Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1042 Allan Gut Klaus D. Schmidt
Amarts and Set Function Processes
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Authors Allan Gut Department of Mathematics, University of Uppsala Thunbergsv~.gen 3, 75238 Uppsala, Sweden Klaus D. Schmidt Seminar f(~r Statistik, Universit~t Mannheim, A 5 6800 Mannheim, Federal Republic of Germany
AMS Subject Classifications (1980): 6 0 G 4 8 ; 6 0 G 4 0 , 6 0 G 4 2 ISBN 3-540-12867-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12867-0 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright.All rights are reserved,whetherthe whole or partof the material is concerned, specificallythose of translation,reprinting, re-useof illustrations,broadcasting, reproduction by photocopying machineor similar means,and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payableto "VerwertungsgesellschaftWort~, Munich. © by Springer-VerlagBerlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214613140-543210
Ama
r t s
set
F u n c t i o n
Allan An
and
Gut:
introduction
asymptotic
Klaus Amarts
Allan Amarts
P r o c e s s e s
D.
to
theor~
of
....................
Schmidt:
- a measure
Gut
the
martingales
and
Klaus
theoretic
D.
- a bibliography
approach
51
Schmidt: ...................
237
AN INTRODUCTION TO THE THEORY OF
ASYHPTOTICHARTINC~ES
By Allan Gut
Contents
page
Preface
4
Introduction
5
I. History
9
2. Basic properties
14
3. Convergence
23
4. Some examples
31
5. Stability
35
6. The Riesz decomposition
40
7. Two further generalizations of martingales
44
References
46
Preface The material of these notes is based on a series of lectures on real-valued asymptotic martingales
(amarts) held at the Department of
Mathematics at Uppsala University in spring 1979. The purpose of the lectures (and now also of these notes) was (is) to introduce an audience~ familiar to martingale theory~ to the theory of asymptotic martingales. A most important starting point for the development of amart theory was made by Austin, Edgar and Ionescu Tulcea (1974), who presented a beautiful device for proving convergence results. In Edgar, and Sucheston (1976a) the first more systematic treatment of asymptotic martingales was made. Since then several articles on asymptotic martingales have appeared in various journals, l~i~ book therefore ends with a list of references containing all papers related to the theory of asymptotic martingales that wehave been able to trace, whether cited in the text or not.
Introduction We begin by defining asymptotic martingales (amarts) and by briefly investigating how they are related to martingales, submartingales, quasimartingales and other generalizations of martingales. This is then followed by a section on the history of asymptotic martingales after which the more detailed presentation of the theory begins. In this introductory part we consider, for simplicity, only the so called ascending case. Let ~= {Tnl 1
(~,9",P) be a probability space and let
be an increasing sequence of sub-c-algebras of ~ .
(The descend-
ing case corresponds to the index set being the negative integers.) Further, let
T
T C T
be the set of bounded stopping times (relative to if and only if
integer
M
T(~) ~ ( ~ ) A net
(depending on
(aT)TE T
for all
n
and
P ( T ~ M ) =I
T) . The convention that
for almost all
only if for every all
{T=n} E T n
~ E ~,
~-}00
T ~ C
if and only if
defines a partial ordering on
of real numbers is said to converge to
g > 0
) , i.e. i for some
there exists
TO E T
such that
a
T .
if and
IaT-a I < E
for
T E T , T ~ TO . For further details about net convergence, see Neveu
(1975), page 96 (and Remark 2.4 below).
Definition. Let adapted to
{Xn}n= I
be an integrable sequence of random variables,
{~n}~=l . We call
and only if the net
(EXT)T ET
{Xn, ~'n}n=l ~
an asymptotic martingale if
converges.
The very first question is of course: Is a martingale an asymptotic martingale? The answer is yes, since, if Doob's optional sampling theorem, net
(EXT)TC T
~ {Xn, Tn}n=l E X T = EX I
is a martingale, then, by for all
T E T,
i.e. the
is constant and hence, in particular, convergent.
However, more is true. Suppose that
{ X n }I~=n
is adapted to
~ {T n}n=l
and suppose that pick
A E ~m
a.s. and
EX 7 = constant for all
7 E T . Let
m < n
be arbitrary,
arbitrarily and define the bounded stopping times
72
=
T2(~0) --
if
By assumption,
~0 ¢ A
EXTI = E X n = ~ X n d P +
71 = n
~ X ndP Ac
EXz2 = ~ XmdP + ~ X n d P . Ac Since
EXTI = EXT2 , subtraction yields
S XndP A
= S XmdP A
for
A C~m,
which is the defining relation for a martingale, i.e.
{Xn,~n}n= I
is a
martingale. The term asymptotic martingale thus enters in a natural way: Martingales, T C T
{Xn,~rn}~= I , are characterized by
and asymptotic martingales,
(EXT)7 C T
EX T = oonstc~zt
for all
{Xn,~n}~= I , are characterized by
being oonvePgent (i.e. "asymptotically constant").
Next, let
1
he a submarti
ale
ust as above one notices
that the classical definition is equivalent to: If EX 7 ~ E X .
It follows that an
7, o C T , 7 ~ ~,
then
Ll-bounded submartingale is an asymptotic
martingale. Similarly for supermartingales. A quasimartingale
(F-process) is defined as an adapted sequence
=o E EIXn- E ~ n ,,X+iI < =o, see Fisk (1965), 0rey n=l (1967) and Rao (1969). Every martingale is thus trivially a quasimartingale.
{X~'~}-~-I'L.-n~
such that
The following computations (see Edgar and Sucheston (1976a), page 200) show that every quasimartlngale is an asymptotic martingale. Choose
E > 0
and
E
.IXn-
n=n 0 and let
T C T,
T > n 0.
no
such that
Xn+ll < Since
T
is bounded there exists
n I,
such that
P ( n o ~ e ~ n I)
=
Now,
1.
nI IEXe-EXnl I= k--noY E(X~-..Xnl)l{e=k} I nl-i nl-I =I
E E E(Xn-Xn+l)l{e=k}l = k=nO n=k nl-i n
=
E E E(Xn - Xn+l) I{T n=n 0 k=n0
=k}
=
I nl-I n ETn 1 Y Z E(XnXn+l)l{T=k} <__ n--nO k--nO
=
n.-I
n
7.
z r EIx n -E ~ n=n 0 k=n o
<
<_ E EIXn-E n=n0 Next, let
< +1 If{e--k} --
nXn+ll <
• l, • 2 E T,
~.
• 1 ~ n o , 72 ~ n o
and choose
n 2 ~maX{el,T 2} .
Then IEXT I -E XT21 e lEXel - mXn2 ] + lEx 2 -EXe21 _< 2E, i.e. the net (EXT)e E T
is Cauchy and hence convergent (cf. Neveu (1975), page 96, see
also Section 2 below), i.e. {Xn,~n}~= I
is an asymptotic martingale.
The following example shows that there exist asymptotic martingales, which are not quasimartingales. Example. Let
I an ffi (_l)n "~'
n = 1,2, .... Set
{Tn }~=I ' Tn E T
arbitrarily, such that
we have
a.s.
XT ~ 0
as
n ~ ~
Xn = an
Tn ~ ~
and since
.
Since
lanl ~ i
a.s., choose an
~
0
as
n
~
it follows that
n
Ex e
~ o
as
n ~ ®.
ie
~X.~=
1
is an asymptotic m a r t i ~ a l e . , o ~
n
2 ever, Zlan-an÷ll ~ z ~ = ÷®, i.e. ~ x , ~ n =
l
is not a quasi~rtingale.
The term asymptotic martingale was motivated above by looking at the behaviour of the net
~XT)T 6 T
(which was constant for martingales and
convergent for asymptotic martingales). Another motivation for the name is
the Riesz decomposition
(see Section 6), according to which an asymptotic
martingale can be represented as a sum of a martingale and a "potential". There are of course other ways of generalizing martingales.
Let us,
instead of regarding the values of the stopped process, consider the differences
An,m = Xn -
An, m = 0
a.s.
martingale
E~ n
for all
Xn+ m , n
(see above). If
n ~ i, m ~ I . If
and
m.
A
~ 0
n,m
If
{Xn,~n}n= I
_~ IAn, l[ < ~
a.s.
as
n,m ~ ~
is a martingale
we obtain a quasiwe obtain a
martingale in the l@mit, see Mucci (1973, 1976), Blake (1978), Edgar and Sucheston (1977a) and Bellow and Dvoretzky (1980). If lity as
n,m ~ ~
we obtain a game fairer~th
A ~ O m,n
in probabi-
time, see Blake (1970),
Mucci (1973), Subramanian (1973), Edgar and Sucheston (1976e). The two latter concepts will be mentioned in the last section, but only in brevity because it turns out that there are several "useful properties" which are not satisfied for martingales in the limit and games fairer with time, see Edgar and Sucheston (1977a) and Bellow and Dvoretzky (1980). One has the inclusions m a r t i n g a l e =
quasimartingale = asymptotic martingale =martingale
in the limit = game fairer with time, but it seems as if the two last objects generate classes which are too wide to be really useful. At this point we also mention that many classical proofs for martingales are built up around the use of stopping times, and that asymptotic martingales are, roughly speaking, those objects for which these proofs remain valid. This fact may also serve as an explanation for the defining relation for asymptotic martingales - in terms of stopping times.
i. Histor 7 The first amart-like results can be found in Meyer (1971), Mertens (1972) and Rao (1970), where, essentially, a continuous time version of the following result is proved. Theorem i.i. A uniformly bounded sequence converges almost surely if and only if it is an asymptotic martingale. Remark 1.2. The names asymptotic martingale (amart) were not used and the class of stopping times was
~,
the f~n4te stopping times.
The theorem will be demonstrated in a more general form in Section 3, (Theorem 3.2). Motivated by gambling theoretic considerations Sudderth (1971) proved Theorem 1.3. Let
{Xn}~= I
be an adapted sequence and let
~
be the set
of finite stopping times. T h e n (i.I)
E lira sup X n < lira s u p E X T n -~== T6 T
(1.2)
E lim inf Xn --> lim inf EX T , n~= T£
provided the occurring expectations are well-defined. Remark 1.4.
lim sup E X T
lim i_nf E X T T6 T
correspondingly.
Remark 1.5. If
P(Xn~O)
is to be interpreted as
inf ( sup __EXT) and
= I , the second inequality together with the
ordinary Fatou lemma yields (1.3)
> lim inf E X T . lim inf E X n _> E lim inf X n -n~ n~ T6
Proof. We follow the proof given in Sudderth (1971). It is clearly enough to prove (i.i). Put
X ~ -- lira sup X n n-~¢o
and
~n = a{Xl ..... X n} , n =1,2 ....
10
If
EX* = - ~
there is nothing to prove.
Now, suppose that
EIK*I < -
large values, i.e. to construct that, of course, stopping at Choose
e > 0
= inf {n; n ~ U ( ~ ) theory we know that T E T,
T > ~
EX* =
=
The idea is to stop T £ ~
sup X n n
such that
at
gets large. Note
does not yield a stopping time.
and
o 6 T.
Define
and
E ~n X* < X n + e}
E yn X* ~ X*
XT
1
a.s.
T
by letting
~ £ 2.
for each (and in
T(~) =
L I)
as
From martingale n ~.
