

00

dFj,(z2) f dG(j2+ 1,m,y2)... I7

v2

12=1

OD

x

Jvt

m-1

x-

100

dFj,(x1)

1

f d(GQ"(zk)) E G(1,jk - 1,Vk) f dFjk(xk) f dG(jk + 1,m,yk) jk=1

xk

vk

The inequalities (24.7) and m-1

m-1

EG(1,j-1,z)(1-Fj(z))=1- fF1(z) < 1-G(z) j=1

!=1

imply that (24.9) m-1

dF,,(xi)

l3,>o

Z'MZOO dG(ji+1,m,yi) r-0 d(GI`(z1))EG(1,j1-1,vi) j,=1 ,

0

M-1

5 E G(1,j, - 1,v1) fmo d( j,=1

1

1--(z,

) f (1 -G(x,))dF,,(x,) ,,

2. RECORD TIMES AND RECORD VALUES

106

( (1 - G(vl))2 \ 1 -G(zi)J 1

00

d( no

(1 - G(z1))2 = 1.

1

1 - G(xi)

Now applying the arguments used for the proof of (24.9), we obtain from (24.6) and (24.8)

E ...E0k>0P{7ja, > 0, ... , 7ja, > 0} < 1,

0,>_0 hence

EE r

Er<(m-1) r

r

l

rl! ... rk! k=1 rl+...+rk=r r

_ (m - 1)r E kr < r(m - 1)rrr = (m -

1)rrr+l

k=1

As a consequence of this lemma we obtain the following result.

LEMMA 24.3. For any t < 1/2e(m - 1) we have C

(24.10)

,

1 - 2et(m - 1) '

where c is an absolute constant. PROOF. This inequality follows from (24.4) and the elementary inequality r! >

ce-rrr. Indeed, o0

E eXp{t(771 +

+ 77n) l

trE (n1 + ... + 71n)r

+

r.

r=1

oo

tr(m - 1)rrr+l

r=1

r

<1+

I

00

<1+cEr(te(m-1))r< r=1

c

1 - 2et(m - 1)

In turn, (24.10) implies the following result. COROLLARY 24.4. For any e > 0 we have

5 c

exp(te)(1 - 2et(m - 1))

Since Ni(n) and N2(n) are nonnegative random variables, the following result is obvious.

LEMMA 24.5. For any x and e > 0 (24.11)

P{Nl(n) < x - e} - P{N2(n) > e} < PIN, (n) + N2(n) < x} < P{Nl (n) < x}.

25. RECORD 'LIMES IN THE F°-SCHEME

107

Now (24.3), (24.10), (24.11), and the elementary inequality e(27r)-1/2

14,(x) - (D(x - E)I <

yield the bound (24.12)

A(n) = I P{N(nm) - logn < x(logn)1/2} - 4D(X)

I

< c (e(log n)-1/2 + (log n)-1/2 +exp(-te)(1 - 2et(m - 1))-1) ,

which holds for any e > 0 and t < 1/(2e(m - 1)). Taking t = 1/(4e(m - 1)) in (24.12) we obtain E

A(n) (m - 1)2, then setting

e=4e(m-1)log( vIlogn m-1 we obtain the conclusion of the theorem. Otherwise, if log n < (m - 1)2, the bound (24.2) is obvious. The proof is completed. One more nonstationary model of records will be considered in the next lecture.

Lecture 25. RECORD TIMES IN THE F°`-SCHEME

Yang's Model. All three nonstationary models presented in the previous lecture involved stationary components. The models with trend of the form Yn = Xn +c(n), n = 1, 2, ... , involved i.i.d. underlying random variables X1, X2,... . In Pfeifer's model the random variables in each series between two successive records were identically distributed. In the Balabekyan-Nevzorov model the maximal results in each run

Yk = max{Xr(k)+1, ... , Xmk},

k = 1,2,...,

formed a stationary sequence. The first model which could be regarded as truly nonstationary was proposed by Yang (1975). Analyzing the dynamics of the Olympic records Yang observed that the time periods between setting subsequent records do not agree with inter-record times in the classical model of records. He assumed that

the best performance shown in some event of Olympic games can be interpreted as the best performance in this sport over the whole world population. Thus Yang proposed to consider the records in the sequence Yk = max{Xk.1, ... , Xk.n(k) },

k = 1, 2, ... ,

where

{X.i,j },

j = 1, 2, ... , n(i),

i = 1, 2, ... ,

are i.i.d. random variables with a common continuous d.f. F depending on the event and n(k) is the population size of the world at the kth Olympic games. The numbers

n(k) = .1k-1n(1),

k=

1, 2, ... ,

2. RECORD TIMES AND RECORD VALUES

105

represented the geometric growth of population of our planet, and the coefficient A = 21/9 = 1.08 was chosen in view of the fact that during 9 four-year periods between successive Olympic games from 1900 through 1936 the world population doubled. Actually in Yang's scheme one considers records in a sequence of independent random variables Y1i Y2,... with d.f.'s

Fk(x) = (F(x))n(k', where n(k) = ,1k-'n(1), k = 1, 2, ... , and F is a continuous distribution function. Yang obtained the following results for the inter-record times

0(n) = L(n) - L(n - 1),

n = 2, 3, ... .

THEOREM 25.1. Let S(k) = n(1) + n(2) +

k = 1,2,...

+ n(k),

Then

P{0(1) > j} =

n(1)

S(j + 1)

and

P{0(n) > j}

_E E ... 00

00

k1=2 k2=ki+1

00

n(1)n(kl)...n(k,1-1) S(k11) S(kn-1 - 1)S(kn-1 +j)' k_j=k_2+1

j =0,1,...,

n=2,3,....

THEOREM 25.2. For any j = 1, 2, .. . (25.1)

pj = lim P{A(n) = j} = (A - 1)A-l.

Yang used the limiting distributions (25.1) with \ = 1.08 to analyze the frequencies of occurrence of Olympic records between 1900 and 1936. The actual inter-record times turned out to be much shorter than theoretical. Of course, it can be easily explained by different rates of growth of the total world population and the number of athletes, as well as by many other circumstances that enhance the progress in sports. Although Yang's model was not very suitable for the analysis of sport records, it stimulated appearance of new record models. Yang's model originated from a concrete problem related to the sport statistics. This was the reason for taking the d.f.'s F of the underlying random variables X1, X2, ... as Fk = FnM, where n(1), n(2), ... was a specific sequence of integers which formed a geometric progression. It is natural to generalize this scheme by taking arbitrary positive numbers instead of integers n(k). But the main reason for the passage to the F"-scheme is the independence property of record indicators which is characteristic for this scheme.

DEFINITION. A sequence of independent random variables X1iX2,... with d.f.'s F1, F2, ... forms an F"-scheme if F k = F"(k),

k = 1, 2, ... ,

where F is a continuous d.f., and a(1), a(2).... are some positive constants.

25. RECORD TIMES IN THE F°-SCHEME

109

REMARK 25.3. The underlying d.f. F can always be taken to equal Fl. Therefore, without loss of generality, we can assume that a(1) = 1. If all exponents a(n), n = 1, 2, ... , are equal, the F°-scheme reduces to the classical scheme determined by a sequence of i.i.d. random variables.

Consider the record indicators f;1i12..... It turns out that Renyi's Lemma can be carried over with minor modifications to the F°-scheme. Let

n= 1,2,... .

S,6

LEMMA 25.4. In the F°-scheme, the indicators S1, 2, ... are independent and (25.2) P{G = 1} = 1 - PIG = 0}

_

S(n) - S(n - 1)

a(n) a(1) +

n = 1, 2, ... .

S(n)

+ a(n)

PROOF. First we prove the second assertion. Since the d.f. Gn_1 of the maximum M(n - 1) = max(XI, X2, ... , Xn_1) has the form Fs("-1),

Gn-1 = F1F2 ... Fn_1 =

and the random variables Yn_1 and Xn are independent, we obtain that

P{Cn = 1} = P{Xn > max(XI, X2, ... , Xn-1)} =

fx

xLGn_l(x)dFn(x)

'00

_

dF,(n)(x)

= Jo XS(n-1) d(x°(n)) a(n) _ a(n) a(n) + S(n - 1) S(n) Fs(n-1)

1

Now in order to prove independence of the indicators it suffices to show that for any 1 < k(1) < k(2) < . < k(r) we have =

r

II P{ek(m) =1}

P{G(1) = 1, G(2) = 1, ... , k(r) =1}

(25.3)

M=1

_

-

r

a(k(r)) S(a(k(r))) .=1

The probability integral transformation allows us to assume, without loss of generality, that F(x) = x

and

Fn(x) = x°("),

0 < x < 1.

Then (25.4)

l

tt 1,Sk(2) = 1,...,Sk(r) = 1} CC

= P{Xk(I) > )VI(k(1) - 1), Xk(2) > M(k(2) - 1), .... Xk(r) > M(k(r) - 1)}

10U'1

(k(1)-1) d(,ui (k(1)))

l

1

S(k(2)-I)-S(k(1)) d(u

Jug

1

UyS. (k(r)-1)-S(k(r-1)) d(1lr°(k(r)) ).

X

fur-11

l

2))1 1

x ...

2. RECORD TIMES AND RECORD VALUES

110

Successive integration in (25.4) leads to the required expression for the probabilities

P{G(1) = 1,...,G(r) = 1}. The following result is closely related to Lemma 25.4. THEOREM 25.5. Let d.f.'s F1, F2i ... , F, of independent random variables XI, X2i ... , X,, be continuous and

1 < j < n - 1,

0 < Fj (a) < Fj (b) < 1,

(25.5)

for some a and b, oo < a < b < oo. If the vector

and the indicator Sn are independent for any choice of the d.f Fn, then there exist positive

constants a(2),..., a(n - 1) such that

2<j
Fj=(F1)°(j),

and 6, b,

, Sn-1 are mutually independent.

PROOF. Let

Hk(x) = Gn-1(x) = Fk(x) ... Fn-1(x), Gk_1(x)

k = 2, ... , n - 1,

and

Hn(x)=1. It follows from independence of indicators ek and n that (25.6)

Hk+1(u) fW00 Gk-1(y) dFk(y) dFn(u) 00 w Gk-1(y)Fk(y)Hk+i(y) dFn(y), = Gk-1(y) dFk(y)

f

f

00

cc

k=2,...,n-1. Since (25.6) must hold for any distribution function, taking Fn(u) = 11,.>xl we obtain the equality z

(25.7)

Hk+1(x) f Gk-1(y) dFk(y) = c(k)Gk-1(x)Fk(x)Hk+1(x),

where

c(k) = P{£k = 1} = f : Fi(x) ... Fk-1(x) dFk(x). 00

Condition (25.5) implies that

0
2<j
Indeed, one can easily check that

c(j) <

f F1(x)...Fj_I(x)dFj(x)+1-Fj(b)+F,(a) 6

a

< F1(b) ... Fj_1(b) (Fj(b) - Fj (a)) + 1 - Fj (b) + Fj (a)

= 1 - (Fj (b) - Fj (a)) (1 - Fi (b) ... Fj_I (b)) < 1

25. RECORD TIMES IN THE F°-SCHEME

111

and b

c(i) > f F ,

> 0.

F ,

F ,

a

Let a = inf{x: F1(x)... Fn_1(x) > 0}.

For x > a, (25.7) implies

fCk_l(Y)dFk(Y) = c(k)Gk-1(x)Fk(x) = c(k) and (25.8)

f

\

f

LFk (u) dGk-1(u) )

Gk-1(u) dFk(u) + 00

x

Gk-1(u) dFk(u) = 7(k) TOO Fk(u) dGk -I (U),

where 0 < ry(k) < oo,

7(k)

j = 2,3,...,n.

= 1 c(c(k),

We see from (25.8) that

fGk(u)d(logFk(u)) =7(k) f xGk(u)d(logGk-I(u)) J. and

(25.9)

f

x

a

Gk (u) d( log

Fk(u) ) =0 (Gk-1(u))7(k)

x > a.

Since G, (u) > 0 for all u > a, we conclude from (25.9) that Fk(x) = d(k) (Gk-1(x))''(k),

where d(k), k = 2,. .. , n, are some constants. As x tends to infinity we have d(k) = 1, k = 2,3,...,n, (Gk-1(x))1+,(k)

Gk(x) = Gk-1(x)Fk(x) =

=

(G1(W))(1+-y(k))...(1+-y(2))

=

(Fl(x))(1+'y(k))...(1+'y(2))

and (25.10)

Fk(x) = (Gk-1(x))ry(k) =

(FJ(X)),y(k)(I+-e(Ac-I)) --- (1+-t(2)),

k = 2,3,...,n. Thus we have shown that the assertion of the theorem holds for x > a with exponents

c(j) a(j)='Y(j)(1+-t(j-1))...(1+7(2))= (1 - c(2))...(1 - c(j))

j=2,...,n.