Thus
and it follows that
E EX*I{T=n} n=l
=
~ E(I{T=n} -/n n=l
Z EE
{T=n})
=
n=l
X*) <
Z E(I{T=n}(Xn+g)) n--i
=EXT+E.
In view of Remark 1.4 this proves (i.I). Finally, if E X * = == we consider the random variables
X ^c,
where
c
is a real number. Then, by (I.I),
n
lira sup E X T >__ lira sup E(X T AC) ~ E l l m sup(XnAC) = E ( X * A c ) ° TCT TCT n~ The conclusion now follows by letting
c ~
.
By modifying the proof of Fatou~s lemma, Sudderth also proves the following result. Theorem 1.6. Suppose further that there exists an integrable r.v.
X < Z
such that
n--
(1.4)
for all
n.
Then
E lim sup X n = lim sup E X T . n~ TC T
Similarly, if there exists an integrable r.v. for all (1.5)
Z ,
n,
W,
such that
W~X
n
then E lira inf X n = llm inf E X T . n-~== TE T
Remark 1.7. Note that Theorem 1.3 is a Fatou type l e m m a w i t h inequalities in the "wrong" direction. To prove Theorem 1.6 one therefore needs the
#!
inequality the usual way. For details see Sudderth (1971), Theorem 2. Suppose that for all
n.
If
{Xn}~= I X
~ X
is adapted and that
a.s.
as
n ~ ~
IXnl ~ Z,
integrable,
dominated convergence yields
n
EX n ~ E X
and thus, by Theorem 1.6, the net
sely, if the net converges, then
~XT)TC ~
0 ~ E(lim sup X n - lid inf Xn) = 0 , i.e. n ~
X
converges. Conver-
n ~
converges almost surely. We thus obtain the following corollary. n
Corollary 1.8. Let
{X n}n=l
for all
Z
n,
where
(1.6)
Xn
(1.7)
The net
be an adapted sequence such that
is an integrable r.v.. The following are equivalent:
converges almost surely as (EXT)T E T
The implication Remark 1.9. If
Z m c
IXnl ~ Z
n ~ m.
converges.
(1.6) ~ (1.7)
is the corollary of Sudderth (1971).
we rediscover Theorem i.I (with
T) .
Remark i.i0. Corollary 1.8 has been slighly generalized by Chen (1976b). Under the assumptions of the corollary it is, for example, shown that lim sup EX = lim sup EX (and similarly for llm inf) , i.e. under those TCT T TET T assumptions (1.6) and (1.7) are also equivalent to (1.8) i.e.
The net {X n}n=l
(EXT)T C T
converges,
is an asymptotic martingale. We shall return to the equiva-
lence of (1.6) and (1.8) in Section 3 (Theorem 3.2). It is clear that the statement implies that
"~X }~ n n=l
"{X }
is uniformly integrable"
is uniformly integrable". An example showing that
the converse does not necessarily hold is given at the end of Sudderth (1971). We shall return to this matter towards the end of Section 2 (for T). The next step in the evolution came with the papers Chacon (1974), Baxter (1974) and Austin, Edgar and lonescu Tulcea (1974). The results of Chacon and Baxter are stated for historic reasons, but
12
without proofs. Theorem 1.11 (Chacon). Let
{Xn}~= 1
be an
Ll-hounded,
adapted sequence
of random variables. Then (1.9)
E (lim sup X n - l i m
Furthermore,
if
{XT}T £ T
is
inf Xn) ~ lim sup E ( X - X T) . Ll-bounded,
then
lim sup X n
and
n ~
lim inf X
are integrable.
n
n ~
Remark 1.12. Note that if the
RHS
of
(1.9)
is
{Xn}~= 1
is an
Ll-bounded asymptotic martingale
0 , which shows that an
Ll-bounded amart is almost
surely convergent. Theorem 1.13 (Baxter). Let metric space, Xn: ~ * M,
~(M)
(~,~r~)
be a probability space,
the space of continuous functions on
n = 1,2,...,
(I.I0)
Xn
(I.ii)
( S~(XT)dB)T £ T
M
M
a compact
and
random variables. The following are equivalent:
converges pointwise
a.e.
on
converges for all
~. ~ E~(M) .
It is moreover shown that it suffices to check the net convergence in (i.ii) for a collection of functions which generate a separating class in
~(M)
(for details see Baxter (1974), page 396).
One reason for mentioning the last result is that it is an example of a more abstract result on asymptotic martingales. Apart from their own interest results of this kind seem to have applications for example in Banach space theory, see e.g. Bellow (1976a, 1978),
Brunel and Sucheston
(1976 , 1977), Edgar and Sucheston (1977b). The article that finally stimulated the last few years work on the theory of asymptotic martingales was the above mentioned paper of Austin, Edgar and Ionescu Tulcea (1974), in particular their leones 1 and 2. The former provides us with an important and elegant tool for proving convergence results. The idea is that the cluster points of a sequence of random
13 variables can be arbitrarily well approximated by stopping the sequence suitably. The precise formulation and a proof will be presented in Section 3 (Lemms 3.1). The term as~ptotic martingale first appeared in Chacon and Sucheston (1975). Since it turns out that there exist several varieties of asymptotic martingales the t e r m ~ n ~ t
is introduced in Edgar and Sucheston (1976 a, b,
c, d, e). An important variety of amarts, semiamarts, (an adapted sequence, {Xn}~= I , such that the net
(EXT)T C T
is bounded), was introduced in Edgar
and Sucheston (1976 c) and will also be discussed in the sequel. A related object called
6-amart is considered in Edgar and Sucheston (1976 a) . For
the theory of pramarts, subpramarts, orderamarts, uniform amarts, weak amarts, strong amarts etc. the interested reader may consult the papers mentioned in the third part of this volume.
2. Basic properties We follow the notation of Edgar and Sucheston (1976 a) . Let be a probability space, put denote either
N
or
of sub-o-algebras of and (TN, if
~r= -
T(m) < o(~)
(T_N)
{~rn}nED
~r
if
i.e.
~nC~m
for almost all
since
D
be an increasing family
n ~m,
TD). The convention that, for any
particular, TN
-N . Further, let
and let
= O{ U ~'n} n£ N The set of bounded stopping t~nes will be denoted by T
N ~n" nE-N
T_N,
N = {1,2,...} , -N = {..., -2,-1}
(~,~',P)
T v O C T
~
defines
and
~
z,o q T, T ~ O a
T ^ o C T
and set
if and only
partial ordering on if
T,O C T,
T . In
it follows that
is filtering to the right (left).
Definition 2.1. The net
of real numbers converges to
(aT)T C T
a
if and
only if V~ >0
3 T O E TD
T ~ TO)
If no means that
we have
such that
with
VT C TN
T ~ TO
(VT C T_N
Iaz - a I < c.
O-algebras are mentloned, the statement that Xn
is
~r-measurablen
for all
n C D
with
{Xn}nE D
is adapted
~n = °{Xk; k _< n}.
For a finite stopping time, T, we define the random variable through the relation
with
XT
(XT)(w) = XT(~)(~ ) .
Definition 2.2. Let
{Xn}n C D
be an integrable family of random variables,
which is adapted to
{~rn}nCD " We call
{Xn'~rn}nED
(i)
a martingale if and only if
EX T = constant for all
(ii)
an asymptotic martingale (umuPt) if and only if the net
Z q T ~XT)T E T
is convergent (iJ/) a sem~umc~t if and only if the net
(EXT)T C T
Remark 2.3. It was shown in the introduction
is bounded.
(for the case
D =N)
that the
15
definition of a martingale given above is equivalent to the usual definition. The reason for using thing in terms of the net
(i)
here is that it allows us to define every-
(EXT)T £ T "
Remark 2.4. It follows immediately from Neveu (1975), page 96, that
{xn'Tn}n c D (ii')
is an amart
if E X T
converges as Inl~ ~
for every
sequence of
n
bounded stopping times
{Tn}n£D,
such that
ITnl ~ ~
~XT)TCT
is Cauchy, i.e. if
as Ini~ ~ ,
and also (ii")
if and only if the net such that for all
•,o~
o)
T,O E T N
we have
with
IEX~-EX~I <
T,O ~ T O
VE > 0 3 T 0 6 T D
(T,O £ T_N
with
~.
We immediately observe that every martingale is an amart (cf. Remark 2.3) and note that it seems reasonable to guess that every amart is a semlamart.
Theorem 2.5. Every martingale is an amart and every amart is a semiamart.
Proof. The first statement was proved in the introduction (for
D =N). For
the second statement we follow Edgar and Sucheston (1976 a) , L~mraa 1.2. Only the case
D = N
is considered, the case
Because of the convergence of that
IEXno
T v no~n
0
EXTI < I
for all
(EXT)T C T
D = -N
being the same.
we can choose
T > n o . Let
T C TN
Xn O
it follows that
IEXTI<_I xT^nol ÷I<_E max
IXkl ÷I<--,
1 <_k<.n 0 i.e.
sup ~XTI < co, which proves the assertion. T
such
be arbitrary. Since
and
X T = XT ^ n O + X ~ v n o
no 6 N
16
It follows immediately from the definition that linear combinations of amarts (semi~marts) are amarts (semiamarts). The following results establish (essentially) that the positive and negative parts of amarts (semiamarts) are amarts (semiamarts) and some consequences thereof. The problem of whether or not powers of amarts are amarts is more complicated than in the case of martingales (cf. Bellow (1976 b, 1977 ), Gut (1982)) and will be dealt with in Section 5. + Set x = x v 0 and x- = - ( x ^ O )
for all real
Lemma 2.6. Let
{~n}n6D " If
{Xn}nED
be adapted to
further that the sequence is (a)
Assume, for
(EX~)T £ T
(b)
If
EIX_I I < ~
(EX~)T 6 T " If
(EX~)T£ T
{Xn,Tn}nC D
{Xn,~rn}nE D and
and
. If
~XT)T C T
is bounded above. If
(EX~)~E T,
D = N,
suppose
Ll-bounded.
D = -N , that
above, then so is
x.
(EXT)T C T
(EXT)T £ T
is bounded
is bounded below, then is bounded, then so are
(EIXTI)T i T "
is a semiamart, then so are
{Xn+, ~ n } n C D ,
{[Xn],~n}nE D.
(c>
if
.
and
{IXn],~C'n}ne D .
is ana
then soare
I x ^ y = ~(x+y
From the relations = ½ ( x + y + Ix-yl)
r,,
Cx+.r#nCO.
- Ix-yl)
and
.
x v y =
and the preceding results the following corollary is
innnediate.
Corollary 2.7. Let
{Xn,~n}nE D
Suppose further that they are and
{XnAYn'~n}n E D
are
and
{Yn,~rn}nED
Ll-bounded if
Ll-bounded m a r t s
D=N.
be amarts (semiamarts). Then
{XnvYn,~rn}n£D
(semi~m~rts).
Remark2.8. Lemma 2.6.c is stated and proved in Austin, Edgar and Ionescu Tulcea (1974), Lemma 2
for the case
D = N . For Corollary 2.7, see Edgar
17
and Sucheston (1976 a) , Proposition 1.3, where a direct proof is given (which then is used to prove a) and c ) o f
Lemma 2.6 through relations like
+ x
= x v O , ef. their Corollary 1.4). See also Baxter (1974), page 397.
Remark 2.9. If
{Xn'~n}n6 D
{Xn'~rn}n£D+ ' {Xn'~n}nED {Xn,~rn}n6 D
is a martingale it is well known that and
{IXnl,~n}n6 D
is a submartingale,
then so is
are submartingales and if {X+,~rn}n£D,
is guaranteed for the others. However, since every is an amart, it follows e.g. that if
rtingule, then
D and
{Xn,~n}n6 D
whereas nothing
Ll-bounded submartingale is an
Ll-bounded sub-
IXni, }ne D are
rts. +
Proof of Lemma 2.6. (~) Since is
a real
D = N.
number,
Let
it
T C TN
x- = -(-x) +
suffices
to prove
and choose
and
the
net.
first
Ixl = x
+ x-,
where
statement.
(This is possible since
t
is
bounded). Define
It
on
{X t
Z O)
In
on
{X t < 0}.
G
Clearly, G 6 T N
and
E X ~ = EX t • I{X t > O }
+ E X n. I{X < O} . Thus
--
t
o <_EX~ = E X t • I{Xt >_0) --EX o - E X - I { X t < O }
<_
-< sup E X O + sup EIXnl < ==. ~6 T N n
D = -N.
Let
T 6 T N
~ G =
Here
-I
and define
on {xt ~ O) on {x~ < O}.
plays the rSle of
OeEX
n
=EX a - E x
above. Calculations as above yield
I I<Xt
sup
EX a+Elx_ll <--.
a6 T N (~)
Immediate from the definition of a semiamart and
(~)
Again, it suffices to consider
{X~,~n}nC D .
(a).
x
18
D = N.
We follow the proofs given in Baxter (1974), page 397 and Austin,
Edgar and lonescu Tulcea (1974), pages 20-21. Since the net
(EXT)T 6 T
is convergent it follows from Theorem 2.5 + that it is bounded and hence, from (a), that the net (EXT)T E T is bounded. Thus, given
E > 0
(2.1)
IEXT - E X
(2.2)
EX: ~ E X ~ I + e
Now, given
there exist [< e
~ ~ TI
TO, T I £ T
with
for all
T, a ~ TO
for all
a ~ T1 .
arbitrarily, we define
c
on
{XTI ~ O}
TI
on
{XTI < 0}.
T I ~ T0,
~i E T
such that
by
cI =
Then (2.3)
EXTI = E X +TI +EXTI • I{XTI < O}
(2.4)
gxg I = E X g " I{XTI >_ O} +EXTI • I{XTI < O} . By subtracting
(2.4)
from
(2.3)
we obtain
EXTI - E X I =EX~I - E X I{XTI > O} , which together with
(2.1)
yields
E X+TI_< E+EXI{XTI_>O}_<~+EX÷t{X~l->°}-< E+EX+.o We have thus proved that (2.5)
EX + < e +EX: T I --
which together with
(2.6)
(2.2)
for
~ > T1 ,
yields
IEX: -EX~I <__e for ~ > T 1 > TO • 1 By performing the same calculations with
and
~1
replaced by
~' we obtain 1
~
(2.6) with
replaced by ~
~' > T 1
replaced by
a'
and
19
thus that (2.7)
] ~ x ~ - ~ x ~ ,+ I ~ 2 ~ + ~XT)T CT
i.e. the net
for
o, a' ¢ T,
~, ~'_> t o ,
is Cauchy and hence convergent
(cf. Remark 2.4).
This proves the assertion. D = -N.
In several instances the proofs for the cases
D = N
and
D = -N
are identical except for "obvious" changes. This time, however, this is not so, which is seen as follows. If, given such that and
e > 0,
(2.1)
o I 6 T_N
one chooses
and
(2.2)
TO, T I
hold for
O
T, ~ ~ T 0
as above, it turns out that
This is so because the order between
and
T1
and
T_N, @ ~ TI
with
TI~T 0
respectively
is no% a stopping time.
OI and
in
~
has been reversed.
To prove the desired results we thus have to modify the above proof so that rSles of fact that
~I o
will be a stopping time. This is accomplished by reversing the and
(EX~)T E T
Thus, given (2.8)
T1
in the definition of
(and by using the (trivial)
is bounded beZow).
E > 0
IEX T -EX~I
there exists
tO £ T
such that
~ £
T, ~ T
0.
for all
~ X +T)T £ T
Further, since
@i
is bounded below there exists
such that (2.9)
EX + >EX + - e ~-TI Now, choose
01 =
~ < T1
{
for all
o < T I.
arbitrarily and define
~
on
{x o > o}
TI
on
{X o < O} .
~i 6 T
Calculations like those above yield (2.10)
EXTI =EXTI.I{X c >__ 0} +EXTI'I{X O < O}
(2.11)
EX~I = E X ~
+ E X T I - I { X ~ < O} ,
by
T 1 < tO
20
from which it follows by subtraction and (2.12)
(2.8)
that
E X $ < E X + + e. TI - -
By combining
(2.12)
and
(2.9)
IE xa+ -Ex~ 1 I ~ e
(2.13)
we obtain
for all
Finally, to prove that the net
C ~ T I ~ TO •
(EX~) T E~T_N
is Cauchy, one proceeds
exactly as in the ascending case. The proof is complete. The second part of the following result is a "maximal" lemm~, cf. Chacon and Sucheston (1975), Lemm8 1 for
D = N
and Edgar and Sucheston
(1976 a) , Le~ma I.i. Le~8 is
2.10. Assume that
Ll-bounded if
(i)
is a semi~m, rt, which, in addition,
{Xn'~n}n 6 D
D = N . Then
sup Eix~i < ® T
X- P( sup IXnl >X) e sup EIXTI
(ii)
nED (iii)
IXnl < ~
sup nCD
Proof. (i)
a.s.
is immediate from Lermm 2.6.a.
The proof of set
T
A = {
(ii)
sup IXnl >I}
follows "the usual pattern". Let
O 6 T
= k - P(
and
o
o
on
Ac
and
IXnl > k} on
A
o
if
D = -N.
sup EIXTI [E[Xql ~ EIXsI'I{A} ~ I P ( A ) = T IXnI > k) . The conclusion follows by letting n o
sup
be fixed,
and define
Inl~n° I min{n C D; Inl ~ n 0
Then
nO £ N
~>
increase
Inl ~no to infinity. (iii)
follows immediately from
(ii)
by letting
1
tend to infinity.
21
From the theory of martingales martingales
(D =-N)
it is well known that reversed
behave more "nicely" than ordinary ones
(D = N).
In
contrast to the latter ones they are always uniformly integrable and converge almost surely and in
L I . It is therefore not surprising that in
the results above the assumption about the case
D = N
the case
D = -N
Ll-boundedness was made only for
and that this condition is automatically satisfied for (el. e.g. Lemma 2.10 (i), according to which
sup EIXnl n
sup EIXT[ < ~) . We further observe that in the proof of L e ~ 2.6.c T the fact that {X +n'~n}nE D is a semiamart was explicitly used only for D = N , since for ÷ ~ X T ) T E T_ N
the (obvious) existence of a lower bound of
was used (formula (2.9)).
Further, if {Xn}n C - N
D = -N
{Xn'~n}nE-N
is a (super) martingale,
uniformly integrable,
in fact
{XT}T E T_ N
then not only is
is uniformly inte-
grable (see e.g. Meyer (1966), page 126). The object of the final result of this section is to establish this uniform integrability for descending semiamarts, but before stating the result we make the following definition and some comments. Definition 2.11. Let {Xn}nCD
is
integrable,
(2.14)
{Xn}nE D
be adapted to
T-uniformly integrable i.e. if for any given
if the set
E > 0
sup EIX~I" ~IxTI > ~ < c
{~n}nED " We say that (XT}T E T
there exists
for all
is uniformly
%0 ' such that
~ > ~0"
T It is trivially seen that every
T-uniformly integrable sequence also
is uniformly integrable. For the ascending case we further know that every uniformly integrable (super)martingale
is, in fact, T-uniformly integrable
(see Meyer (1966), page 126) and also that every uniformly integrable amart is
T-uniformly integrable
(see Edgar and Sucheston (1976 a), page 210). For
the descending case, however, more is true. Theorem 2.12. I) = -N. Every semiamart is
T-uniformly integrable.
22
This is Theorem 2.9 of Edgar and Sucheston (1976 a) . Proof. It follows from Lemm~ 2.10 that for every TO C T_N
e > 0
there exists
such that
EIx~I ~ EIX~oI + c for all T c T_N,
(2.1s)
and, further, n o ~ T O
(2.16)
E
Now, let
and
X0
such that
max IXnl.l{suplXn[ >l} < E no<_
be arbitrary and
for all
~ > %0"
I > A O.
Since
EIXcI'I{Ix~I > ~} = = EIX I-I{IX I >%, ~ > no } + EIX~I-I{IXal >X,
~E
o ~ n O}
max Ixi-I{suplxl>A} + Elxo^nol'I{lXc^noi>X} no~
n
~ + ElXo^nol-X{IXo^nol > ~ } it suffices to prove uniform integrability for To this end, let
T E T_N,
T ~ n O ~ T0
T E T_N,
where
be arbitrary, let
T~n E
0 . and
be as above and define
{
T
TI =
on
{IxTI >~}
30 on { I E T I ~ } .
Clearly, T I 6 T_N.
Furthermore,
EIX~ll = EIx, I'~Ix~I >x) ÷ EIx%I.~IxTI ~ x} and hence, by
(2.15)
and
(2.16)
it follows that
+ zlXTol.~{Ixl >~} _< ~ + E,~<~__lmaxIx=l-~{s~plxl > x} _< e + C which completes the proof.
= 2c,
3. Convergence The first result of this section is the "elementary but extremely useful" approximation lepta which was mentioned in Section 1. It first appeared for the case
D=N
in Austin, Edgar and lonescu Tulcea (1974),
page 18. For the extension to the case (1976 a) , page 202.
D = -N , see Edgar and Sucheston D = N , see also Billingsley (1979),
For the proof when
problem 35.23, pages 428 and 503. L~-,,e 3.1 (a) D f N . for every
~ C ~
Let
Y
~r-measurable random variable, such that
be an
the number
is a cluster point of the sequence
Y(~)
{Xn(~)}nE N . Then there exists a sequence Tk+ I _> Tk
and
T k > k,
X
-~ Y
{Tk}kEN'
Tk C T N,
where
such that
a.s.
as
k-~.
Tk (b) D = - N . every
Let
~ E ~
~__-measurable random variable, such that for
be an
the number
{Xn(~)}nE_N. Tk_ I ~ Tk
Y
Y(w)
is a cluster point of the sequence
Then there exists a sequence
and
Tk ~ k,
X,rk ~ Y
{Tk}k6_N,
(3.1)
r 6 T,
where
such that
a.s.
as
k ~-~
.
Proof. In both cases it suffices to show that for each there exists
TkCT_N,
with
n 6 D
and
£ > 0
IT[ t In[ , such that
P([X -Y[ ~ ~) > I-
~.
The le-,,a then follows by induction by generating an increasing (decreasing) sequence of bounded stopping times {Tn}n C N with the property that P XT---~Y as n ~ . The conclusion is then obtained by taking a subsequence n (cf. Austin, Edgar and lonescu Tulcea (1974), page 19 and Edgar and Sucheston (1976 a) , page 202). We begin with the proof of
(b), since it is easier.