By the definition of a there exists k such that Fk(a) = 0. Then (25.10) and continuity of d.f.'s F1,.. . , Fn imply that Fk(a) = 0 for any k, hence (25.10) holds

not only for x > a, but also for x < a. The assertion on independence of the indicators follows by Lemma 25.4.

2. RECORD TIMES AND RECORD VALUES

112

REMARK 25,6, Condition (25.5) means that P{X, < X,} > 0 for any i # j, i, j = 1, ... , n - I. There are simple examples showing that without this condition the theorem may hold not only for the F°-scheme. For example, let Xj have

the uniform U((aj, aj+l]) distribution, j = 1, ... , n - 1, with a1 < a2 <

.<

an. Obviously, then 1, 6,. . ., Sn_1 have a degenerate distribution and the vector (c1, 2, , Sn_ 1) is independent of the indicator 1;,,, though the underlying random variables do not form an F°-scheme. Many results for the F°-scheme can be stated as a slight modification of the corresponding results for the classical record model, since by Lemma 25.4 the number of records N(n) can be represented as the sum of independent indicators

n= 1,2,... .

N(n) =SI +l;'2+...+.£n-1, Denote n

A(n) = EN(n) = E pi,

(25.11)

j=1

where

Pn = P{Sn = 1} =

a(n)

a(l) +

+a(n)

P{N(n)noo -+ oo} = 1 if and only if A(n) - oo. By Dini's criterion, both sequences S(n) and A(n) tend simultaneously either to a finite limit or to infinity as n -+ oo. Hence we have the following lemma. LEMMA 25.7. The equality

P{N(n) n-oo -+ oo} = 1 holds if and only if S(n) -+ oo as n -+ oo.

This statement means also that each record time L(n), n = 2, 3, ... , exists (is finite) almost sure if and only if S(n) -+ oo as n -+ oo. To see this, it is enough to recall the equality

P{L(n) < m} = P{N(m) > n}. Note that for sequences of i.i.d. random variables there was no problem of existence

of record times because S(n) = n in the classical scheme. In general, to deal with records in the F°-scheme we need the restriction on the exponents a(n),

lim S(n) = oo.

(25.12)

n-.oo

As we pointed out, the classical record model can be embedded into the F°scheme. It turns out that the kth record times L(n, k) can also be embedded into the F°-scheme. Indeed, the distributions of these random variables are determined by the sequence of independent indicators .1(k),Wk),... such that 1} = 0

for j = 1,...,k- 1, P{ek(k)=1}=1, and

j>k.

25. RECORD TIMES IN THE F'°-SCHEME

113

Now take the exponents in the FI-scheme as follows: k

a(n)

+n-2 n = 1,2,...

k - 1

Then pn

P{Gn

a)

11

k

PIG+k-1(k) = 1}.

n+k a(1) + Therefore, for any n = 1, 2.... we have the equality

(L(1, k), ... , L(n, k)) _ (L(1) + k - 1, ... , L(n) + k - 1),

where the record times L(n) correspond to the F°-scheme with exponents a(n)

_

k+n-2

n = 1, 2, ... .

k-1

Since the study of the record times in the FO-scheme uses the arguments which

were used in the classical record model, we leave the proof of some additional relations to the reader. EXERCISE 25.1. Show that f o r any n = 1, 2, ... and 1 < m(1) < m(2) < . - . < m(n) we have

P{L(1) = 1, L(2) = m(2),..., L(n) = m(n)} =

l a(m(r)) S(m(n)) 1 2 S(m(r) - 1) nn

EXERCISE 25.2. Show that L(1), L(2),... form a Markov chain with

P{L(n) = j I L(n - 1) = i} = S(i) ( and

P{L(n) > j I L(n - 1) = i} =

1

S(.

1

1)

S(:), SW

)

S(j)

>i.

EXERCISE 25.3. Let

A(n) = EN(n) = -1 a(1) a(j) +....}. a(j) j and let

.Fn = a{L(1), L(2), ... , L(n)} denote the a-algebra generated by the record times L(1), L(2),. .., L(n). Assume that as n --.oo. Then the random variables V(n) = A(L(n)) - n,

n = 1,2,...,

form a martingale with respect to the sequence of o-algebras Fn, and

E A(L(n)) = n,

n = 1,2,....

2. RECORD TIMES AND RECORD VALUES

114

EXERCISE 25.4. Let

n +)1) -, 1 as n -+ oc. S(n Then for any fixed k = 2, 3, ... the random variables S(L(n + k - 1)) S(L(n + 1)) S(L(n)) ' S(L(n + k)) S(L(n + 1))' S(L(n + 2))' are asymptotically independent and S(L(n)) S(n) -+ oo and

lim P nco S(L(n + 1))

<x =x,

0 < x < 1.

We will restrict ourselves to the results stated above, though there are yet a number of limit theorems for record and inter-record times in the F°-scheme which generalize the corresponding results for the classical record model. Lecture 26. INDEPENDENCE OF RECORD INDICATORS AND MAXIMA

In the previous lecture we discussed some problems concerning independence

of the record indicators in the F°-scheme. The relationship between indicators S1, b, ... , S. and maxima M(n) = max{Xl, X2, ... , Xn} is also of interest. We have the following result.

THEOREM 26.1. In the F°-scheme the record indicators {1i 2i ... , Cn and the random variable M(n) = max{Xl, X2, ... , X,.} are independent for n = 1, 2,... . PROOF. Since the probability integral transformation does not affect the ordering of random variables and hence preserves the distributions of record indicators,

it is sufficient to prove the theorem for F(x) = x, 0 < x < 1. One can easily see that in this case

P{M(n) < x} = xs(n),

(26.1)

0 < x < 1,

and for any r and 1
P{G(1) = 1,G(2) = 1, ... , Sk(r) = 1, M(n) < x} US(k(I)-1) d(ui(k(1))) l

0 X

r

-

fx c

T

uS (k(2)-1)-S(k(Id(k(2))) 2

U2

X ...

us(k(r)-1)-S(k(r-I)) d( u°(k(r)))XS(n)-S(k(r))

(k(r))

11 m=1 S(a(k(r)))

S(n)

0<x<1.

Now the theorem immediately follows from (25.3), (26.1), and (26.2).

0

It turns out that independence of maxima M(n) and indicators 6, 6, ... , G characterizes the F°-scheme under some additional restrictions on the underlying random variables. The following theorem is closely related to Theorem 25.5. Condition (26.3) of this theorem, with n replaced by n-1, coincides with condition (25.5).

26. INDEPENDENCE OF RECORD INDICATORS AND MAXIMA

115

THEOREM 26.2. Let independent random variables X1 , X2, ... , X have continuous d. f. 's F1i F2, ... , Fn such that for some a and b, -oo < a < b < oc, 1 < j < n.

0 < Fj(a) < Fj(b) < 1,

(26.3)

I f the vector (6, 1;2, ... , Sn) and the maximum M(r) = max(X1,... , X,.) are independent for any r = 2, 3, ... , n then there exist positive constants c(2),. .. , a(n - 1) such that

Fj=(F1)"(j),

2<j
and the record indicators i,1;2, ... , Sn are mutually independent.

PROOF. Since M(n) and 6, 6, ... , bn are independent, for j = 1, 2, ... , n we have

P{M(n) < x, Cj = 1} = c(j)Fi(x) ... Fn(x),

(26.4)

where CU)

11 = f00 F1(x)...F-1(x)dF x .

P

It was shown in the proof of Theorem 25.5 that 0 < c(j) < 1. One can easily check 00

that (26.5)

P{M(n) <

1

Fj+I(x)

Fn(x)

f

00

F1 (u) ... Fj - 1 (u) dFj(u )

Let a = inf{x: F, (x) ... Fn(x) > 0}. Obviously, -oo < a < a. As follows from (26.4) and (26.5), f o r x > a the following equalities hold f o r j = 1, 2, ... , n: x

J

F1(u) ... Fj(u) dFj(u) = c(j)F1(x) ... Fj(x) c

z

G- u dF u

F u dG

u

where Gj(u) = Fl(u) ... Fj_1(u). Then Jz

Gj-1(u) dFj(u) = 7(j)

f

x

F'j(u) dGj(u), 00

where -y(j) = c(j)/(1 - c(j)) with 0 < y(j) < oo, j = 2,3,...,n. The relation we have obtained coincides with (25.8). Proceeding further as in the proof of 0 Theorem 25.5 we complete the proof. EXAMPLE 26.3. One may ask if we necessarily have an F"-scheme when M(n) and Sn+1 are independent. Consider the following simple example. Let X1 and X2 be independent random variables with d.f.'s F1 and F2, and let M(1) = X1 and 6 be also independent. Then

P{M(1) < x, 2 = 0} = P{X2 5X1 < x} = f F2(u) dFi(u), and independence of M(1) and 6 implies that

f(F2(u)-c)dFi(u)=0,

-oc < x < oo,

2. RECORD TIMES AND RECORD VALUES

£16

where

c = P{6 = 0} = J

F2 (u) dFi(u). 00

Obviously, in this case the supports of X1 and X2 are disjoint and both indicators have a degenerate distribution. Thus even for n = 1 independence of M(n) and Sn+1 does not lead to an FI-scheme. Independence of record indicators b1, C2, , In and maxima M(n) allows us to prove the following relation for record times L(n) and record values X(n) in the F°`-scheme.

LEMMA 26.4. Let X 1,X2, ... be independent random variables which form an

F°-scheme with d.f. F and exponents a(1),c (2),.... Then

P{X(n) < x} = E

(26.6)

(F(x))S(c(n))

where

S(n) = a(1) +

+ a(n).

PROOF. Recall that the event {L(n) = m} coincides with the event

{S1+...+&m-1=n-1, Sm=1} which is determined by the indicators 6, 1;2, ... ,' m and hence does not depend on the maximum M(n). The following equalities are obvious:

P{X(n) < x} = P{M(L(n)) < x} 00

_ E P{M(L(n)) < x I L(n) = m}P{L(n) = m} m=n 00

_ E P{M(m) < x I S1 +

+ 1'm_1 = n - 1, em = 1}P{L(n) = m}

m=n 00

_ E P{M(m) < x}P{L(n) = m} m=n 00

(F(x))S(m)P {L(n)

_

= m} = E (F(x))s(L(n))

m=n

0

Lecture 27. ASYMPTOTIC DISTRIBUTION OF RECORD VALUES IN THE f°-SCHEME

It was shown in Lecture 18 that the limiting d.f.'s for record values X(n) in the classical record model have the form G(x) = 4'(9(x)),

27. ASYMI''I'OTIC DISTRIBUTION OF RECORD VALUES

117

where I is the standard normal distribution function and g may belong to one of the following three types: 91(x) = x;

7 log x, 92 (x)

=

-oo y log(-x),

9s (x) = {

for x > 0,

7 > 0, ry > 0,

oo

for x < 0; for x < 0, for

x > 0.

It turns out that when we pass from the classical model to the F°-scheme, the set of possible limiting distributions for the record values X(n) under fairly general conditions on the exponents a(1), a(2).... remains unchanged. We will formulate the corresponding results and briefly sketch their proofs. For that we have to recall the material of Lecture 25. Recall the notation n

S(n) = a(1) +

A(n) = EN(n) _ Epj,

+ a(n),

j=1

where Pn =

1} =

a(1) +a(n) - + a(n)'

and introduce the sequences

n

D(n) _

Pi !=1

and

n = 1, 2,... . B(n) = Var N(n) = A(n) - D(n), We mentioned in Lecture 25 that there are a number of limit theorems for record times in the Fe-scheme which generalize the corresponding results for the classical record model. Here we formulate one of them without proof. THEOREM 27.1. Let A(n) - oo,

0,

and Pn

0

as

n

oc.

as

n -+ co.

Then

(27.1)

sup IP{logS(L(n)) - n < xn112} -.t(x)I -0 X

First we consider the F°-scheme with exponential d.f. F. THEOREM 27.2. Let X1, X2, ... form an F'-scheme with F(x) = 1 -exp(-x),

x > 0, and exponents satisfying the conditions of Theorem. 27.1. Then for any x and y we have (27.2)

P{X(n) - log S(L(n)) < x; logS(L(n)) - n < yn112} -+ A(x) = exp{- exp(-x)}.

2. RECORD TIMES AND RECORD VALUES

118

PROOF. Since for any r = 1, 2,... the maximum M(r) and the indicators r are independent, the events {M(r) - log S(r) < x} and

{L(n) = r} = {fl +"+r-1 =n-

1}

are independent for any x, r, and n. If F(x) = 1 - exp(-x), then for r -+ 00 exp(r) ) s(r)

P{M(r) - log S(r) < x} = (1 -S(T)

= exp { - exp(-x) + O ( 1r)) } S(

Consequently, we obtain (27.3)

P{X (n) - log S(L(n)) < x; log S(L(n)) - n < yn1/2} 00

_

P{M(L(n)) - log S(L(n)) < x; log S(L(n)) - n < yn112 1 L(n) = r} r=n

x P{L(n) = r} 00

_ E P(M(r) - logS(r) < x}1(log s(L(n))-n
_

exp (-

e-x

l

+O(S(r)))1{Iogs(L(n))-n
r=n

_=

or.

exp

( -e`+0 (S(n) / /

1 {Iog s(L(n))-n
=exp(-a_x+O(S(n)))P{logS(L(n))-n
n-+oo.

By assumption A(n) -# oo, hence by Dini's criterion S(n) -- oo as n -i oo. Now

0

(27.1) and (27.3) imply (27.2).

By simple arguments we can derive from (27.2) that

P{X(n) - n < xn1/2} -+ fi(x),

n

oo.

Thus Theorem 27.2 entails the following result. COROLLARY 27.3. Let X1, X2,

form an F* -scheme with

F(x) = 1 - exp(-x),

x > 0,

and exponents satisfying the conditions of Theorem 27.1. Then (27.4)

P{X(n) - n < xn1/2} -+ F(x),

n -p oo.

When we studied in Lecture 18 the structure of limit distributions for record values in the classical scheme, we also considered first the exponential distribution and obtained relation (18.1). Then using the probability integral transformation

we obtained the three types of limit distributions for X(n). Now, starting with relation (27.4), which coincides with (18.1), and using the same arguments we arrive at the same set of limit distributions in the Fa-scheme. Thus we have the following result.

2s, RECORDS [.'Oil. DEPENDENT RANDOM VARIABLES

II)

THEOREM 27.4. Let X1,X2,... form an F"-scheme with continuous d.f. F and exponents which satisfy, as n

oo, the conditions

pn -+ 0,

S(n) -' oc,

(n)))

and

(A

Then the set of all limiting d.f 's T(x) for suitably centered and normalized record times X (n) comprises the following three types: Ti(x) = 'D (x);

for x > 0, for x < 0;

14) (-Ylogx), T2 , 7

(x) =

7's ,7

(x)

t f

0

(D(-y log(-x)),

for x < 0,

1

for x > 0,

where -y > 0.

REMARK 27.5. It can be shown that Theorem 27.4 remains valid for an F'scheme with exponents a(1), a(2).... satisfying the following condition: there ex-

ists p,0
Pn -+ P,

V'P - Pr -+0 r=1

n

1` (P - Pr)'

and

n

r=1

p.

n

Lecture 28. RECORDS IN SEQUENCES OF DEPENDENT RANDOM VARIABLES

So far we studied record times and record values in a sequence of independent random variables X1, X2,... . Now we abandon the independence condition on the underlying X's. How will this affect the properties of the records? Let us consider some settings.

Records for symmetrically dependent random variables. Here we give an example of random variables X1, X2, ... for which the distributions of the record times L(n) are the same as in the classical model. DEFINITION 28.1. Random variables X1, X2, ... , Xn are called symmetrically dependent if all n! permutations (Xr(1), Xr(2), , Xr(n)) of these random variables have the same n-dimensional distribution. Random variables which form an infinite sequence X1, X2, ... are said to be symmetrically dependent if X1, X2, - - . , Xn are symmetrically dependent for any n. -

Note that i.i.d. random variables X1, X2, ... form an infinite sequence of symmetrically dependent random variables. For record indicators we have the following result. THEOREM 28.2. Let X1, X2, ... , Xn be symmetrically dependent random variables and let (28.1)

P{X1 = X2} = 0.

Then the indicators k, k = 1,2,...,n, are independent and P{£k = 1} = 1/k.

2. RECORD TIMES AND RECORD VALUES

120

PROOF. First we will show that

i,

P{G =1}=

k=1,2,...,n.

Under the conditions of the theorem the probability

P{G = 1} = P{Xk > max(XI,...,Xk_1)} coincides with any of the probabilities

P{Xj > max(XI,...,Xj_l,Xj+1,...,Xk)},

j = 1,2,...,k.

Taking into account (28.1) and the property of symmetric dependence of our random

variables we obtain that P{Xi = Xj} = 0 for any i 34 j. Therefore

P{Xj = max(X1,... , Xj-1, Xj+l, ... , Xk)} = 0,

j < k,

and we have the equality k

1 = EP{Xj > max(X1,...,Xj-1,Xj+1,...,Xk)} j=1

= kP{Xk > max(X1i...)Xk_1)} = kP{G = 1}, which implies that 1

P{G=1}= 1 For the proof of the first assertion of the theorem it suffices to show that for

any2
(282)

, n(r)

=1}=-

1

f1jr=1

n(j)

It follows from (28.1) and symmetric dependence that (28.3) P{en(1) =

1} _ E P{Xr(1) < Xm(2) < ... < X,(n)} MEo

=NP{X1<X2<...<Xn}, where the summation is extended over all elements of the set o which consists of all permutations M = (m(1), ... , m(n)) of the numbers 1, 2, ... , n such that

{Xn(1) > max(Xj,...,Xn(1)-1),...,Xn(r) > max(X1,...,Xn(r)-1)}

with N, denoting the cardinality of the set o. Note here that N, coincides with the corresponding value of N, in a sequence of i.i.d. random variables X1..... Xn, where it is equal to n!

rjj=1

n(j) In our setup all events {Xm(1) < Xm(2) < . . < X,n(n)} corresponding to all possible n! permutations (m(1), ... , m(n)) of numbers 1, 2, ... , n are equiprobable. .

Therefore, as in the classical scheme, n!'

28. RECORDS IOR DEPENDENT RANDOM VARIABLES

121

and we obtain from (28.3) that

P

n

1} _ No = Fjr

1,..., nr

1

j=1

i.e., (28.2) holds. Thus the proof is completed.

REMARK 28.3. Condition (28.1) cannot be replaced by the continuity condition on the underlying d.f. F. Indeed, let X1, X2, ... have the uniform distribution on [0, 1] and let P{XI = X2 = .. = Xn} = 1 f o r any n = 1, 2, .... Then X1,X2,... are symmetrically dependent, but (28.1) does not hold. In this case the sequence X1, X2i ... , X,, contains the only record value XI, and 1} = 0 for

k=2,3,....

EXERCISE 28.1. Construct a finite sequence of discrete symmetrically dependent random variables XI, X2i ... , Xn which satisfy condition (28.1) and for which Theorem 28.2 is valid.

REMARK 28.4. One can easily verify that the distributions of N(n), L(n), and 0(n) are completely determined by the distributions of the record indicators I, 2..... Therefore all results for these record statistics which were obtained in the case of i.i.d. random variables X1, X2, ... with continuous V. remain valid for symmetrically dependent X's satisfying condition (28.1). Now we consider distributions of record values X(n) in sequences of symmetrically dependent random variables. Recall the following result by de Finetti (see, e.g., Galambos (1978, 1987), Theorem 3.6.1). REPRESENTATION 28.5. Let X1, X2, ... be an infinite sequence of symmetri-

cally dependent random variables. Then there exists a family of d.f.'s Fe, -oo < 0 < oo, and a d. f. G such that (28.4)

P{X1 <X1i...,Xn <xn}= f F9(x1)...Fe(xn)G(dO) 00

for any n > 1 and any xl, x2, ... , xn.

REMARK 28.6. The right-hand side of (28.4) is a mixture of distributions of the n-variate vector (X1, X2i ... , Xn) with i.i.d. components having the common d.f. Fe. Assume additionally that d.f.'s Fe are continuous for almost all 0 (with respect to G). Then, as we know from Lecture 15, for any fixed 9

P{Xo(n) < x} =

1

(n

1)!

f

log(1-Fe(y))

so that (28.5)

P{X(n) < x} = f P{Xg(n) m < x} G(dO) 00

_00 (n

1

- 1)

1-Iog(l-Fe(x))

u"-1e

du,

2. RECORD TIMES AND RECORD VALUES

122

Note that in this case the joint distributions of record values X1, X2.... in an infinite sequence of symmetrically dependent random variables are also representable in the form

P{X(1) <xii...,X(n) <xn} =

r00

J

Fn.e(xi,...,xn)G(d9),

00

where

Fn,9(x1,... , xn) = P{X9(1) < xl,... , X9(n) < xn},

n = 1, 2, ... ,

are the d.f.'s of record values X9(1), X9(2), ... in a sequence of i.i.d. random variables with common d.f. F9. EXAMPLE 28.7. Consider a stationary Gaussian sequence X1, X2, ... with moments

O
EX1=0, VarX1=1, and EX1Xn=p, This sequence admits the following representation: (28.6)

{Xn}°n°

1

= {Y + Zn}n

1'

where Y, Z1, Z2, ... are independent normal random variables with parameters

E Y = E Z1 = E Z2 = . . . =0,

Var Y = p,

and

Var Zn = 1-p,

n=1,2,....

The random variables X1, X2, ... are symmetrically dependent and (28.4) and (28.5) hold with d.f.'s F9 (x) _ t (

and G(B) _ -t1/P_

Moreover, (28.6) implies that (28.7)

{X(n)}°° n=1

{Y+Z(n)

n=1 ,

where Z(n), n = 1, 2, ... , are record values in the sequence of i.i.d. random variables Z1, Z2,..., and Y and Z(n) are independent.

Records in Markov sequences. Now we consider Markov sequences of random variables X1, X2, .... We will show that not only the corresponding record values X (n), n = 1, 2, ... , but also the sequences of two-dimensional random vectors (X(n), L(n)) inherit the Markov property of the underlying random variables. We will assume that X1, X2, ... form a stationary irreducible Markov chain

with state space S, where S = (-oo, oo) or S = [a, b], -oo < a < b < oo. If S = [a, b], we assume additionally that b is not an atom of the underlying distribution. This last assumption ensures that any record time L(n) and any record value X(n) exist almost sure. Define the transition probabilities

F(x, y) = P{Xn+l < y I Xn = n},

n = 1, 2, ... ,

The following result is due to Adke (1993).

-oo < y < oo,

x E S.

28. RECORDS FOR DEPENDENT RANDOM VARIABLES

123

THEOREM 28.8. The random vectors (L(n), X (n)), n = 1.2, ... , form a Markov chain with transition probabilities

Q(j, x, y) = P{L(n + 1) = r + j, X (n + 1) < y I L(n) = r, X (n) = x}, which do not depend on n and for x < y are given by the formulas Q(1, x, y) = F(x, y) - F(x, x), =Jpx

(F(u, y) - F(u, x)) F(x, du),

Q(2, x, y) 00

Q(j,x,y) =

(28.8)

J

... fx0C F(x,dul)(F(ui-1, y)

x

j-2

- F(uj-1, x)) fl F(ui, dui+1) for j > 2. i=1

For x > y and j = 1, 2, ... we have Q(j, x, y) = 0. PROOF. For any

1 = m(1) < m(2) < . . < m(n + l) and t1
we have the equality (28.9)

P{L(n + 1) = m(n + 1), X(n + 1) < to+1 L(1)

= m(1), X(1) = t1,...,L(n) =m(n), X(n) =tn} = P{Xm(n)+l < tn, ... , Xm(n+l)-1 < tn, to < Xm(n+l) < to+l I L(1) = m(1), X(1) = t1i... , L(n) = m(n), X(n) = to}. The event

{L(1) = m(1), X(1) = t17..., L(n) = m(n), X(n) = tn} belongs to the a-algebra generated by the random variables Xl, X2, ... Xm(n)Hence the right-hand side of (28.9) equals

P{Xm(n)+l < tn, ... , Xm(n+l)-l < tn, tn <_ Xm(n+l) < tn+1 I Xm(n) = tn} and

P f L(n + 1) = m(n + 1), X(n + 1) < t,,+, I L(n) = m(n), X (n) = to } = P{Xm(n)+l < tn,...,Xm(n+l)-1 < tn, tn < Xm(n+l) < to+1 I Xm(n) = to}, so that (28.10)

P{L(n+ 1) = m(n+ 1), X(n+ 1) < to+1 L(1) = m(1), X(1) = t1,...,L(n) = m(n), X(n) = tn} = P{L(n+ 1) = m(n+ 1), X(n + 1) < to+l I L(n) = m(n), X(n) = to}.