24
D = -N. Fix
n O E -N
{Xn(W)}nE-N
(3.2)
and
for each
e > O.
w E ~
Y(W)
Since
is a cluster point of
we have
~ = {~0; [Xn(~0)-Y(w)l <__ ~
for some
n <_no}.
A = {~; [Xn(W )-Y(w)[
for some
n,
Set (3.3) Since the set pick
nI
A
! ~
increases towards
(~ no)
such that
~
as
P(A) > i - E.
rain{n; n l ! n ! n
0
nI
n I < n < no } .
decreases we can (and do)
Define
T
as follows:
and
for
~CA
T = no Since
Y
is
for
~_-measurable
~-measurablen fop a ~ i.e.
~ ~ A .
T C T_N.
and
nE-N.
Further,
which we conclude that
~ - ~ = n ~ - N ~n
Therefore,
IXT(w)-Y(~)I
it follows that
{T =n} 6 ~rn for all
~ e
if and only if
P(IXT- YI ~e) > i - e , which is
Y
is
n C -N,
w E A,
from
(3.1).
D = N. The idea here is the same, but in this case we cannot conclude that Y nO
is
7-measurable for all n and E > O , there exist
measurable and such that is a cluster point o f
(3.4)
n , since n' ~ n O
P([Y-Y'I
{Xn(W)} n E N
~ and
= ~( U ~n) . However, given n£ N Y' such that Y' is ~n' -
~ ~/3) > i - ~/3 . Further, since it follows that
{m;Iy'(~)-y(m)l< E I 3 } c { ~ ; I X n C W ) - Y ' ( w )
and consequently there exists A ={~; Now, define
T
n" > n'
I <2C/3
for some
n>n'}
P(A) > i - 2£/3 , where
for some
n,
n' < n < n"}.
by
I min{n; n' < n < n " n" and
such that
IXn(0~)-Y'(~0)l <__2~/3
T=
Then, T E T N
Y(~)
if
w ~ A.
and
IXn(W)-Y'(w)l
!2E/3}
if ~ E A
25
P(IXT--YI >e) ~ P(IX T -Y'I >2e/3) +
P(IY'-YI
>e/3) < e.
This concludes the proof.
Theorem 3.2. Let
{Xn}n 6 D
E sup IXnl < ~ . n
(i)
Xn
(ii)
{Xn}n C D
be an adapted sequence and suppose that
The following statements are equivalent:
converges
a.s.
In I " "
as
is an asymptotic martingale.
Remark 3.3. This is Proposition D =N
2.2 of Edgar and Sucheston (1976 a) . For
the result was earlier proved by Austin, Edgar and lonescu Tulcea
(1974), page 19. Compare also Baxter (1974), (Theorem 1.13 above). Note that, for uniformly bounded sequences of random variables the supremum is trivially integrable and thus Theorem I.I is an immediate corollary.
Proof. D = N . {Tn}n C N as
(i) ~ (ii)
Suppose that
Xn~
Y
a.s.
as
n ~.
Let
be a sequence of bounded stopping times increasing to infinity
n ~.
~ Y a.s. as n ~ , which together with the inten grability of the supremum (IX T I ~ supIXnl) and dominated convergence n n implies that E X T ~ E Y as n ~ , from which the assertion follows in n view of Remark 2.4. (ii) ~ (i)
Then,
XT
X* = limsup X n and X, = liminf X n . According to Lemma n~ n~ 3.1 there exist two sequences of increasing bounded stopping times, {rn}nEN (3.5)
Set
and
{Gn}n£N' X
T
~ X*
such that
and
X
n
~ X,
a.s.
as
n ~.
n
Since
IX~ - X T I <_. 2 suplXnl , which is integrable it follows by n n n dominated convergence and the amart property that
(3.6)
O < E(X*-X,) --
and hence that
X* = X,
= lim E(X_ -Xon) = O , n-,o= tn
a.s.
26 The proof for the case
D •-N
Remark 3.4. In the proof of
is the same.
(ii) ~ (i)
order to conclude that the limit in
the amart property was used in
(3.6)
is
{Xn'~n}n6D
is a semi~mgrt we obtain
(3.7)
E(X* - X.) ~ limsup E(X O - X T) , T,O E T
0 . If we only assume that
which is a slightly weaker form of the theorem of Chacon (1974), i.e. Theorem i.ii above (see also Chen (1976 a,b) ). Note further that holds for the case
D •-N
(3.7)
too (see also Edgar and Sucheston (1976 a) ,
page 213). We are now ready to prove the amart convergence theorem.
{Xn,~n}ne D be an asymptotic martingale and suppose, if
Theorem 3 . 5 . Let DffiN,
in addition that it is
Remark 3.6. For
D =N
Ll-bounded. Then
X
n
converges
a.s.
as
this is Theorem 2 of Austin, Edgar and lonescu
Tulcea (1974). Our proof follows Edgar and Sucheston (1976 a) , page 203. Proof. Pick
~ > O
and set
Y
{Yn'~n}n C D
is an amart by Corollary 2.7, which, furthermore is uniformly
n
= -~ v X
n
bounded. It follows from Theorem 3.2 that
Inl
~ ~.
Finally, since
{gn}nC D
and
^ ~
Y
n
for
n E D.
Then
converges almost surely as
{Xn}nED
differ on the set
{sup IXnl > ~} , whose measure can be made arbitrarily small by choosing n % large enough (see L ~ a 2.10), we conclude that X n converges almost surely too. Some immediate corollaries are: 3.7. Every reversed martingale and every
Ll-bounded reversed submartingale
is almost surely convergent. 3.8. Every
Ll-boundedmartingale
almost surely convergent.
and every
Ll-bounded submartingale is
27 3.9. Every descending amart converges in
L I . This follows from the
uniform integrability (Theorem 2.12). It is worth mentioning that the proof of the amart convergence theorem differs from the proofs used to prove martingale convergence. The following result (see Gut (1982), Theorem 4.1), which will be used in the sequel, is a minor strengthening of Theorem 3.2. Theorem 3.10. Let
{Xn}nED
be adapted to
{~n}nED
and
T-uniformly inte-
grable. The following assertions are equivalent: (i)
Xn
converges
(ii)
{Xn'Yn}nC D
a.e.
as
Inl
~
is an asymptotic martingale.
For a related result for the case
D =N,
continuous time and finite
stopping times, see Mertens (1972), T Ii. Corollaire. Since, by Theorem 2.12, the above uniform integrability condition is satisfied for every descending semiamart, the following corollary is immediate. Corollary 3.11. Let
{Xn'~rn}ne -N
{Xn'~rn}n£ -N
be an
a.s.
convergent samiamart. Then
i s an amart.
Remark 3.12. There exist
a.s.
convergent ascending semi~m~rts that are
not amarts. See Austin, Edgar and lonescu Tulcea (1974), page 19 and Example 4.2 below.
Remark 3.13. Since
{XTI <_ s u p ] X I
for all
TC Z
it is clear that
n
E supIXn] < ~
implies that
{Xn}ne D
is
T-uniformly integrable. However,
n
according to Blackwell and Dubins (1963), Theorem 2, there are
a.s.
con-
vergent martingales (and hence amarts), {Xn,~n}nC D , which are uniformly integrable, and hence for which
E supIXnl n
T-uniformly integrable (see Meyer (1966), page 126), is not finite. For
D=-N
see also Remark 3.16 below.
28 Remark 3.14. The examples of Section 4 below show that Theorem 3.10 and Corollary 3.11 cannot be (essentially) improved. Proof of Theorem 3.10 (i) ~ (ii).
D =N. Let
sequence of bounded stopping times. Since it follows that {XTn}~=I
Xn~
X , say,
a.s.
as
n ~
XT
~ X a.s. as n ~ ~ and since, in particular, n is uniformly integrable we conclude that E X T n ~ E X as n ~ m ,
which in view of Remark 2.4 implies that The proof for the case (ii) ~ (i). is
{Zn}n C N ' be an increasing
D = -N
{Xn'Jrn}nC N
is an ~m~rt.
is the same.
The uniform integrability implies in particular that
Ll-bounded and thus, by Theorem 3.5, a.s.
{Xn}n C D
convergent.
As an application an amart related to the Marcinkiewicz strong law of largenumbers
(see Lo~ve (1963), pages 242-243) is presented. Recall that
in the case of the classical Kolmogorov strong law of large numbers the sequence of arithmetic means constitutes a reversed martingale. Example 3.15. Let
b e independent, identically distributed (i.i.d.)
{~n}~=l
random variables. Suppose that that
E~I = O
if
[n -I/r .
~
k=l k
if
I <_ r < 2
~'-n
O < r < 2
and further
Ir
if
0 < r < i
-n n i. I k
and
for
i < r < 2 . Put
=~ X
EI~I Ir < ~
o{~;
k < -n}
for
n = 1,2,
Then
{Xn,~
is an
nm~Tt.
Proof. Let
r = 1 . Since
{Xn'~n}n£-N
is a martingale and since mar-
tlngales are amarts we are done. Let
i < r < 2.
Then
E suplXnl < ~ by Klass (1974), page 904. The n Marcinklewlcz law implies that X * 0 a.s. as n * - ~ and so an applin cation of Theorem 3.2 yields the desired conclusion.
2g n
Now, let where
0 < r < I.
Set
Y-n = n-l" I I~k Ir and ,.=I z% L
Z
-n
--E(i~iIrISrn) '
n = 1,2, .... n
Since
~ ~kl r -< X_n = n-l.I k=l
n-l.
n
~ l~kl k--I
r
-- Y -n
is adapted it follows (recalling interchangeability)
X
-n
7- n
=E
X
<E -n
7- n
Y
m
-n
=Z
and since
{Xn}ne_N
that
-n
Thus, (3.8)
O < Xn < Z --
for
n = ..., -3, -2, -I .
n
Now, {Zn,~n} n C - N
is a martingale and thus
(see Meyer (1966), page 126), which in view of {Xn}n £ -N
is
T-uniformly integrable (3.8)
implies that
T-uniformly integrable. The 8msrt property now follows from
Theorem 3.10. Remark 3.16. In Example 3.15 we used Theorem 3.2 for the case
i < r < 2.
n_i/r, n This was possible because E sup I E ~kI < ~ . For the case O < r < i n k=l n Theorem 3.2 is not applicable because E sup n -I. I E ~k [r < ~ if n k=l and 0nly if E[~l[r'log+l~l I < ~ (see Gut (1979), Theorem 3.2) and the latter condition was not assumed. We thus have obtained an example (for D = -N) , where Theorem 3.10 applies and where Theorem 3.2 does not. Remark 3.17. Example 3.15 can be used to illustrate the importance of the a-algebras relative to which a sequence is adapted. Consider the case 0 < r < i D -- -N
and suppose further that
and
~-n = ~ { ~ ;
k <__-n}
~i' ~2"'"
all are non-negative.
we thus know that
{Xn'~rn}nE-N
If
is an
n
amart. However, let
= n-ll Y ~k Ir and ~n = ~ { ~ ; k < n } , n k=l n--l,2, .... According to Krengel and Sucheston (1978), Corollary 4.4, {Xn'~rn}n £ N
X
is a sem~-m~rt if and only if
only assumed that (1979) ,
D=N,
E~
Since we have
< o~ it follows from the previous remark (i.e. Gut
Theorem 3.2) that
sup X n
in the ascending case
E sup X < co. n n
{Xn'~n}n E N
is not necessarily integrable and so n
is not even a semlamart.
30
Except for special examples constructed like those of Section 4 below, Example 3.15 seems to be the first example of an amart which is not a martingale,
or a submartingale etc. Unfortunately,
used the strong Marcinkiewicz
though~we have
law in the proof. It would therefore be
interesting to obtain a direct proof of the amart property, and thus, as a corollary, a new proof of the Marcinkiewicz
strong law. This would in
a natural way generalize the martingale proof of the Kolmogorov strong law of large numbers to an amart proof of the Marcinkiewicz
strong law.
4. Some exampl,es This section contains some examples which illustrate certain pathologies. They will be used in Section 5 in the investigation of stability properties (cf. Gut (19 82)). Throughout
(G,~rp)
is the Lebesgue measure space and for any
sequence of random variables, { X n } n e D , Example 4.1. For
~nffiG{~; k ~ n},
n e D.
n £ N , define
{~ n
Xn(~) =
if
w E (O,2 -n)
if
~ £ [2-n,l) •
This yields an a.s. convergent martingale (and thus also an a.s. convergent amart) which is not uniformly integrahle and E X if
n
= i
for all
(Xn~ 0
a.s.
as
n ~
n) , see Doob (1953), pages 347-348. Recall that,
D ffi-N , every martingale is uniformly integrable (see also Theorem
2.12). Example 4.2. For
p > i
X (p) s 0 2n
When
p=l,
and
and
n E N,
X~Pn)+I(~)
..(I), ~r(1)} {An n n£N
let
Io
if
~0 C [2-n,l) .
is an a.s. convergent semiamart that
fails to be an amart, see Austin, Edgar and lonescu Tulcea (1974), page 19. E suplXnl ffi Y. 2n/P. 2-(n+l) < ~o. Since the sequence n kffil is a.s. convergent, an application of Theorem 3.2 shows that If
p > I, then
Y(P) ~ P ) "~n " n }nEN
is an smart.
Example 4.3. This example is related to that of Sudderth (1971), page 2145. Let
p > 112
and define [| n I/p
if
co C
y n(~)
for
i=l,2,...,n
ffi I 0
otherwise
and
n =1,2, ....
2 '
32
The s e q u e n c e 2 YI'
{x(P)}nn EN
Y22 ''''' Y In '
(4.1)
x,p,C~ -~ O n Define
i s now d e f i n e d a s t h e sequence
Y2n ''''' ~n ' .... It is easily verified that a.s.
{Tn}nEN
and in
+ min{k ~ n ;
Tn(~) ffi n ( n + l )
among
if
C T
and
n
(4.2) Let
~k(~) ÷ 0~
X
Y
among
T 7~
for
1 < k < n.
which corresponds to the first
that is non-zero and
that corresponds to the last zero. Clearly, T
n ~o.
~k(w) = O
equals the index of the
Yln ,..., Ynn
as
as follows:
(n-2 1)n
Thus, Tn
LI
as
Tn
equals the index of the
YIn ~ ' " ' n ~.
ynn
Y X
if the latter are all
Furthermore,
n
EX T (p) ffi E max{Y~ . . . . . Y~} ffi n ( 1 / p ) - I n 1/2 < p < 1 .
yields
Then E x ( P ) - ~ + ~
as
n ~,
an example o f a u n i f o r m l y i n t e g r a b l e
which t o g e t h e r w i t h a.s.
(4.1)
c o n v e r g e n t s e q u e n c e which
fails to he a semiamart. Next, let that
{X~ 1)
a.s.
as
'
p = 1 . Then
E X (1) = I and in view of Tn fails to be an amart. Further,
~(I)} n nCN
n ~ =,
it follows that
(4.1)
it follows
since
X (I) ~ 0 Tn
= -~X(1)~ T n -nffil ' and hence that
{X$1)}T E T ' i~ not uniformly integrable. Finally,
let
it follows that
E suplX~P)l < ~ , n
Theor. 3 2 shows that Example 4 . 4 . L e t
p~
P)>nCN 1 , nE-N
x ( p ) ( ~ ) ffi I 2 n / p -n
-
1 P(sup]X~(p) i ffik l / P ) ffi n which together with (4.1) and
p > I . Then, since
[O
We first note that
is an
and d e f i n e
if
~ £ (0,2 -n)
if
~ E [2-n,1).
33
(4.3) Let
x,p,t~ , 0 n p=i
a.s.
as
n ~m
and that
~_{x,p,} _ t n n £ -N
and introduce the (finite) stopping time
T
is
Ll-bounded.
by
inf{kq-N; x~P)(~) # O}
if an and the sequence
{Tn}nCN'
Tn C T_N
Tn = T v ( - n ) . A simple compu-
by
tation yields (4.4)
E X (I) = n + l
Tn
which shows that If
---~
~
n ~ ~,
as
{X~ I) , ~ n l ) } n C _ N
p > i , then
it follows that
~
is not a semiamart.
E s~p IXn(P) I < ~
~r(P) ~nE-N {X~ p) '~ n ~
and so, by
(4.3)
and Theorem 3.2,
is an amart.
Remark 4.5. Note the difference between Example 4.1 and Example 4.4 with p = I . Just as in Remark 3.17 the different behaviours for D •-N
D =N
and
are due to the different sets of (bounded) stopping times. In the
present case we observe in particular that it is possible to stop at
sup X (I) n n
D=-N
~X(1)~ " n ~£ D
(which is not integrable) with a finite stopping time if
(i.e.
%
in Example 4.4), something that is not possible when
D =N.
A difference between the present case and the case discussed in Remark 3.17 is, however, that here the "better" behaviour occurs when Example 4.6. Let 4.3 and define
p > 1/2 , define {X (p)
n
{Y~; I < i < n, n ~ l } •
as
}nE-N
D=N .
yn
"" '
n
n
as in Example 2
2
i
n''''' Y2 "YI ''''' Y2 'YI 'YI "
We have (4.5)
X (p) ~ 0
a.s.
and in
LI
as
n ~-~.
n
To continue the analogy with Example 4.3 define n
equal the index of the
Yln ,... , Ynn
X
which corresponds to the ~z8t
that is non-zero and by letting
that corresponds to the ~irst
{Tn}n6 N
Y
among
Tn
by letting Y
among
equal the index of the
YIn ''''' yn n
X
if the latter are all
34
zero. Thus
Tn C T_N,
Tn + - ~
(4.6)
E X (p) = n (I/p)-I T n Just as above the case
and
1/2 < p < I
yields an example of a uniformly
integrable a.s. convergent sequence which fails to be a semiamart. When p=i
•
~X (I) }
~ n
n E -N
is uniformly integrable but not
grable, in particular, {X(1) n '~n i) } n C - N 2.12). Finally, for
T-uniformly inte-
is not a semiamart (by Theorem
p > i , {X(p)n'~nP)}nC-N
is an amart.
For the construction of amarts and semlamarts we also refer to Krengel and Sucheston (1978), pages 217-223.
5. Stability This section deals with the following problem: Given an smart ~: R ~
and a function
R,
{~(Xn)'~n}nED
when is
an ~m~rt?
The first result of this kind is that the conclusion holds for +
~O(x) = Ixl , x (L~a
and
x
, provided
2.6). For the case
D •N,
and sufficient conditions on Here the case
D =-N
{Xn}nCD
is
Ll-bounded when
D=N
Bellow (1976 b, 1977 ) gives necessary
~
for the conclusion to hold in general.
will also be covered. The proofs differ slightly
from those given in Bellow (1977) . We also investigate which further assumptions on the amart one needs for the conclusion to remain valid when the necessary conditions on
~
no longer are satisfied.
Following Bellow (1976b, 1977 )
such problems are called stability
problems. Theorem 5.1. Let that
{Xn}nEN
(5.1)
is
~
(5.2)
{Xn'~n}nED
be an amart. If
Ll-bounded. Let
D = N , assume, in addition, be a function such that
~: R ~ R
is continuous and
lim
~(x)
and
lim
~(x)
X
exist and are finite.
X
X ~
X ~ --~
Then, {~(Xn),~n} n C D
is an
Ll-bounded amart. +
Remark 5.2. The cases obviously included. For
~(x) = Ix[, D = N,
x
and
x
mentioned above are
Bellow (1977) , Theorem 2, shows that
(5.1) and (5.2) are necessary and sufficient for an
{~(Xn)'~n}nCN
to be
Ll-bounded amart.
Proof. We first assume that
X
> O, n
~(O) = 0
and
lim ~,x~ = O.
--
x x ~ m
By the amart convergence theorem we know that In[ ~ m (5.3)
and thus, by o(x)
(5.1), also that
converges
a.s.
as
Ini '" ® .
In
converges a.s. as
36 By invoking Theorem 3.10 and Corollary 3.11 it therefore remains to show that (5.4)
{~(Xn)'~n}nE-N
(5.5)
{~(XT)}T q T N
is a semiamart.
is uniformly integrable.
We first consider the
Ll-boundedness of
By assumption, x-l-l~(x)l < E if
0 < x < M.
if
{~(XT)}Z E T "
and
x > M
I~(~)[ ~ 0 '
say,
Thus,
E[~(XT) [ = E[~(XT)['I{X T ~ M} + EI~(XT)['I{XT>M} ~ ~o'P(XT ~M) + EEXT'I{XT>M} ~ 0
+ E supEX T < =, T
since every ~m~rt is a semiamart. Thus, {~(Xn),~n} n E D and, in particular, if Now, let
D = N.
D =-N
is a semiamart
we are done.
A similar argument together with the maximal inequa-
lity, Lemma 2.10 (ii), yields
Z[~(Xz)['X{[~(Xz) [ > A} = s[~(xz)[.z([~(xz) [ > A,
= x z ~ M} + z[~(xz)[.z{[~(xz) [ > A,
~ o ' P ( I ~ ( X T ) I > A) + S E X T ~ o ' A - I ' s u p T
x z >M} <
E[~(XT) I + e s u p Z X T
Consequently,
lim sup E[~(Xz)I.Z{I~(XT) I > A) ~ E sup ZX T A~ and, since
e
T
may be chosen arbitrarily small, (5.5) follows.
It now remains to remove the restrictions lim ~(x) x
= 0
made a b o v e ( c f .
Assume that Set
X
> O, n--
~(x) = ~(x) - ax.
X
> 0, n--
~(0) = 0
Bellow (1977)).
~(0) = 0
Then, since
and that
x-l.~(x) ~
x-l.~(x) ~ 0
as
~ 0
as
x ~.
x ~,
{~ (x) '~.}n c D is an ~m~rt and because of the linearity is too.
and
{~(Xn)'~n}n E D
37
+ ~(0) = 0 . Be Lemma 2.6, {X n,~rn}ncD
Next, suppose only that {Xn'~n}nE D
are non-negative a~-rts. From what has been shown so far, it
follows that that Since
{~(X~),~n}nC D
{~l(Xn),Srn}nC D ~(X)
and
and
is an amart, where
= ~(X~) + ~l(Xn)
{~(Xn) ' ~ } n C D Finally,
(because
are amarts and thus a l s o
~l(X) ffi~(-x) ~(0) = O)
for all
x C R.
we conclude that
is an a m a r t .
if
{*(Xn)'~n}neD
{~(Xn),~n}nE D
~(0) ~ 0
we p u t
~ ( x ) = ~ ( x ) - ~ ( 0 ) . Then
is an amart and thus
{~(Xn) ,~n}neD
~(0)
= O,
is too.