2. RECORD TIMES AND RECORD VALUES

124

The Markov property of the sequence (L(n), X (n)) follows directly from (28.10). Furthermore, observe that (28.11)

Q(j, x, y) = P{L(n + 1) = r + j, X (n + 1) < y I L(n) = r, X (n) = x} = P{Xr+1 < x, ... , Xr+j-1 < x, x < Xr+j < y I Xr = x}

=P{X2<x,...,Xj <x,x<Xj+1
Q(x, y) = P{X (n + 1):5 y I X (n) = x} = E Q(j, x, y), j=1

where Q(j, x, y) are given by (28.8).

PROOF. Using (28.10) and (28.11) we obtain

P{X(n + l) < y I X(1), ..., X(n -1), X(n) = x} = E (1{x(n+l)
X(1),...,X(n- 1), X(n) = x) I X(1),...,X(n- 1), X(n) = x} 00

= E { E E (1{L(n+1)=L(n)+j,x(n+l)
X(1),...,X(n- 1), X(n) = x) I X(1),...,X(n- 1), X(n) =x} 00

_ E E {E (1{L(n+1)=L(n)+j. X(n+i)
X (n) = x) I X (1), ... , X (n - 1), X (n) = x} 00

_ E E {P (L(n + 1) = L(n) + j, X (n + 1) < y I L(n), j_1

X (n) = x) I X(1),. .. , X (n - 1), X(n) = x} 00

_

00

E {Q(j, x, y) I X(I),..., X (n - 1), X(n) = x} _ j=1

Q(j, x, y) j=1

29, RECORDS AND THE SECRETARY PROBLEM

12;

EXAMPLE 28.10. Let X1i X2, ... be i.i.d. random variables with a common d.f. F. This sequence is a stationary Markov chain with transition probabilities F(x, y) = F(y). Then for any y > x Q(1, x, y) = F(y) - F(x), Q(2, x, y) =

f

(fly) - F(x)) F(du) = (F(y) - F(x)) F(x),

Q(j, x, y) = 00

f

X

j-2 F(dul) rl F(dui+1) (F(y) - F(x))

00

i=1

= (F(y) - F(x)) (F(x))j-1

for j > 2.

Consequently, in this case

- F(x) Q(x, Y) = 00 E Q(j, x, y) = F(y) 1 F(x) j-1

y>x.

REMARK 28.11. It follows from the properties of the record times that in the classical model L(1), L(2).... form a Markov chain. Adke has shown that for Markov sequences X1, X2,. .. the record times L(n), n = 1, 2, ... , need not possess the Markov property.

Lecture 29. RECORDS AND THE SECRETARY PROBLEM

One of popular problems in the probability theory is the optimal selection problem, or the so-called "secretary problem". There are n applicants (with n > 2 fixed in advance) for a vacant secretary position. The employer meets them successively and after each interview he has to decide, based on this and previous interviews, but without any knowledge about the remaining applicants, whether the present applicant should be accepted or rejected. In case of acceptance the selection procedure is terminated, otherwise it continues without returning to rejected candidates. The problem is to find the strategy maximizing the probability of the optimal selection, i.e., finding the best out of n candidates. There are various versions of this problem. We will formalize one of them. Let one of n! permutations (a(1), a(2), ... , a(n)) of the numbers 1, 2, ... , n be chosen at random. We can observe only sequential

ranks r1, r2.... of the random variables a(1), a(2),... For m = 1, 2, ... , based on r1, r2, ... , rm, we sequentially test the hypothesis that the number at the mth position in the random permutation is n. If we accept the hypothesis, the procedure is terminated, otherwise, if the hypothesis is rejected, the sequential rank r,n+1 is added to the set of known sequential ranks and we make another attempt to guess whether rm+l corresponds to n, and so on. We search for the strategy maximizing the probability to determine the position of n correctly. Note that the number n may occupy any of n positions with probability 1/n. Hence the random selection taking no account of the ranks r1i r2, ... can be successful only with probability 1/n. Knowledge of the ranks rl, r2i ... enables us to increase substantially the probability of the correct selection. We are primarily interested in this problem with regard to the theory of records. Let X1, X2,. .., X.n be i.i.d. random variables with common continuous d.f. F, which

RECORD TIMES AND RECORD VALVES

126

is unknown to us. By symmetry, the classical ranks R1, R2, .... R of these variables satisfy the relation

P{R1 = a(1),..., R, = a(n)} =

n!

for any permutation (a(1), a(2), ... , a(n)) of the numbers (1, 2, ... , n). We observe the random variables X1, X2 ... , X, sequentially and we want to stop when the maximal value occurs. In other words, we have to guess the last record time, i.e., the value of L(N(n)). In this setup, based on the sequential ranks r1, r2, ... , rm of the random variables X1, X2 ... , Xm, we need to figure out whether the event Am,n = {Rn = n} occurs. Note that for m < n this event cannot be expressed in terms of events generated by the ranks r1, r2, ... , rm, but it is expressed in terms of the ranks rm, rm+1, ... , rn as follows: (29.1)

Am,n = {rm = m, rm+1 < m + 1, ... , rn < n}.

If we recall the record indicators em = 1{X,,,>max(Xt,...,Xm-1)} = 1{rm=m},

we can easily see that (29.2)

Am,n = {Sm = 1, Cm+1 = 0, ... , Cn = 0}.

Thus the event Am,n does not depend on the sequential ranks r1, r2i ... , rm_1, and its dependence on rm amounts virtually to the dependence on the event

{rm = m} = {fm = 1}.

Actually the optimal selection uses the information contained in the sequence of record indicators 1 , 6, ... rather than in the sequence of ranks r1, r2, ... , and the only information one needs to know about an applicant is whether or not he is better than any of the previous ones. For instance, the knowledge that the seventh applicant is better than any of the first five, but worse than the sixth one, does not add any useful information to the knowledge that the seventh one is no better than the best of the first six. Thus we have to detect the last unit when it appears in the sequence of indicators £1, 2, ... , n The optimal strategy (see, e.g., Dynkin and Yushkevich (1967) f o r justification) belongs to the set T = {Tm, m = 1, 2, ... , n - 11, where the strategy Tm consists in taking the first indicator among m+1, ... , Sn which is equal to one. Out of the strategies T1,. .. ,Tn_1 we will choose the one maximizing the probability of the correct selection. It can be easily seen that under the strategy Tm the probability pm,n of selecting the last unit in the sequence 1, has the form (29.3)

Pm,n = P{Cm+l + ... + n = 1}.

We can find the exact and asymptotic expressions for Pm.n It follows from (29.3) that 1

PO,n=P{51=1,C2=0,...,G =0}= n

20. RECORDS AND 1'ltE SEC'RET'ARY PROBLEM

127

and

(29.4) p..,, _ E

=0,...,G-1 =Mk = 1,G+1 =0,...,l;n =0}

k=m+1 n

k- 2 1

m

11

r m'

k=m+l

_mn-11 n

n- I

k

T-1 1r L--L1

n

m=1,2,...,n-1.

k

Moreover, we have the equality n-1

1

n(Pm,n - Pm+1,n) = 1 -

(29.5)

F,

k=m+1

which implies that the probabilities Pm,n first increase with m and from some point on become decreasing. Hence p,n,n takes the maximal value for m = m* satisfying the inequalities n-1

n-1

<1<

(29.6)

k=m,

k=m*+1

and the maximal value equals

m" n-1

Pn =

(29.7)

E

k=m.

1L

The values of m* and pn for n = 1, 2, ... ,10 are given in the following table: n

1

2

3

4

5

6

7

8

9

10

m*

0

0

1

1

2

2

2

3

3

3

pn

1

0.5

0.5

0.458

0.433

0.428

0.414

0.410

0.406

0.399

For large n we can use approximate formulas. From (29.6), (29.7), and relation n

1

E ti log n, k=1

which holds for n equality

oo, we obtain that m* can be found from the approximate log

Thus

m- nand e

n M.

p"n

1.

-.1

e

as

n - oc.

The last result can be obtained without deriving (29.6) and (29.7). Indeed, using the Poisson approximation for sums of independent record indicators we obtain that +Sn = 1} N P{t = 1}, Pm.n = P{Sm+1 +

128

2. RECORD TIMES AND RECORD VALUES

where rl is a random variable having the Poisson distribution with parameter (29.8)

1

1

Arvin =V", E+1 +...+W = \m+1 +

+

n/

Consequently, for n -+ oo, (29.9)

pm,n - An,m exp(-)\n,m)

The right-hand side of (29.9) is maximized for An,,n = 1, and the maximal value is equal to 1/e. It follows from (29.8) that An,m 1 if m - n/e. Hence an asymptotically optimal strategy for large n can be described as follows: reject the first m' = [n/e] candidates and accept the first out of the remaining (n - m') candidates which is better than any of the rejected m' candidates. The probabilities prz of selecting the best candidate under the above strategy fulfill the asymptotic relation

= 0.368... . Recall that under random selection we guess the best among n candidates with pn -+

1

C

probability 1/n, which tends to zero with growing n. The strategy described above for sequences of i.i.d. random variables can be extended to the Fe-scheme with an unknown d.f. F in the case where the exponents

al, a2.... are given. The only difference from the previous setting is that the probabilities P{Sn = 1} for the indicators i;l, 6.... in the Fa-scheme are equal to an/A(n) with A(n) = al + a2 + + an rather than 1/n as it was in the classical scheme. Hence formulas (29.4) and (29.5) become n-1 (29.10) m = 1, 2, ... , n - 1, pm,n = 4 > a(A(k)1) k=m

and (29.11)

1: a(k + 1)

A(n)(pm,n - pm+1,n) = a(m + 1) 1C

k=m+1

A(k)

It follows from (29.11) that Pm,n first grows and then decreases with in. Hence Pm.n

attains maximum for m = m' satisfying the inequalities n-1 (29.12)

E

a(k + 1) < 1

k=m-+1

(k)

<

a(k + 1) k=m,

A(k)

and the maximal value is equal to

.n-1 (29.13)

pri =

E a(A(k)1) A(n) ) k=m

where m* is determined by (29.12). Thus if we observe sequentially the values x1, x2, ... , x, of the random variables X1, X2, ... , Xn which obey an F' -scheme with known exponents a1 , a2, ... , an and unknown d.f. F, then the probability to identify the value xn.n = max{x1,x2, .... xn } when it appears is maximized by the following strategy: observe the values X1, x2, ... , xm with m* given by (29.12) and take the first value in the sequence xn exceeding max{xl, x2, ... , xm. I. xm'+l,

30. STATISTICAL. PROCEDURES RELATED TO RECORDS

129

EXERCISE 29.1. Analyze the asymptotic behavior of m' and p;, given by (29.12) and (29.13) if the exponents al, a2.... are uniformly bounded from above and bounded away from zero, i.e., there exist constants 0 < c < C < oo such that

c < a, < C for any n = 1, 2,.. .. Lecture 30. STATISTICAL PROCEDURES RELATED TO RECORDS

In this last lecture we will discuss the relation of records to various fields of the probability theory and mathematical statistics. Already in two years after the first paper on records by Chandler (1952) there appeared papers that proposed using records for testing some statistical hypotheses (see Foster and Stuart (1954), Foster and Teichroew (1954), which was followed by Stuart (1956, 1957) and Barton and Mallows (1961)). They treated testing for randomness, for homoscedasticity, for trend against natural alternatives. For example, the nonparametric statistic SS(n) = N, (n) - N2(n),

which is the difference between the number of upper and lower records in the sequence of observations x1i x2, ... , xn, was used to detect trend in the sequence

Xk=Yk+6k,

k=1,2,...,

where Y1, Y2, ... , are i.i.d. random variables. It is clear that if 6 > 0, then the number of upper records is stochastically larger and the number of lower records is smaller compared to the case of 6 = 0, with opposite relationships when 6 < 0. It can be easily seen that for 6 = 0 the above statistic can be represented as the sum S(n) = 772 + ...+ 77n

of independent indicators 77k, k = 2, 3, ... , n, where lk = -1 if Xk is a lower record value, 77k = 1 if Xk is an upper record value, and 71k = 0 in other cases. Hence

E S(n) = 0,

Var S(n) = E

2 log n

k=2 k

and the statistic

S(n) (2 log n) 1/2

is asymptotically normal for 5 = 0. If 6 > 0, then E S(n) > 0, and E S(n) < 0 for 6 < 0. Consequently, when testing the hypothesis of no trend (5 = 0) against right-sided (6 > 0) or left-sided (6 < 0) alternatives the level a critical regions are

{s(n) > (2logn)1/2z,}

and

{s(n) < -(2 logn)1/2zQ}

respectively, where z,, is determined by 1 - 4i(z,,) = a. The statistics based on records have not found wide use for testing statistical hypotheses. It is largely related to the fact that the record values appear seldom and their number in a sample of size n is of order log n. This drawback can be compensated to some extent by using kth records which appear more frequently. An essential reason for studying and using the record statistics is that the statistical data often comprise only the record values (e.g., in sports). The record statistics have been more often used for parameter estimation. Again, it is largely connected with sport records. In order to predict future record

2. RECORD TIMES AND RECORD VALUES

1311

values one has to build a record model and estimate its parameters. Let us give some

examples. Berred (1992) considered record values X(1),X(2),... in a sequence of i.i.d. random variables with d.f.