This terminates the proof. Now, suppose that
~: R ~ R
is a function for which
hold. As pointed out in Bellow (1977) , sequence of real numbers
(5.1)
does not
page 286 one can always find a
{an} , which is an amart and such that
{~(an)}
is not. We therefore turn to the problem of finding what additional assumptions on the amart are needed (together with Theorem 5.1 to remain valid when Theorem 5.3. Let
{Xn'~n}nED
tinuous functions such that
(5.2)
(5.1)) for the conclusion of
no longer holds.
be an amart and let lim q0(x) x x-++~
and
~0: R-~ R
be a con-
lim q~(x) do not exist x x-~ - ~o
(finitely).
(a)
D =N. Assume in addition that {~(XT)}TC T
{Xn}nE N
is
Ll-bounded and that
is uniformly integrable. Then, {~(Xn),~n}nE N
is an
Ll-bounded amart.
(b)
D =-N. Assume further that {~(Xn)'Yn}nE-N
{~(Xn)'~n}nE-N
is a semiamart. Then,
is an amart.
Proof. The amart convergence theorem and the continuity of imply that
~(Xn)
converges a.s. as
n ~ ~
(n ~ - ~ )
~
. The conclusion now
follows immediately from Theorem 3.10 and Corollary 3.11. Remark 5.4. After reduction to the case
X
together
> O , ~(0) = O n--
and
38
lim x-l-to(x) = 0 x ~ +~ validity of (5.4)
the proof of Theorem 5.1 consisted of s h o ~ n g the and
(5.5)
above. In the present theorem the corre-
sponding properties are ~ p p o s e d to hold. However, following these remarks, some examples are presented to show that the theorem is (essentially) the best possible. Remark 5.5. D =N. It is easily seen by an estimate related to those used to show
(5.4)
and
(5.5)
that the assumption that
{Xn}nE N
is
Ll-bounded can be dropped if
~I -- lim inf Ix-l.to(x)[ and u 2 = x -~+~ = lim inf Ix-l-to(x)l both are positive, because the Ll-boundedness then x-~-~ follows from the uniform integrability of {to(XT)}TC T . However, if
C~l = ~2 = 0 Let
this cannot be done as is seen by the following example:
{~n}nEN
be a sequence of i.i.d, random variables such that 1 n Y ~k and ~n = G{Xk; k < n}, P(~n = w) = P(~n=-W) = ~ . Put X n = k=l n=l,2,....
Then
ZiXnl ~ /~n
as
{Xn,Tn}nC N
n~--,
is a martingale (and hence an amart),
{ X } n £ N i s not
Ll-bounded. Now, choose
~I =(12 = O
lim suplx-l-to(x)I = i) .
i.e.
tO(x) = I x . sinxl , (for which
and
Ixl
- ®
Clearly, to(Xn) m O
for all
n,
in particular, {to(XT)}T E T
is uniformly
integrable. Remark 5.6. If one of the limits
lim X~
x-l-to(x)
and
--~
lim
x-l-to(x)
X~
exists, finite and the other does not, then, by considering the positive and negative parts separately, the assumptions on {to(Xn)}nC-N ' for
D =N
and
D =-N
{to(XT)}T E T
and
respectively, can be reduced to
assumptions on one part only, by applying Theorem 5.1 to the other part. Similarly, if e.g.
~i > 0
and
~2 = 0,
where
~I
and
~2
are defined
as in Remark 5.5. As an example, consider x
if
x ~ 0
if
x < 0 .
to(x) = Then, f o r
D=N,
if
{Xn,~n}nE N i s an amart, {to(Xn),~n}nC N i s an
39
Ll-bounded amart, provided
{Xn}ne N
is
Ll-bounded and
{(X~)2}Te T
is
uniformly integrable. In the remainder of this section we use the examples from Section 4 to produce the examples that were promised at the end of Remark 5.4. First, let
D =N .
Suppose that the assumption that is replaced by the assumption that
{~(XT)}T £ T
{~(Xn)}n E N
is uniformly integrable
is uniformly integrable and
consider Example 4.3 together with the function ~(x) = Ixlp , p > 1 . Then, {X (p) ~ P ) } n ' n n£N
(with
p > I)
is an
~(X~ p)) = X(1)n it follows that that
{~(X~p)) ,~nP)}nqN
{~(Xn)}n E N
Ll-bounded amart. Further, since
{~(x~P))}n6 N
is uniformly integrable and
fails to be an amart. The condition that
is uniformly integrable is thus not sufficient for Theorem 5.3
to hold in general (if
D =N) .
Next, consider a possible replacement with the assumption that {~(Xn)'~rn}nq N
is an
Ll-bounded semlamart (or, equivalently, that
{~(x)}~ £ T is Ll-bounded) and apply Example 4.2 together with the function ~(x) ~ [ x l P ,
and
~Y(P) p > 1 . Then, ~ - n "~nP ) ~" n e N
{,~,cx(P)~n "'~P)}nn EN
is an
(p > I)
Ll-bounded ~m~rt
Ll-bounded semiamart but not an amart.
Note that, since none of the conditions integrable" and "{~(XT)} T E T
is an
is
Example 4.1 and Example 4.3 with
"{~(Xn)}n 6 N
is uniformly
Ll-bounded'' imply each other (combine 1/2 < p < i) both conditions had to be
investigated. Now, let
D=-N.
Suppose that the assumption that
{~0(Xn),~n}nE_N
weakened (cf. Theorem 2.12) to the assumption that
is a semiamart is
{~0(Xn)}nC_N
is uni-
formly integrable and consider Example 4.6 together with the function ~0(x) -- Ix[p , p > i. Then, ^'X p) } ~ n ~ nE-N an amart.
;X (p) ~ P ) } n 'n n6-N'
is uniformly integrable but
where
p > i
is an ~m~rt,
{~P(Xn(P)),~P)} n n6-N
is not
6~The
Riesz decomposition
The Riesz decomposition theorem for amarts was first proved in Edgar and Sucheston (1976 a), Theorem 3.2 for the case
D=N
and, independently,
in Krengel and Sucheston (19781 (except for the problem of uniqueness) and Gut (1982) ,
Theorem 6.1 for the case
of semi~m~rts,
D =-N.
For the Riesz decomposition
see Ghoussoub and Sucheston (1978) and Krengel and Sucheston
(1978). We consider amarts only. First, let
D = N . Instead of presenting the
original proof we use the following lemma from Astbury (1978), where a Riesz decomposition theorem for amarts indexed by directed sets was proved. Lemma 6.1. Let T0 6 T
{Xn'grn}nCN
be an smart and let
E > O . Then there exists
such that
zlx~ - S ~'~XoI
(6.1) Consequently,
the net
~E
for
(E~XT)T C T
~ > T > TO .
converges in
LI
for any
~ E T.
Remark 6.2. Here we have used net convergence in a more general form than described in Section 2. For details, see Neveu (19751, page 96. Proof. Since the net (6.2)
(EXT)T £ T
IEX 0 - E X a l ~ ~/2 Let
T E T,
T O < T < ~, T
p = where
converges we can choose for all
~, O ~ T
TO C T
such that
O-
and define
on
A
on
AC
A £ ~T" It follows from
(6.2)
that
IE I{A} (X T - EgrTxG) 1 = 1E I{A}XT -E I{A}Xffl =
= IE (I{A)XT + I{Ac}X~) -E(IfAC}x~÷ I{A}X~)I = ZEX~- EXal ~ E/2. Set
A = {X T - E ~ X ~
O} . Then, by applying the previous inequality
41 to the sets
A
and
E IXT-E which proves
A c , we obtain
"Xol = E I { A } ( X T - E
(6.1).
As to the second conclusion of the l e m m a w e use that for
~
T, TO, P £ T T -E
which, since
< E,
Xo)- EI{AC}(x T - E ~ X O )
LI
such that
(6.1)
to observe
~ > • > TO, p
X~I =EIETO(XT - E
XG) I e E IXT -
XO
_ E ,
is complete, completes the proof.
We are now ready to state and prove the Riesz decomposition theorem. Theorem 6.3. Let written as {Zn'%}n£N and in
be an amart. Then
Xn = Yn + Zn " where is a
{Yn'~rn}n£N
X
n
(E~OXT)T £ T
is a martingale and
T-uniformly integrable amart, such that
p 6 T
Zn ~ O
a.s.
be arbitrary. It follows from Lemm~ 6.1 that the net
converges in
LI
e > 0 , there exists
(6.3)
to TO
E[Yo -E~0XTI < e
YO ' say. In particular this implies that, such that
for all
T > T 0.
Our next goal is to show that for
p E T,
(6.4)
O £ T , such that
Let
can be uniquely
LI .
Proof. Let
given
{Xn'~n}n6N
Y 0 < p < T
= E vY and
EIYa-
O
for all
T ~ TO.
In view of
eEIYo - E XTI + E I E
fixed,
(6.3)
XT -
~ < 0-
we obtain
01 =
= EIY-E~r~XTI + m IE~(E ~0XT-yp) I ~ e + E I E ~-OXT-YoI ~ 2e, from which
(6.4)
follows because of the arbitrariness of
We have thus e s t a b l i s h e d
that
{Y ' ~ " } n E N n n
g.
is a martingale.
42
. Since (~-%'Zn"n~n£ N is the n difference of two amarts it is itself an amart. Next, given e choose T O To complete the proof, set
and
c > r > TO
that
such that
EIYT- EYTXGI ~ e
Zn = X n - Y
(6.1)
is satisfied for
and such
(ZT)T£T
(cf. (6.3)). Then, since
E~TZ = E~(X
-Y~) = EYTX -yT
we obtain
ZlZTI'I IZTI> }_<EIZ I_<EIZ-E ZoI +zlE Xo-Y I_<2 which proves that
{ZT}T q T
is uniformly integrable.
To show uniqueness, suppose that Then
{[y~l) -Y(2)In ' ~ n } n E N
X n = y(1)n + g(1)n = y(2)n + Z(2)n "
is a submartingale, in particular
EIY~ I)-Y(2) I increases with
n.
However,
EIY~ I)-Y(2) 1 = E Z (1)-z(2) I <
n
n
< EIZ~I) I + EIZ~2) I " O
as
n ""
and consequently
n
n
y(1) = y(2)
a.s.
--
n
for all
for all
n.
Therefore,
Z (I) = Z (2) n
a.s.
for all
n
n
and the theorem
n
is proved. We now turn to the case Let
{Xn'Yn}n£ -N
theory that
D = -N.
be an amart. It is well known from martingale
{E~n X-I'~n}nE-N
is a uniformly integrable martingale and
thus that it converges a.s. and in theory we know that n
-~-~+
(6.5)
n ~ - ~ • Also, from amart
converges a.s. and in
LI
to
X _ ~ , say, as
n C -N,
y n
Since
(6.6)
as
.
Set, for
for all
Xn
LI
. ~ n X_I + X_oo- E
X_oo and
n E -N,
E
X_I
are
X_I
n
Yn =
Zn = X n - Y n "
~r -measurable, i.e. ~--measurable n
can also be written as
Y
and
EJn(X_I+X_~-E ~-= X_l ) .
--
43 It follows that is an amart. Also, Z
n
~
0
a.s.
and
{Yn'~n}nE-N {ZT}T C T
in
L1
as
is a martingale and that
{Zn'~n}nE-N
is uniformly integrable by Theorem 2.12 and n ~
~
The following theorem is thus obtained. Theorem 6.4. D = -N . Let written as
{Xn'~n}nC-N
X n = Y n + Z n , where
{Zn'~n}n C -N
(Yn,~n}nE_N
is an amart such that
and (automatically) such that
be an amart. Then
Z
~ 0
n
{ZT}~ C T
The proof of the uniqueness when
Xn
is a martingale and a.s.
and in
LI
D = -N
as
n~-~
is uniformly integrable.
D = N
is essentially based upon the
fact that there are no non-trivial martingales that converge to When
can be
0
in
LI •
this is no longer the case and the following example shows
that there need not be uniqueness (cf. Gut (1982), Section 6). Example 6.5. Consider Example 3.15 with
I < r < 2 . The following decompo-
sitions are possible: X
=O+X -n
-n n
x
Since
1
-n = ~
{Xn'~n}nC-N
Z Ek + (X-n-
k=l
n
E 5k).
k=l
is an smart which satisfies the properties of the
"potential part", the first decomposition is obvious. For the second one, n
set
Y-n = n-I
martingale, Yn
Z ~k " Then, {Yn,~n}n£_N is a uniformly integrable k=l O a.s. and in L I as n ~ - m and X -n - Y -n is an
amart that has the properties of the "potential part" as described in the theorem.
7.
Two further ~eneralizations
of martingales
In this last section we shall briefly mention two further generalizations of the martingale concept. The main reason for brevity is that, as will be seen below, there are several negative results; viz. there are several properties which hold for amarts hut fail to hold for these other concepts (see Edgar and Sucheston (1977 a) and Bellow and Dvoretzky (1980)). Throughout
D =N .
Definition 7.1. An adapted process,
~ t h time
if and only if for every
P(IE~rmXn - Xml >E) ~ 0
{Xn} n C N ' is called
a game fairer
e > 0
as
n,m ~--
with
n ~ m.
Games which become fairer with time were introduced in Blake (1970). Later Muccl (1973) and Subr~m~nian (1973) (independently) formly integrable games fairer with time converge in
proved that uni-
L I . The idea of the
proof in Subr~msuian (1973) is to show that the sequence converges in
LI
to
Ym'
say, for every
Riesz decomposition theorems), that martingale, which hence converges in 3E-argument that
X
n
converges in
m
(just as in the proof of
{Yn'~rn}n £ N LI LI
to to
{E m X n } n £ N
is a uniformly integrable
Y , say, and finally, by a Y .
It is immediate from the definition of a martingale that
every
martingale is a game fairer with time. Further, it is shown by Edgar and every
Sucheston (1976 e) that
time
(real valued)
amamt is a game fairer with
(but not necessarily vice versa). Mucci (1973) also introduces the concept of a martingale in the limit
as follows. Definition 7.2. An adapted process, {Xn} n C N ' is called a
the limit
martingale in
(m.i.l) if and only if
E~mx
- X n
~ 0 m
a.e.
as
n,m ~
with
n > m.
45
It is clear from the definitions that every martingale is a and that every
m.i.l,
m.i.l.
is a game fairer with time. Further, Blake (1978)
and Edgar and Sucheston (1977 a) show that
e~ePyumuPt is a mc~ti~gule in
the limit. Further, Mucci (1976) proves that every
Ll-bounded
m.i.l.
converges almost surely. Since amarts and martingales in the limit both considerably generalize the notion of amartingale, ing a
one is naturally led to the problem of develop-
m.i.1.-theory just as has been done for amarts. We have, for example,
noticed above that some kind of convergence theorems remain valid. However, it turns out that there areseveral other basic and useful properties which hold for martingales and amarts which do not hold for martingales in the limit (and games which become fairer with time), such as the maximal ineuqality,
the Riesz decomposition theorem. For details,
see Edgar and
Sucheston (1977 a) and Bellow and Dvoretzky (1980). See also Blake (1981). For other generalizations of martingales and amarts we refer to the references given in the third part of this volume.
References: Astbury, K., (1978), Amarts indexed by directed sets. Ann. Probability 6, 267-278. Austin, D.G., Edgar, G.A., and lonescu Tulcea, A., (1974), Pointwlse convergence in terms of expectations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 30, 17-26. Baxter, J.R., (1974), Pointwise in terms of weak convergence. Proc. Amer. Math. Soc. 46, 395-398. Bellow, A., (1976a), On vector-valued asymptotic martingales. Proc. Nat. Acad. Sci., U.S.A. 73, 1798-1799. Bellow, A., (1976b), Stability properties of the class of asymptotic martingales. Bull. Amer. Math. Soc. 82, 338-340. Bellow, A., (1977), Several stability properties of the class of asymptotic martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 37, 275-290. Bellow, A., (1978), Some aspects of the theory of vector-valued amarts. Springer Lecture Notes inMathematics 644, Springer, Berlin, 57-67. Bellow, A. and Dvoretzky, A., (1980), On martingales in the limit. Ann. Probability 8, 602-606. Billingsley, P., (1979), Probability and measure. Wiley, New York. Blackwell, D. and Duhins, L.E., (1963), A converse to the dominated convergence theorem. Illinois J. Math. 7, 508-514. Blake, L.H., (1970), A generalization of martingales and two consequent convergence theorems. Pacific J. Math. 35, 279-283. Blake, L.H., (1978), Every smart is a martingale in the limit. J. London Math. Soc. 18, 381-384. Blake, L.H., (1981), Tempered processes and a Riesz decomposition for some martingales in the limit. Glasgow Math. J. 22, 9-17. Brunel, A. and Sucheston, L., (1976), Sur les smarts faibles ~ valeurs vectorielles. C.R. Acad. Sci. Paris, S~r. A 282, 1011-1014.
47
Brunel, A. and Sucheston, L., (1977), Une caract~risation probabiliste de la s~parabilit~ du dual d'un espace de Banach. C.R. Acad. Sci. Paris, S~r. A 284, 1469-1472. Chacon, R.V., (1974), A"stoppe~'proof of convergence. Adv. in Math. 14, 365-368. Chacon, R.V. and Sucheston, L., (1975), On convergence of vector-valued asymptotic martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33, 55-59. Chen, R., (1976a), A simple proof of a theorem of Chacon. Proc. Amer. Math. Soc. 60, 273-275. Chen. R., (1976b), Some inequalities for randomly stopped variables with applications to pointwise convergence. Z. Wahrscheinlichkeitstheorie verw. Gebiete 36, 75-83. Doob, J.L., (1953), Stochastic processes. Wiley, New York. Edgar, G.A. and Sucheston, L., (1976a), Amarts: A class of asymptotic martingales. A. Discrete parameter. J. Multivariate Anal. 6, 193-221. Edgar, G.A. and Sucheston, L., (1976b), Amarts: A class of asymptotic martingales. B. Continuous parameter. J. Multivariate Anal. 6, 572-591. Edgar, G.A. and Sucheston, L., (1976c), Les amarts: Une classe de martingales asymptotiques. C.R. Acad. Sci. Paris, S~r. A 282, 715-718. Edgar, G.A. and Sucheston, L., (1976d), The Riesz decomposition for vectorvalued amarts. Bull. Amer. Math. Soc. 82, 632-634. Edgar, G.A. and Sucheston, L., (1976e), The Riesz decomposition for vectorvalued amarts. Z. Wahrscheinlichkeitstheorie verw. Gebiete 36, 85-92. Edgar, G.A. and Sucheston, L., (1977a), Martingales in the limit and amarts. Proc. Amer. Math. Soc. 67, 315-320. Edgar, G.A. and Sucheston, L., (1977b), On vector-valued amarts and dimension of Banach spaces. Z. Wahrscheinlichkeitstheorie verw. Gebiete 39, 213-216.
48
Fisk, D.L., (1965), Quasi-martlngales. Trans. Amer. Math. Soc. 120, 369-389. Ghoussoub, N. and Sucheston, L., (1978), A refinement of the Riesz decomposition for amarts and semiamarts. J. Multivariate Anal. 8, 146-150. Gut, A., (1979), Moments of the maximum of normed partial sums of random variables with multidimensional indices. Z. Wahrscheinlichkeitstheorie verw. Gebiete 46, 205-220. Gut, A., (1982), A contribution to the theory of asymptotic martingales. Glasgow Math. J. 23, 177-186. Klass, M.J., (1974), On stopping rules and the expected supremum of Sn/a n
and
ISnl/a n . Ann. Probability 2, 899-905.
Krengel, U. and Sucheston, L., (1978), On semiamarts, amarts and processes with finite value. Advances in Prob. 4, 197-266. Lo~ve, M., (1963), Probability theory, 3rd ed., Van Nostrand, Princeton. Mertens, J.F., (1972), Th~orie des processus stochastiques g~n~raux; applications aux surmartingales. Z. Wahrscheinlichkeltstheorie verw. Gebiete 22, 45-68. Meyer, P.A., (1966), Probabilit~s et Potentiel. Paris, Hermann. Meyer, P.A., (1971), Le retournement du temps, d'apr~s Chung et Walsh. S~minaire de probabilit~s V, Lecture Notes in Math. 191, SpringerVerlag, Berlin, 213-236. Mucci, A.G., (1973), Limits for martlngale-like sequences. Pacific J. Math. 48, 197-202. Mucci, A.G., (1976), Another martingale convergence theorem. Pacific J. Math. 64, 539-541. Neveu, J., (1975), Discrete parameter martingales. North-Holland Publ. Co. Orey, S., (1967), F-processes. Proc. 5th Berkeley Symposium on Math. Stat. and Prob., Vol. If.l, 301-313. Rao, K.M., (1969), Quasi-martingales. Math. Scand. 24, 79-92.
49
Rao, M., (1970), On modlfication theorems. Preprint, Arhus univ. Subramanian, R., (1973), On a generalization of martingales due to Blake. Pacific J. Math. 48, 275-278. Sudderth, W., (1971), A "Fatou equation" for randomly stopped variables. Ann. Math. Statist. 42, 2143-2146.
Klaus
D.
A m a r t s
a
Schmidt:
-
m e a s u r e
t h e o r e t i c
a p p r o a c h
I-'-
,¢
I-'-
0
C o n t e n t s
I.
Introduction
2.
Real
2.1.
Measures
2.2.
Set
2.3.
Martingales
.
amarts
55
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
.... .............
function
processes
.... .........
...........
............................
.......................................
2.4.
Submartingales
2.5.
Amarts
and
quasimartingales
...............
............................................
2.6.
Semiamarts
2.7.
Remarks
3.
Amarts
in
3.1 o
Vector
measures
3.2.
Set
3.3.
Martingales
.... ....................................
Strong
3.5.
Uniform
3.6.
Weak
62 69 74 81 89 106 112
a
123
Banach
..........................
................................... .................
, ..........
125 136
.......................................
142
.....................................
145
....................................
149
amarts
amarts,
space
processes
amarts
weak
.
...........................................
function
3.4.
and
......................................
un{form
sequential
weak amarts
amarts, ........................
157
...........................................
165
a
167
3.7.
Remarks
4.