F(x) = 1 - x-'L(x), where the slowly varying function F(x) and the parameter a > 0 are unknown. He proposed the statistics Rk,n =

logX(n)-logX(n-k) k

k logX(n-j+1)

and Tk, = Fj=1

n(k)

where 1
2

j=1

He considered the asymptotic properties of Rk,n and Tk,,,. It was shown that

when k-+ooand k/n-+ ooasn -+oo, (a) Rk,n converges to 1/a in probability; (b) the random variables ak1/2(Rk,n - 1/a) are asymptotically normal. The statistic Tk,n for a fixed k has similar properties: (a) Tk,n converges to 1/a in probability; (b) the random variables ant/2(Tk,n - 1/a) are asymptotically normal. As we mentioned already, parameter estimation was often needed for prediction of the future record values. For example, Ahsanullah (1995) considered the sequence of Olympic records in women's freestyle swimming of 100 m. There were 11 records registered during the period from 1912 through 1988 (time in seconds): year 1912 1920 1924 1928

1932

1936

1952 1956 1972 1976 1980

time 79.80 73.60 72.20 71.00 66.80 65.90 65.50 62.00 58.89 55.65 54.79

The underlying distribution was taken to be the three-parameter generalized extreme value distribution given by the formulas:

F(x) = 1 - exp { - (1 + rya-1(x -.a))"' } for -y 54 0, a > 0, and x such that 1 + ya(x - u) > 0, and

F(x)=1-exp{-exp

l-Ill or

forty=0, a > 0. Use of the generalized extreme value distribution in the record model was mo-

tivated by the fact that each record under consideration, being the time of the Olympic winner, is the minimum min{Y1, Y2, ... , YN} of large number N of random variables, where N is the total number of people practicing the particular sport in the year of the Olympics. For this particular example, Ahsanullah obtained the following best (minimal mean squared error) linear estimates: ry = 0.60,

2 = 78.33,

a = 3.51.

30. STATISTICAL PROCEDURES RELATED TO RECORDS

131

Then, based on the 11 records, the estimates for future Olympic records for this sport were computed. These predictions were as follows: X(12) = 54.36,

X(13) = 53.91,

X(15) = 53.01,

X(16) = 52.26.

X(14) = 53.46,

Note that the actual value of X(12) was 54,48 (this twelfth record was set in Barcelona in 1992). We have already pointed out in the previous lectures that for various reasons the classical record model does not describe adequately the dynamics of sport records.

Hence a number of other record models were proposed (see, e.g., Yang (1975), Ahsanullah (1980, 1992a), Dunsmore (1983), Ballerini and Resnik (1985, 1987a,b), Smith and Miller (1986), Smith (1988), Basak and Bagchi (1990)), where records in various sports were analyzed and predicted. The theory of records has connections not only with statistical applications,

but also with a number of probabilistic problems. For example, some results of this theory can be used in insurance, see Teugels (1984), in the study of random trees, see Devroye (1988). There are relationships between inter-record times and lengths of cycles in random permutations of numbers 11, 2, ... , n} (DeLaurentis and Pittel (1985), Goldie (1989)). There is a curious relationship between records and the process of platoon formation in a single-lane traffic. Indeed, if n vehicles (with desirable speed V1, . . . , VV,) start in a random order on a lane, then after some time some of them will catch up with those going ahead and will follow them.

In this way the vehicles will split up into groups whose leaders' speeds form the sequence of lower record values. The number of groups is random and is distributed as the number of records N(n), while the sizes of successive groups of vehicles, M1, M2, ... , MN(n) have the same joint distribution as the vector 7

(L(2) - 1, L(3) - L(2),..., L(N(n)) - L(N(n) - 1), n - L(n) + 1)

where L(1) < L(2) < ... are the lower record times in the sequence V1, V2,..., V,,. This problem is discussed in more detail in Shorrock (1973) and Haghighi-Taleb and Wright (1973). We complete hereby the exposition of the basic results of the theory of records. In Appendices we give a brief historic and bibliographical review as well as the solutions to the exercises. The reader interested in a deeper study of the mathematical theory of records is referred to the extensive list of references on the subject.

APPENDIX 1

Theory of Records: Historical Review 1. The first paper on records was by Chandler (1952). Almost immediately after its publication, there appeared papers where statistical procedures based on records were proposed; see Stuart (1954, 1956), Foster and Stuart (1954), Foster and Teichroew (1955). Among the publications of the fifties, let us mention the paper by Wilks (1959) on the number of additional observations which are needed to exceed the maximum of n observations. A related result is contained in Gumbel (1961). The decade of the sixties was not rich in publications about records, but two papers, Renyi (1962) and Tata (1969) of this period gave an impetus to an extensive attack on records in the seventies. R6nyi showed that the record indicators are independent, which allowed him to express the distributions of the record times through the distributions of sums of independent random variables and to obtain various limit theorems for record times. Tata obtained a similar representation in the form of a sum of independent terms for exponential record values. We also mention the results for inter-record times obtained during this period (Neuts (1967), Holmes and Strawderman (1969)). Moreover, that decade was the time of active development of the theory of extremal processes, which is closely related to records; see Dwass (1964, 1966), Lamperti (1964), Tiago de Oliveira (1968). The early seventies were marked by the appearance of series of papers by Shorrock (1972a,b, 1973, 1974, 1975) and Resnick (1973a,b,c, 1974, 1975), who significantly advanced the theory of records. They have studied record values for discrete and continuous distributions, described the set of possible limit distributions for records, introduced and explored the record processes, discussed the connections between maxima, record values, and extremal processes. Strawderman and Holmes (1970), Siddiqui and Biondini (1975) obtained a number of new limit theorems for inter-record times and record values. Vervaat (1973) and Dziubdziela and Kopocinsky (1976) introduced weak records and kth record values, respectively, and studied their properties. There were a number of results on the characterization of distributions (first of all, the exponential distribution) by properties of record values; see Ahsanullah (1978, 1979), Nagaraja (1977, 1978), Srivastava (1979). Freudenberg and Szynal (1976), Grudzien (1979a) obtained distributions and moments of record statistics with random indices. In several papers some classical assumptions were dropped: Biondini and Siddiqui (1975) and Guthrie and Holmes (1975) omitted the independence assumption on the underlying random variables, and Yang (1975) did not require them to be identically distributed.

By the end of the seventies there were about 75 publications on records. The classical theory of records was by that time in many respects completed. As a result, 133

1:11

Al i'FNOIX

1

in 1978 there appeared the first reviews of the "record" results: see Click (1978) and Galambos (1978). These reviews gave rise to a new upsurge of interest in records.

The next review by Nevzorov (1987) already contained 166 references. By 1998 (i.e., in 10 years after the reviews by Nevzorov (1987) and Nagaraja (1988)) this list practically doubled again. We cannot give here even a brief account of the main results on records which appeared during last 20 years (there were about 250 papers published for this time). So we will review only the main directions of the studies in the eighties and nineties. During this period there appeared many characterizations of distributions by the properties of records; see Ahsanullah (1981a, 1982, 1987a, 1990b, 1991a), Ahsanullah and Holland (1987), Ahsanullah and Houchens (1989), Ahsanullah and Kirmani (1991), Balakrishnan and Balasubramanian (1995), Ballerini (1987), Dallas (1981, 1989), Deheuvels (1984b), Gupta (1984), Huang and Li (1993), Iwinska (1986, 1987), Kakosyan, Klebanov, and Melamed (1984), Kirmani and Beg (1984), Korwar (1984), Lau and Rao (1982), Lin (1987), Mohan and Nayak (1982), Nagaraja (1988b), Nagaraja, Sen, and Srivastava (1989), Nayak (1981), Nevzorov (1986, 1992, 1993), Nevzorov and Rannen (1992), Pfeifer (1982), Rao and Shanbhag (1986), Roy (1990), Srivastava (1981a,b), Srivastava and Bagchi (1985), Stepanov (1989), Tallie (1981), Too and Lin (1989), Westcott (1981). Witte (1988, 1990, 1993), Yanushkevichius (1993).

There were studies on moment properties and generating functions of record statistics; see Ahsanullah (1986a, 1991b,c,d, 1992a), Balakrishnan and Ahsanullah (1995), Balakrishnan, Ahsanullah, and Chan (1995), Balakrishnan, Chan, and Ahsanullah (1993), Deheuvels and Nevzorov (1993, 1994), Huang (1989), Kamps (1995b), Lin and Huang (1987), Nagaraja (1994), Nevzorov (1985, 1986a, 1987c, 1989, 1990), Nevzorov and Stepanov (1988), Pfeifer (1981, 1984a), Stepanov (1987).

Many papers dealt with statistical problems, such as estimation, hypothesis testing and, prediction of future records; see Ahsanullah (1980, 1986b, 1989, 1990a, 1991a, 1992a), Balakrishnan, Ahsanullah, and Chan (1995), Ballerini and Resnick (1987a), Berred (1991, 1992, 1994, 1995), Dunsmore (1983), Gulati and Padgett (1994), Samaniego and Whitaker (1986, 1988), Smith (1988), Smith and Miller (1986), Tryfos and Blackmore (1985). The studies of limit theorems for record times, inter-record times, and record values were continued; see Balabekyan and Nevzorov (1986), Ballerini and Resnick (1987a,b), Deheuvels (1981, 1982a,b,c, 1983a,b, 1984a,c.d, 1988), Deheuvels and Nevzorov (1994a), Deushel and Zeitouni (1993), Ennadifi (1995), Gajek (1985), Gut (1990a,b), Haiman (1987a,b, 1992), Imlahi (1993), Nayak (1984), Nayak and Wali (1992), Nevzorov (1981, 1984, 1985, 1986c,d, 1988, 1995), Pfeifer (1984a, 1985b, 1986, 1987), Pfeifer and Zhang (1989), Rannen (1991), Yakimiv (1995), Zhang (1988). Multivariate records were studied by Kinoshita and Resnick (1989), Goldie and Resnick (1994).

Various types of processes generated by or related to records (extremal, Fextremal, record processes) were studied; see Ballerini and Resnick (1987b). de Haan (1984), Deheuvels (1981, 1982a,b, 1983a), Engelen, Tommassen. and Vervaat (1988), Goldie and Rogers (1984), Ignatov (1981, 1986). Pfeifer (1986, 1989a), Rogers (1989), Stam (1989). During these years nonclassical record models received considerable attention.

Records in stationary sequences of dependent random variables were considered

I'IIEORY OF flR(ORDS: HISTORICAL. REVIEW

137,

by Adke (1993), Andel (1990), Ballerini (1994), Haiman (1987a,b); see also the monograph by Leadbetter, Lindgren, and Rootzen (1983), Chapter 5. Following Yang (1975) various nonstationary record schemes were developed; see Alpium (1985), Balabekyan and Nevzorov (1986), Ballerini (1994). Ballerini and Resnick (1985, 1987a,b), Borovkov and Pfeifer (1995), Deheuvels and Nevzorov (1993, 1994a), Dziubdziela (1990), Nagaraja (1994), Nevzorov (1984. 1985, 1986b,d, 1990, 1993, 1995), Nevzorov and Rannen (1992), Pfeifer (1982, 1984c, 1989a, 1991). Rannen (1991), Smith and Miller (1986).

2. In the first part of the book we considered order statistics. Since that was auxiliary material for the study of records, we confined ourselves to exposition of the most relevant results of the theory of order statistics. Lectures 2-4, 6. and 7 contain a number of classical results on this subject. The major part of them along with many others can be found in the monograph by David (1970, 1981). Lecture 5 contains material from Nevzorov (1984) and Tikhov (1991). Lectures 8 and 9 are based on the results by Egorov and Nevzorov (1977). Lectures 10-12 include classical results of the asymptotic theory of extremal order statistics; a more detailed exposition of this theory can be found in the monograph by Galambos (1984). The basic parts of the proof of Theorem 10.5 are taken from de Haan (1976). Whereas many topics related to order statistics have been presented in the monographs by David (1970, 1981), Galambos (1978, 1987), Gumbel (1958), Leadbetter, Lindgren, and Rootzen (1983), the theory of records presents more problems in this respect. The reader who, after reading our introduction, will be interested in a deeper study of this theory, will have to look for the material in various publications. In Chapter 6 of Galambos (1978, 1987) a number of results of the classical theory of records is given with proofs. There is only one chapter about records in the book of Arnold, Balakrishnan, and Nagaraja (1992), while the book by the same authors Arnold, Balakrishnan, and Nagaraja (1998) is entirely devoted to

records. A special attention in this book is paid to characterizations of distributions by properties of records, use of records in statistical procedures, and some processes generated by records. The recent book by Ahsanullah (1995) is composed as follows. First, definitions and some basic properties of records are given. In the subsequent chapters the record times and record values for specific distributions are studied. The author presents the distributions and moment characteristics of order statistics, estimates of parameters of the underlying distributions, and statistical procedures for prediction of future records from a sequence of past records. This book is largely based pn author's papers of the eighties and nineties. A more detailed account of "record" results can be found in two reviews, Nevzorov (1987) and Nagaraja (1988). But these papers contain no proofs, and the latter is made up as follows: a brief title of the topic (e.g., Characterizations of the Exponential Distribution) is followed by the relevant references. Recent papers which appeared during the last decade have been reviewed by Nevzorov and Balakrishnan (1998). Here we list the papers which were used when writing the second part of this book. In Lecture 13 we referred to the results of Wilks (1959) and Renyi (1962). Exercise 13.1 uses the results of Gumbel (1961). The distributions of record times, the generating function, and some asymptotic relations for them (Lecture 14) were obtained by Chandler (1952) and R6nyi (1962). The relation (14.6) is due to Williams (1973). Inequalities (14.22) and (14.23) can be found in Nevzorov (1986b). Theorem 14.2 is due to Tata (1969), and the assertions given in Exercises 14.2

113E

APPENDIX I

and 14.3 were taken from Shorrock (1972a) and Galambos and Seneta (1975), respectively. The main result of Lecture 15, namely Representation 15.6, which enhanced the studies of record values and record processes, had been obtained by Tata (1969). Theorem 16.1 proved by Shorrock (1972a) clarified the structure of records for discrete distributions. The martingale properties of records in the discrete case (equalities (16.10) and (16.11)) were studied by Deheuvels and Nevzorov (1993). Theorem 16.8 is due to Stepanov (1992). Relation (17.1) was obtained by R6nyi (1962). Theorem 17.1 and relation (17.7) were proved by Shorrock (1972b). Lecture 18 uses the results by Resnick (1973a). Dziubdziela and Kopocinsky (1976) introduced the kth records. Lecture 19 is based on that paper and those by Nevzorov (1986a,b). Lecture 20 contains the results by Nevzorov (1990). The martingale method proposed by Nevzorov (1987c, 1989) enabled us to obtain the main results of Lecture 21. Theorem 22.3 was proved by Dziubdziela and Kopocinsky (1976). Theorems 22.5 and 22.6 are taken from Deheuvels (1984b). Theorem 22.8 and the assertions in Exercises 22.1 and 22.2 are due to Ahsanullah and Nevzorov (1996). The main result of Lecture 23 was obtained by Nevzorov (1992). Lecture 24 reviews some nonstationary schemes; see Balabekyan and Nevzorov (1986), Ballerini and Resnick (1985, 1987a), Pfeifer (1982, 1984c), Rannen (1991). Theorems 25.1 and 25.2 were proved by Yang (1975). Lemma 25.4

and Theorem 25.5, as well as the assertions in Exercises 25.1 and 25.2 are due to Nevzorov (1984, 1985, 1986b,d). The assertion in Exercise 25.3 is taken from Deheuvels and Nevzorov (1993). Theorems 26.1 and 26.2 were obtained by Ballerina and Resnick (1987b) and Nevzorov (1990), respectively. Lecture 27 contains the results of Nevzorov (1995). Theorems 28.8 and 28.9 were proved by Adke (1993). Lecture 29 uses the ideas and results of Nevzorov (1987d) and Pfeifer (1989a. 1991). The example in Lecture 30 is taken from Ahsanullah (1994).

APPENDIX 2

Hints, Solutions, and Answers PART 1. ORDER STATISTICS EXERCISE 2.1 (hint). It suffices to check that both sides of (2.2) vanish for y = 0 and their derivatives are equal.

EXERCISE 2.2 (solution). The event

A = {Xi,n < x, Xi,n < y} can be represented as a union of the disjoint events

exactly s out of X1, ... , Xn he on the left of y, and exactly r of them are less than x},

Ar,,

where i < r < s and j < s < n. Since F(x),

F(y) - F(x) and 1 - F(y)

are the probabilities for any X to fall into (oo, x),

[x, y),

and [y, oc),

respectively, we obtain that

P{A,,,} =

n!

r! (s - r)!(n - s)!

(F(x))r(F(y) -F(x))' r(1 - F(y))n-s

and then (2.5) becomes obvious. EXERCISE 2.3 (solution and answer). If a random variable X has the geometric distribution with parameter p, then

P{X >m}=p"

M =0,1.....

Therefore n

n

m}=HP{X, >- m}=fipm=pm J=1

J=1

and

P{Y, = m} = P{Yn > m} - P{Yn > m + 1} = ; I - P)Pm, where p =

ra=1 Pi. 137

M = 0,1....,

Arr'ENDIX 2

188

Answer: the random variable min{Xi, X-2,..., X,) has the geometric distribution with parameter 71

P = f Pj. j=1

EXERCISE 2.4 (solution and answer). Let us find the probabilities P{Mn < x, Mn+1 < y}. Since Mn+1 > M,,, we have y} = P{Mn+1 < y} = Ff+l (y),

P{lbin < x,

ify<x. Fory>xwehave P{Mn, < x, Mn+1 < y} = P{Mn < x, X,,+1 < y} = Fn(r)F(y). Answer: P{Mn < x, Mn+l < y} = Fn (min(x, y)) F(y). EXERCISE 2.5 (solution). Using (2.1) we find that

P{-Xn_k+l,n < x} = P{Xn-k+l,n > -x} = 1 - P{Xn_k+1.n !5 -x}

=1-E

(fl)Fm(_X+O)(l_F(X+O))n_m

m=n- k + 1 m k

_

Fm(-x+0)(1

m=o Cn/ m

-

F(-x+0))n

m.

Since F(-x + 0) = 1 - F(x), 1 - F(-x + 0) = F(x), and (m) = (n nm), we obtain that

n-k

E m=U

(n) Fm(-x+0)(1 -

F(-x+0))n

M

_

F(r))n-m

(n)Fm(x)(1 -

m m=k

= P{Xk.n < x},

i.e., Xk,n and -Xn-k+1,n have the same distribution. Taking 2n + 1 instead of n and n + 1 instead of k, we see that d

Xn+1,2n+l = -Xn+1,2n+1,

which means that the sample median Xn+1,2n+i in this case has a symmetric distribution. EXERCISE 3.1 (solution). The uniform distribution U([0,11) is symmetric about the point 2. Hence by reasoning similar to that in Exercise 2.5 we obtain Ukn

k=1,...,n.

1-Un_k+l,ni

Now (3.7) and the probability integral transformation imply that

Wr= ( Ur,n \r d (1-Un-r+l,nlr d 1 - Un_r,n J `Ur+l,n J = exp { - r(Zn-r+1,n - Zn_r)}

d

(exp(-Zn-r+i.n)lr

I\

exp/(-Zn-r.n) J

exp{-vr}

d

1 - Ur

d

Ur,

HINTS, SOLUTIONS. AND ANSWERS

1319

i.e., the random variables W1, W2,. .. , W,, are independent and uniformly distributed on (0, 11.

EXERCISE 3.2 (solution). We know from Remark 3.7 that the joint density , vn.f 1), of the sums S1, ... , Sn, Sn+l in the domain 0 < vi < function, f (vi , < vn+I is given by v2 < , , .

f (vl,... vn+1) = exp(-vn+i ) ,

As is well known, the sum Sn+I has the gamma distribution with density

g(x) =

xn exp(-x) n!

Hence the conditional density of the first n sums given that Sn+1 = 1 equals

f('Ui,Vn,1) =n!

if 0
,9(1)

i.e., it coincides with the joint density of order statistics U1,n, ... , Un,n (see (2.11)). EXERCISE 3.3 (hint). Rewrite (3.17) as Uk,n d (Wk/k W11(i +1)

. .. W,ri

m)(WmTm+1)

...

The rest is obvious. EXERCISE 3.4 (solution). It follows from Representation 3.6 that n

7'

n

d EcmUm,n=E

b,n ym ,

m=1 Sn+1

M=1

where bm = cm + + cn. Denote by b(l) < arranged in ascending order. Let

< b(n) the coefficients b1, b2,. .., bn

d1 = b(i), d2 = b(2) - b(i), ..., do = b(n) - b(n_l). Clearly, d2,

do are nonnegative. It remains to note that bmvm d

E

m=1 Sn+1

E

m=1

S+I

d E dm Um,n M=1

EXERCISE 4.1 (solution). The assertion follows from Theorem 4.1. One must

only observe that the d.f. Fn in that theorem has the form (for simplicity, we consider the standard exponential distribution) F,, (x) =

exp(-u) - exp(-x) = 1 - exp{-(x - u)}, exp(-u)

i.e., it corresponds to the sum u + v, where v is a standard exponential random variable.

APPENDIX 2

140

EXERCISE 4.2 (solution). We have to show that for any r = 2, ... , n and any , x2, - , , ;c which forma monotone sequence of zeroes and ones, the following equality holds xI

Xn,n = xn, .... X,-,n = Zr I X.r_1 n = xr-1.... , X1 = X11

(1)

= P{Xn,n =xn,...,Xr,n = Zr I Xr-1,n =xr_I}. If xr_1 = 1, then both sides of (1) are equal to 1. Otherwise, if xr_1 = 0, the events {Xr- I,n = 0, ... , Xl,n = 0}

(X,-,.,, = 0)

and

coincide and (1) holds. EXERCISE 5.1 (hint). Use Corollary 5.4. Observe that the equality therein can be rewritten as

Z,,,,. - Zn-1,n)

(Z1,n, Z2,n - Z1,n, where

n

_d

(vi, R2,

., Vn

v2

d

(R2,...,Rn) _

flr(2) +...+A !=t

Rn),

(n)'...a

k=I

t

EXERCISE 5.2 (hint). Use the fact that in this setting for any permutation (r(1), r(2), ... , r(n)) of the numbers (1,2,. .. , n) in Corollary 5.5 the coefficient of the random variable vl vanishes, and v1 is the variable which determines the distribution of the order statistic Z1,n. EXERCISE 6.1 (hint). It suffices to use (6.3) and integration by parts.

EXERCISE 6.2 (solution). Since F(x) = 1 - x-' for x > 1, the order statistics Xk,n can be expressed in terms of uniform order statistics as Xk,n

d

(1 -

Uk,n)-p

(Un-k+1.n)-a, where /3 = 1/ry. By comparison with (6.9) we see that E Xka,n = E (Un-k+1 , n)-

n!r(n-k+1-a/3)

n!r(n-k+1--)

r(n-a,0+1)(n-k)! = r(n- °- +1)(n-k)!

and this moment exists provided n - k + 1 - a/ry > 0, i.e., for

a<7(n-k+1). In the same way one can find the product moments. For example, since

EXr,nXs,n = E(Un-r+t,n)-0(Un-s+1)_we obtain from (6.12) that for r > s

,

r(n+1)r(n-r+1-f)r(n-s+1-2/3)

r(n-r+1)r(n-s+1 -a)r(n+1 -2i3) r(n + 1) r(n - r + 1 - 1/-y) r(n - s + 1 - 2/y)

r(n-r+1)r(n-s+1 -1/ry)r(n+1 -2/ry)' this moment being finite provided 1

7> min{n-r+1,(n-s+1)/2}'

III

IIIN'I'S. SOLU'T'IONS. AND ANSWI-:Its

EXERCISE 6.3 (solution). We have the obvious equalities

X2,1 = max{XI, X2} = X1 + max{0, X2 - X1 } and

EX2,2=EX1+Emax{0,X2-XI}=al-rEmax{0,Y}. where Y = X2 - Xl has the normal distribution with mean

EY=a2-a,=aa and variance

a 2 = Var Y = Var X1 + Var X2 - 2 cov (X, , X2) = ai t a2 - 2pa I a2.

Let a > 0. Write Y = as + aV, where V is a random variable with standard normal distribution. Then EX2,2 = al

+aa+aE

max{-a, V}.

It is easily seen that 00

E max{-a, VI = -aP{V < -a} +

J

xV(x) dx = -a(b(-a) -

Jx d (x)

n

= -al,(-a) + w(-a) = -a(1 - 0(a)) + p(a). Hence we finally obtain that

E X2,2 = al + as + alp(a) - aa(1 - t(a)) = al + ap(a) + aa4D(a). If Var Y = 0, then

E X2,2 = a, + E max{O, Y} = al + E max{0, a2 - a, } = a, + max{0, a2 - a, } = max{a,, a2 l. EXERCISE 7.1 (solution). It follows from the equalities

cov(Xr,n-X)X=0 and that n

E COV(Xr,n, Xs.n) = COV(Xr,n, Xl,n + ,

+ Xn.n)

s=1 1.

EXERCISE 7.2 (solution). For the underlying random variables XI, X2, ... , Xn write the sum (min(Xr, (lnax(Xr, Xs))k

X,))m

1
The structure of this sum can be described as follows: we take all pairs of order statistics Xr,n and X,y,n, r < s, raise the smaller of them, Xr.n. to power r and the larger, X,,n, to power m, and then sum up all terms Xl nX ,, 1 < r < s < n. It follows from the equalities 1
(rrllil(X,.,Xs))k(nlax(Xr,Xs))m =

1
Xr,nX n

APPENDIX 2

112

and

E (min (X rr

X')) k (

max(Xr.,

.Y,))rn

= E (min(.YI, X2));. 2))&(max(Y1 X2))... =

E (Xk 1,2X2.2)

that

Xk,.n.,n _

E

()E(x2x2).

1
Thus

= n(n - 1)E (Xi.2Xz ,)

2 L E (Xk,X,,, I
EXERCISE 7.3 (hint). Consider the order statistics

X1 n-1 < ... < Xr,n-1 < ... < Xn-1,n-I of the random variables X1, ... , Xn_ 1 and an additional observation Xn. The variable Xn is less than Xr,n.-1 with probability r/n, in which case Xr.n-1 equals Xr+l,n In the opposite case, which occurs with probability (n - r)/n, the order statistic Xr,n_1 equals X. EXERCISE 9.1 (solution). In the first case the sample mean X has the normal distribution with variance o1 1 = 1/n, whereas the sample median X(12) is asymptotically normal with variance 2

7r

1

- 2n

1,2 = 4n

Hence the asymptotic efficiency of X(1/2) relative to X is equal to 2 2 1,2

0.636....

7r

In the second case X and X(12) are asymptotically normal with variances 2

2

1

2

1

and a2,2 = 4nfi(a) = n respectively, i.e., in this case it is better to use the sample median for estimation of the parameter a of the Laplace distribution, because the asymptotic efficiency of X(112) relative to X is equal to a2,1 = n

2

2,1

= 2.

(72 2,2

Let us mention also that X and X(1/2) are asymptotically efficient estimates for the location parameter a of the normal distribution with density f1 and the Laplace distribution with density f2i respectively. EXERCISE 10.1 (hint). Find the limit of F, (x) = ((Da(xnl/«))

HINTS. SOLUTIONS, AND ANSWERS

I43

EXERCISE 10.2 (hint). Find the limit of

F. (x) = `n a(Pc (/). EXERCISE 11.1 (hint). Use Theorem 11.5 and the fact that for the standard normal distribution we have

x - oc.

1 - F(x) , L(x) and Fl2i(x) - -xF'(x), x

EXERCISE 11.2 (hint). Use Theorem 11.1 and the fact that for the standard Cauchy distribution

1 - F(x) ti

1 and F'(x);rx21 irx

x

00.

EXERCISE 12.1 (solution). Since

min{X1,...,Xn} _

ifXk=-Yk,k=1,2,...,wehave P( min{X1,... , Xn} < xA(n) + B(n)} = P{ max{Yl,... , Yn} > -xA(n) - B(n)} = 1 - P{ max{Y1i... , Yn} < -xA(n) - B(n) }. Therefore, if for some underlying d.f. F(x) = P{Y < x} and centering and normalizing constants an and bn the sequence of d.f.'s

Fn(x) =P{max{Yi,...,Yn}
F(x) = 1 - F(-x + 0) and constants A(n) = an and B(n) = -bn we obtain

P{max{Y1,...,Yn} <xA(n)+B(n)} - 1-H(-x),

n-. oo.

Hence Theorems 11.1, 11.3, and 11.5 can be reformulated accordingly for the min-

ima. In these theorems F(x) is to be replaced by F(x) = 1 - F(-x+0), constants b,, by -bn, and d.f.'s A(x), 4),(x), %P,,(x) by

A(x) = 1 - A(-x),

;D -.(x) = 1

- 1Q(-x),

&,(x) = 1 - W0(-x).

APPENDIX 2

111

EXERCISE 12.2 (hint and answer). One can use the analog of Theorem 11.1 discussed in the previous solution taking

r < 0.

F(x) = 1 - F(-x) = ex, or verify directly that in this case F(x))n

P{nmin{X1,...,Xn} < x} = 1 - (1 -

1(x)

n

= 1 - exp(-x),

n - oo.

Answer: the centering and normalizing constants can be taken as an = 0 and bn = 1/n respectively. EXERCISE 12.3 (hint). The problem actually consists in verifying the following well-known relationship between the Poisson d.f. and the incomplete gammafunction: k-1

e-x = E xm m! (k

1

.n_0

1)!

fo a-ttk-l dt.

0< x< oo.

PART 2. RECORD TIMES AND RECORD VALUES

EXERCISE 13.1 (solution and answers). Applying (2.3) with k = n - m + 1 we obtain

P{N,n(n) > r} = P{ max{Xn+li

.

Xn+r} < Xn-m-1.n}

f

00

n!

(m-1)!(n-m)!

F' (u)F"'-'(u)(1 - F(u))m-1 dF(u) o0

1t! ,fin-m+r(1 - v)ii-1 dv = n!(n - m + r)! (m - 1)!(n - m)! J0 (n - m)!(n + r)! Note that for m = 1 this equality coincides with (13.1). Furthermore, __

1

00

P{N,n(n) > r}

E Nn (n) _ r=0

=

n!

00 0C

(n - 'rn)!(m - 1)! r//L''0

1

,vn-m+r(1

-

v)'-l dv

0

nl

( 1 7'I-m(1 -v)m-2dv (n - m).(m - 1)1 0 n n! (n - m)! (m - 2)!

(n - m)!(rn - 1)!(n - 1)!

m-1

For m = 2 this expression coincides with the right-hand side of (13.4). Answers: (a)

P{N,,,,(n) > r} =

n!(n - m + r)! (n - m)!(n + r)!'

m = 1,2,...,n. r = 0,1.... n=1,2,..., rrt=1,2..... n.

n = 1,2,..., (b)

-1,

ENn(n)=mn

II

IIIN'tS. SOLU'T'IONS. AND ANS\VERS

EXERCISE 13.2 (solution and answers). By symmetry

P{M" = Xk} _

1

it

and P{N(n) > m, Nl.,, = Xk} =

P{N(n) > m} n

Hence

P{N(n) > m I Mn = Xk} =

P{N(P{Mm=

Xk} Xk}

n

= PIN(n) > m}

n+m

i.e., for any k = 1, 2, ... , n the random variable N(n) and the event {M" = Xk } are independent. Furthermore, P{N(n) > rn I Mn = x} = P{ coax{X.n+1, ... , Xn+.n < x}} = Fm(x). Answers:

n = 1,2,..., m = 1,2,..., k = 1,...,n. EXERCISE 14.1 (solution). Let ao

H(z, s) _

00

P(L(n) = k}z"sk.

Qn(s)z" n=1k=n

n=I

Using the binomial expansion 00

(1 - s)m =

where

F(-1)"

(a) -

a(a - 1) ... (a - n + 1)

n

n=0

n!

we get by (14.11) and (14.12) 00

00

H(z, s) = 1: 1: j'P{N(k - 1) = n - 1}znsk n=1k=n

00 zskk-1P k=1

z z

00

k-1

n z"=z

k n=0

skz(z+1)...(z+k-2) k- 1.

k=1 k

z)(-z) I... (-z - k + 2) = z

1

k=1 z

z

c(-s)k C1 k z

1

k=1 z

((1 - s)1

(exp{(1 - z)log(1 - s)} - 1).

z - 1) =

z-1 z -1 It remains to expand the last expression in terms of z" to obtain Qn(s) = E sL(n) = I - (1 - s)

E

(- log(I - s))k 41

k=0

We have already mentioned that k-1-1

M=0

Xm

m!

e_x

1

(k - 1)!

J

°0

e-atk-1 dt,

0 < x < oc

l J

APPENDIX 2

146

(see, e.g., hint to Exercise 12.3). Hence

log(1 - s))k

x .

1

(n - 1)I

kI

k=o

-

I

e_tti-1 dt log( 1-a)

and

Iog(1-g)

1

t"-le-` dt.

Q" (S) = ( n - 1)!

EXERCISE 14.2 (solution). We restrict ourselves to the proof of asymptotic independence of the ratios Tt.n =

L(n + 1) L(n)

and

T2.n =

L(n + 2)

L(n+

1)

Taking into account the Markov property of the sequence of record times and the equality (14.3) and using some elements of the proof of Theorem 14.2 we obtain that for any u > 1 and v > 1 00

PIT,,. > u, T2,n > v} = E P{Tl,n > u, T2,n > v I L(n) = m}P{L(n) = m} m=n 00

_ E E P{L(n + 1) = r, L(n + 2) > yr I L(n) = m}P{L(n) = m} rn=nr>um 00

_ E E P{L(n + 2) > yr I L(n + 1) = r, L(n) = m} m=n r>um

x P{L(n + 1) = r I L(n) = m}P{L(n) = m} 00

= 1: 1: P{L(n + 2) > yr I L(n + 1) = r} m=n r>um 00

x P{L(n+ 1) = r I L(n) = m}P{L(n) = m}

E

(vr)P{L(n + 1) = r I L(n) = m}P{L(n) = m}

rn=n r>um

E00 P{L(n+ 1) > um I L(n) = m}P{L(n) = m} _ (v +O(n)) m=n = m} (v +0(n1)) n (urn] P {L (n )

-(1+0(1))(1+0(1)) EP{L(n)=m}= 1 +O(1 v

n

u

n

uv

m=n

so that

limP{Tl,n > u,T2,n > v} = 1 , uV

n - oo,

which means the asymptotic independence of the ratios T1,n and T2.n.

n

HINTS, SOLUTIONS, AND ANSWERS

147

EXERCISE 14.3 (solution). To avoid cumbersome calculations, we will prove,

as before, independence of two random variables T(n) and T(n + 1). For any integers u > 1 and v > 1 we have L(n)1)

P{T(n) > u, T(n + 1) > v} = P j L(

> u, L(L(n)1) > V

By the above arguments and using additionally the fact that u and v are integers we obtain L(n)

P L(n

L(n + 1)

L(n)

> u,

> v

- 1) 00

=

E

P{L(n + 2) > yr I L(n + 1) = r}

m=n r>um

x P{L(n + 1) = r I L(n) = m}P{L(n) = m} 00

VV

vrP{L(n + 1) = r I L(n) = m}P{L(n) = m}

m=rt r>um 00 1

V

E P{L(n + 1) > um I L(n) = m}P{L(n) = m} m=n 00

-

= 1 E M P{L(n) = m} = V

m=n um

1uv E P{L(n) = m} = uv 1

m=n

Thus we get the equality

P{T(n) > u,T(n + 1) > v} = v = P{T(n) > u}P{T(n + 1) > v}, which implies independence of T(n) and T(n + 1) as well as the fact that

P{T(n) = r} = P{T(n) > r-1}-P{T(n) > r} =

1

r11 r

r(r 1 1)'

r > 2.

EXERCISE 15.1 (hint). Use (15.7) and Remark 4.3.

EXERCISE 16.1 (hint). Express the probabilities

P{X(1) = il, X(2) = i2i ...,X(n) = in } in terms of the distributions of the indicators ?7k as in Theorem 16.1 and use the independence of these indicators. EXERCISE 16.2 (hint). Use (16.8) to find the probabilities

P{X(n+1)=jn+1 IX(n)=in,...,X(1)=J1} P{X(n+ 1) = j,,+1, X(n) = jn,... X(1) = j1}

P{X(n)=jn,. ,X(1)=j1} and compare the expression thus obtained with (16.6).

APPENDIX 2

118

EXERCISE 16.3 (solution and answer). It follows from Theorem 16.8 and Representation 16.9 that

P{X,,,(n) > 'm} = P{/10 + µ, +

+ Lm < n},

where po, pt, ... are i.i.d. random variables, since in our case

P{µ,,=m}=(1-Pn)Pn for any n, where

P

P{X = n}

-

=(1-P),

P{X >n}

n=0,1,.

.

This implies that the sum µ, + AI + + µ,,, has the negative binomial distribution with parameters (1 - p) and (m + 1). Consequently,

P{X.(n) < nil = P{µ + A, +

+ µ,n > n}

(r+m)(l_p)rpm+l r>n

r

and, in particular, P{X.(n)=0}=E(1-p)rp=(1-p)n.

r>n

Answer: For any n = 1, 2.... and m = 1, 2, .. .

P{X.(n) < m} = pn` 1 (r+rn_ 11(1 - p)r. r>n J EXERCISE 18.1 (answer). With each d.f. F associate the d.f.

F(x) = 1 - exp {- V1-_Iog(1 -- F(x))} I.

Let N(n) = [exp(n)'12]. The following result is due to Resnick (1973a): F belongs to the domain of attraction of the limiting distribution H for maxima if and only if F belongs to the domain of attraction of the limiting distribution

G(x) = fi( - log(- logH(x))) for record values. The centering and normalizing constants for maxima (a(n) and b(n)) and for records (A(n) and B(n)) are related by

A(n) = a(N(n)),

B(n) = b(N(n)).

EXERCISE 18.2 (hint). Combining the result of Exercise 18.1 and Examples 10.3 and 10.4 with d.f.'s

PI(x)=1-x--', x>1, and F2(x)=1-(1-x)', 0<x<1, check that the required d.f.'s F1(x) and F2(x) are determined by the equalities

Fk(x)=1-exp{-

log (1-Fk(x))},

k=1,2.

HINT'S, SOI ('1'1ONS. .AND ANSWERS

11',

EXERCISE 21.1 (solution). We have to calculate the difference L(nlik)L(n3,k) cot = E

L(n3,k)

L(n:,k)

E

L(n2,k)L(n4,k)

L(n2.k)

E

L(n4.k).

Using the martingale property of the sequence T-(1) =

(k - 1)nr(L(n, k) + 1) k r(L(n, k))

n > 1,

(see Corollary 20.4) we obtain that

E L(nl k)

(k -

L(n2, k)

_

E Tn,(1)

kn1-na 1)nl-n-a

T.-I(1)

kn1-n2

(k - 1)n1-n,, kn1-na

/

1

E

(

Tns 1

)

E(Tn,(1)

Tn2(1)))

knt-ns

Tn2(1)

(k - 1)n1-n2 '

(k - 1)n1-n" E F42-(I) kn3-n4

E L(n3, k) = L(n4i k) (k - 1)na-na

Furthermore,

E L(n1, k)L(ns, k) L(n2, k)L(n4, k)

k

) n1 +n3-^2-n4

E Tn' (1)Tn3(1)

(k-1

Tn2 (1)Tn.(1)

and Tn1(1)Tna(1)

E Tna(1)Tna(1)

E CTn1

(Tn1(1)

E \Tn3(1) \E (Tns(1)E

I Tnz(1)) I Tn3(1)) I Tna(1))))

= E \Tna(1)E (Tn.(1) 1 Tn3(1)) I Tn1 (1)) = E lTna(1)E (Tna(1) 1 Tn,(1))) = E Tna(1) = 1,

which implies that n,+n3-n9-n4 k cot=\k1)

(

k

k1)

nt+n3-n2-n4

=0.

EXERCISE 21.2 (hint). It suffices to check that

E (f (L(n, k)) I L(n - 1, k)) = f (L(n - 1, k)) +

1

APPENDIX 2

150

EXERCISE 23.1 (hint). The relation (23.6) can be written as

I(r) =

fI

J0

G2(u)(1 - u)k`1 (- log(1 -u))T-I du < oc.

where G(u) is the inverse function to the d.f. F. For u close to 0 and 1 the integrand is of order G2(u)'u'-1

and

G2(u)(1

- u)k-1( - log(1 -

u))'-1,

respectively. To construct the required example, observe that u'-1 decreases with r

for 0 < u < 1, while ( - log(1 - u))T-1 increases with r for u > I - e-1. EXERCISE 24.1 (solution). It follows from (24.1) and the Markov property of the vectors {0(n), X (n)) that

P{ 0(n) = m,X(n) - X(n - 1) > x 0(n - 1) = X(n - 1) = xn-1, .. , 0(1) = 1,X(1) = x1)} = P{1(n) = m, X (n) - X (n - 1) > x A(n - 1) = mn-1i X (n - 1) = xn-1 } _ (1 - Fn(x + Xn-1)) (F'n(Xn-1))m_1,

x > 0,

and

P{X(n)-X(n-1)>xIA(n-1)='+'nn-1, X(n - 1) = xn-1i...,A(1) = 1,X(1) = x1)} = P{X(n) - X(n - 1) > x 10(n - 1) = inn-1, X(n - 1) = xn-1 }

1-Fn(x+xn-1) 1 - Fn(xn-1)

x>0.

Substituting

Fn(x)=1-exp{-

n},

x>0,

we obtain that the right-hand side of the last equality does not depend on xt, ... ,

xn-l,mn-1,...,ml, and Tn = X(n) - X(n - 1) is independent of the vector {X(1), X (2), ... , X (n - 1) } for any n = 2, 3, ... This implies that the random variables

T(1) = X(1), T(2) = X(2) - X(1), ... are independent. In our case 1 - Fn(x + 1 - Fn(xn-1)

Fn

therefore,

P{Tn < x} = Fn(x). EXERCISE 25.1 (hint). Express the event {L(1) = 1, L(2) = m(2),...,L(n) = m(n)}

in terms of the record indicators t;1i1;2, ... , m(n) and use Lemma 25.4. EXERCISE 25.2 (hint). Use the assertion obtained in the previous exercise.

HIN'T'S. SOLUTIONS, AND ANSWEHS

151

EXERCISE 25.3 (solution). Using the Markov property of the sequence L(1),

L(2).... we obtain

E{V(n) I L(1),L(2),...,L(n- 1) =i} = E {A(L(n)) I L(1), L(2), ... , L(n - 1) = i} - n

= E {A(L(n)) L(n - 1) = i} - n 00

_ E A(j)P{L(n) = j I L(n - 1) = i} - n i=i+l 00 1

1

E A(k)S(i) S(k - 1) k=i+1 \ A(k + 1)

= S(i)

S(k)) - n

- c. A(k) L.

n

S(k)

k=i+1

k =i 00

= A(i + 1) + S(i) = A(i + 1) + S(i)

=A(i+1)+

pI`+1

-n

1

-

k=i+l S(k)

E

( S(k)

1 S(k + 1)k=i+1

) -n

S(i)

S(i+1) -n =A(i+1)-pi+1-(n-1)=A(i)-(n- 1), where Pk

A(k) - A(k - 1). a(1) + (k + a(k) =

Then

E {A(L(n))-n I L(1), L(2),..., L(n-1)} = A(L(n-1))-(n-1),

n = 2,3,...,

hence

E (V(n) I Yn_I) = V(n - 1). The second assertion follows from the equalities

EA(n)-n=EV(n)=E(E(V(n) I=EV(n-1)=... =EV(1)=EA(1)-1=1-1=0. EXERCISE 25.4 (solution). For simplicity, consider only the case k = 2. Let

S(x), x > 0, be a monotone and continuous continuation of S from the set of

APPENDIX 2

positive integers. We see that r

P{

S(L(n)) S(L(n+1)) S(L(n + 1)) < x' S(L(n + 2)) < y

x

E

= E P{L(n) = m}

01 ...

r=ro+l

nn-n

1} ...

X 00

00

m=n

S (r) - S(r

S (m)

P{L(n) = m} 1 =ru+1

1}

-

0}

S (r)

1)

S([h(S(r)/y)])'

S(r)

S(r - 1)

where h(x) is the inverse function to S and rp = [h(S(m)/x)]. Obviously, under the conditions of the exercise we have

S(h(S(r)/y))

S(r)

S(h(S(r)/y)]) =yS([h(S(r)/y)])

y

n -+ oo. Hence, as n -* oo we obtain S(L(n))

S(L(n + 1))

P S(L(n + 1)) < x' S(L(n + 2)) < 00

°O

y > P{L(n) = m}S(m)

r=ro+1

m=n

1

1

(S(r - 1)

S(r)

00

= y E P{L(n) = m} m=n

S([hS(m)lx (Sm)])

00

yx E P{L(n) = m} = yx. m=n

Thus we have, as n -+ 00,

P{

S(L(n))

<

ll S(L(n + 1))

and

x,

P

S(L(n + 1)) < y} S(L(n + 2))

S(L(n)) <x S(L(n + 1))

0 < x < 1,

xy

--r x,

0 < y < 1,

0 < x < 1.

EXERCISE 28.1 (answer). One of the simplest examples is given by a random sample (XI, X2, X3) without replacement from the set {1, 2, 3}. In this case the random vector (X1, X2, X3) is uniformly distributed on the set of 6 permutations of the numbers 1, 2, 3. EXERCISE 29.1 (hint). Use the relation nn-`1 a(k+ 1)

n-1 N

(1

10

g

\

a(k+ 1)l A(k)

/

n-I

to (A(k±1)) g

to

)

g A(m))

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Hilbert C*-Modules (Translations of Mathematical Monographs)

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Semigroups (Translations of Mathematical Monographs 3)

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Semigroups (Translations of Mathematical Monographs 3)

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Embeddings and Immersions (Translations of Mathematical Monographs)

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Theory of Limit Cycles (Translations of Mathematical Monographs)

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Theory of Limit Cycles (Translations of Mathematical Monographs)

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Theory of Limit Cycles (Translations of Mathematical Monographs)

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Function Theory in Several Complex Variables (Translations of Mathematical Monographs)

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Distribution of Values of Holomorphic Mappings (Translations of Mathematical Monographs)

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Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)

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Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)

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Distribution of Values of Holomorphic Mappings (Translations of Mathematical Monographs)

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Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201)

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Methods of Information Geometry (Translations of Mathematical Monographs)

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Calculus of Variations and Harmonic Maps (Translations of Mathematical Monographs)

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Methods of Information Geometry (Translations of Mathematical Monographs 191)

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Geometry of Characteristic Classes (Translations of Mathematical Monographs)

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Introduction to Complex Analysis (Translations of Mathematical Monographs)

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Linear Differential Equations in Banach Space (Translations of Mathematical Monographs)

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Linear Differential Equations in Banach Space (Translations of Mathematical Monographs)

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Variational Problems in Geometry (Translations of Mathematical Monographs)

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An Introduction to Algebraic Geometry (Translations of Mathematical Monographs)

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One-Dimensional Stable Distributions (Translations of Mathematical Monographs - Vol 65)

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Algebraic Groups and Their Birational Invariants (Translations of Mathematical Monographs)

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Some Applications of Functional Analysis in Mathematical Physics (Translations of Mathematical Monographs)

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Applications of Functional Analysis in Mathematical Physics (Translations of Mathematical Monographs, Vol 7)

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Applications of Functional Analysis in Mathematical Physics (Translations of Mathematical Monographs, Vol 7)

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Local Class Field Theory (Oxford Mathematical Monographs)

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