Amarts
in
4.1.
Vector
measures
4.2.
Set
function
Banach
lattice
........................
...................
processes
... . . . . . . . . . . . . .
................
............
169 177
54
4.3.
Submartingales
4.4.
Uniform
4.5.
Weak
4.6.
Order
4.7.
Remarks
5.
Further
amarts
amarts
and
amarts
and
positive
positive
strong
weak
potentials
potentials
.....
..........
......................................
...........................................
aspects
A~endix
of
on Banach
References
Index
....................................
amart
193 196 200 207
...................
211
.......................
215
theory
lattices
179
........................................
.............................................
217
233
I.
I n t r o d u c t i o n
It is the i n t e n t i o n approach
of these notes
to p r e s e n t
to the t h e o r y of a m a r t s w h i c h
integers.
This m e a n s
set f u n c t i o n s than amarts
that we shall
which
stochastic
processes.
This m e a s u r e
form the natural
theoretic
by the very d e f i n i t i o n of s t o p p e d
approach
r a n d o m variables,
role p l a y e d by m e a s u r e
by p a s s i n g
arguments
Radon-Nikodym
derivatives set f u n c t i o n
then p o s s i b l e stochastic
suitable
conditions
to p r o v e p o i n t w i s e
process.
or g e n e r a l i z a t i o n s
for a m a r t s
of r a n d o m variables.
There
is also a h i s t o r i c a l
approach
to amarts.
originated
While
suggested
on the set f u n c t i o n
in f a v o u r
the general
from the p a p e r s by Austin,
methods. to their
a stochastic process,
it is
theorems
for the induced
obtained
in this way are
convergence
of the m e a s u r e
theory of a m a r t s Edgar,
theory
to the u n d e r l y i n g
induces
of the w e l l - k n o w n
argument
of
the structure
set f u n c t i o n s
respect
theorems
of a net
by the i m p o r t a n t
theoretical
process
convergence
The c o n v e r g e n c e
equivalents
with
to
in the i n v e s t i g a t i o n
additive
measure,
Under
counterpart
is c e r t a i n l y
processes,
by p u r e l y m e a s u r e
probability process.
theoretical
and it is a l s o m o t i v a t e d
generalized
rather
of set f u n c t i o n
in terms of the e x p e c t a t i o n s
from b o u n d e d
each
derivatives,
and their expectations.
in the p r o p e r t i e s
of set f u n c t i o n
of amarts c a n be d e v e l o p e d Furthermore,
Radon-Nikodym
measure
theoretical
In the f r a m e w o r k
of b o u n d e d a d d i t i v e
to a m a r t t h e o r y
of amarts
theoretic
are i n d e x e d by the p o s i t i v e
random variables
we will be i n t e r e s t e d
processes,
a measure
study a m a r t s
and their g e n e r a l i z e d
of i n t e g r a b l e
More generally,
amarts.
.
theorems
theoretic
of r a n d o m v a r i a b l e s
and Ionescu
Tulcea
[3] and
56
Chacon
[32] in 1974, amarts of set functions had been studied earlier,
by C h a t t e r j i
[35] in 1971, and by Lamb
[90] in 1973. A l t h o u g h C h a t t e r j i ' s
c o n d i t i o n on a set f u n c t i o n process as well as Lamb's d e f i n i t i o n of an a p p r o x i m a t e m a r t i n g a l e are f o r m u l a t e d w i t h o u t
(explicit)
use of stopping
times, they can easily be seen to be e q u i v a l e n t to the amart condition.
In the literature,
the theory of m a r t i n g a l e s and their g e n e r a l i z a t i o n s
was from an early stage on a c c o m p a n i e d by the i n v e s t i g a t i o n of set f u n c t i o n processes. A m o n g the m o s t i n t e r e s t i n g papers on this topic are those by A n d e r s e n and Jessen 1955, J o h a n s e n and K a r u s h Chatterji Lamb
[35,36,37]
[1,2] in 1946 and 1948, Bochner
[85,86]
in 1971,
in 1963 and 1966, Uhl
1973, and 1976, Rao
[22] in
[128] in 1969,
[107] in 1971, and
[90] in 1973. Most of these authors c o n s i d e r e d c o u n t a b l y a d d i t i v e
set f u n c t i o n processes. studied by B o c h n e r
F i n i t e l y a d d i t i v e set f u n c t i o n p r o c e s s e s were
[22], C h a t t e r j i
[35,36,37], and Schmidt
For the reasons of s i m p l i c i t y and generality,
[111-118].
the finitely a d d i t i v e case
seems to be the really i n t e r e s t i n g one and will be studied in these notes.
The i n v e s t i g a t i o n of p o i n t w i s e c o n v e r g e n c e of stochastic p r o c e s s e s in the f r a m e w o r k of set f u n c t i o n p r o c e s s e s
is m a r k e d by two f u n d a m e n t a l
ideas w h i c h have their origin in the work of A n d e r s e n and J e s s e n
[1,2].
The first of these ideas is to c o n s t r u c t a g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e for b o u n d e d a d d i t i v e set functions w h i c h need not be a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t to the u n d e r l y i n g p r o b a b i l i t y measure, and to study the p o i n t w i s e c o n v e r g e n c e of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a set f u n c t i o n process.
The second idea is to identify
the limit of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a set function process w i t h the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e of the limit of the set f u n c t i o n process.
In a sense, the first idea is a c o n s e q u e n c e
of the second one: Even in the case of a set f u n c t i o n process w h i c h results from i n t e g r a t i n g a stochastic process,
the limit m e a s u r e may
possess a n o n t r i v i a l singular part, and in this s i t u a t i o n the idea of i d e n t i f y i n g the limit of the stochastic process e n f o r c e s the c o n s i d e r a t i o n of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e of the limit of the set f u n c t i o n process.
It is then natural to g e n e r a l i z e one step further,
and to study the p o i n t w i s e c o n v e r g e n c e of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a general set f u n c t i o n process w h i c h n e e d not result from i n t e g r a t i n g a stochastic process. c o n c e r n e d real martingales,
The w o r k of A n d e r s e n and J e s s e n
and it was e x t e n d e d to submartingales,
amarts, and v e c t o r - v a l u e d p r o c e s s e s by J o h a n s e n and K a r u s h Chatterji
[35,36,37], Lamb
and others.
[1,2]
[90], Rao
[107], Schmidt
[85,86],
[113,116,117,118],
57
For f i n i t e l y derivative
additive
was first c o n s t r u c t e d
the p r o b l e m
to the c o u n t a b l y
decomposition. construction
lattice
the g e n e r a l i z e d
Radon-Nikodym
:
with
its g e n e r a l i z e d
the c l a s s i c a l lattice
Radon-Nikodym
the theory
of set f u n c t i o n
consider
By T h e o r e m vector
lattice.
the class
Therefore,
amar t of set functions. derivatives
of
Radon-Nikodym
~v ~
and it follows
=
methods
b a s e d on the d i f f e r e n c e Fatou-Chacon-Edgar
and S u c h e s t o n Riesz
[18,19].
[33]
decomposition
and in d e d u c i n g inequality.
for amarts
For p r o v i n g
Radon-Nikodym
derivatives
X v Y
Bv~
Y
and
~
.
is a
Radon-Nikodym
, the g e n e r a l i z e d
converges
we have
a.e.
theorem,
and an i n e q u a l i t y
of the
[64], and
introduced
It c o n s i s t s
of the p o t e n t i a l
is
[32], and it
[54], E g g h e
m e t h o d was
by C h a c o n
in p r o v i n g
and a p o t e n t i a l
a
part,
part from a m a x i m a l
for the g e n e r a l i z e d
of an a m a r t of set functions,
.
is a b o u n d e d
n 6~
is due to C h a c o n
theorems
and
One of these m e t h o d s
case:
of
,
into a m a r t i n g a l e
convergence
~
the a m a r t c o n v e r g e n c e
in the v e c t o r - v a l u e d
the c o n v e r g e n c e
X
lattice h o m o m o r p h i s m ,
by E d g a r
A different
applicability
the g e n e r a l i z e d
ones.
vector
of the
of set f u n c t i o n s
for all
of
processes.
process
for amarts
further
a wide
amarts
process
This a p p r o a c h
been d e v e l o p e d
B e l l o w and E g g h e
continuous)
. This p r o p e r t y
D n ( B n v ~n )
interesting
property
type.
Since,
for p r o v i n g
there are two p a r t i c u l a r l y
has r e c e n t l y
=
extension
of set functions,
2.5.14,
is a v e c t o r
D n B n v Dn~ n
on the a l g e b r a
of r a n d o m variables,
amarts
that the s t o c h a s t i c
the v a r i o u s
(necessarily
the set f u n c t i o n
Dn
operator
is the u n i q u e
to stochastic
amarts
a.e.
of
transparent.
set f u n c t i o n
guarantees
By C o r o l l a r y
the
is b a s e d
the p r o p e r t i e s
Radon-Nikodym
ba(F, ~)
of all b o u n d e d
converge
operator
X n v Yn
Among
bounded
we o b t a i n b o u n d e d
2.5.3,
to a
processes
in w h i c h
,
of
operator
derivative
also b e c o m e
additive
operator
direct
This c o n s t r u c t i o n
derivative,
on the w h o l e
who r e d u c e d
the Y o s i d a - H e w i t t
a more
from w h i c h
derivative
each b o u n d e d
generalized
By integration,
arguments
• LI(~, ~)
Radon-Nikodym
As an example,
Radon-Nikodym
Radon-Nikodym
homomorphism
present
that the g e n e r a l i z e d
ba(F, ~)
which associates F
we shall
Radon-Nikodym
[35,36]
case by u s i n g
is not needed.
theoretical
In fact, we shall p r o v e
D
by C h a t t e r j i
of the g e n e r a l i z e d decomposition
the g e n e r a l i z e d
additive
In these notes,
Yosida-Hewitt on v e c t o r
set functions,
this m e t h o d
58
seems
to be the a p p r o p r i a t e
Another
a s p e c t of a m a r t
between
the p r o p e r t i e s
and the p r o p e r t i e s For example,
one and will be used
theory
is the c l o s e
of a m a r t s
respectively,
and these c h a r a c t e r i z a t i o n s
More precisely,
certain
can be c h a r a c t e r i z e d for amarts between first
certain
of B a n a c h which
classes
clarify
the r e l a t i o n s
The m e a s u r e
authors
these n o t e s
between
In the f o l l o w i n g
aspects Amart
set f u n c t i o n
chapters,
of these c h a p t e r s
set f u n c t i o n s processes. aspects
Throughout of B a n a c h
will
we shall
be d e v o t e d
processes
of set f u n c t i o n
of rather e l e m e n t a r y
of amart
theory w h i c h
these notes,
theory a c t u a l l y
we shall
Banach
lattices
theory,
lattice.
set f u n c t i o n
amarts
The f i r s t
sections
processes the analysis.
in a
section
of each
discussion
contain
of
all results
in the i n v e s t i g a t i o n
on
of set f u n c t i o n
indicate
some further
the scope of these notes.
frequently
apply
information
results
on B a n a c h
Some d e f i n i t i o n s
may a l s o b e f o u n d
at the end of these notes.
to show in
and f u n c t i o n a l
we shall b r i e f l y
[109].
stimulated
and we hope
serve as a link b e t w e e n
are b e y o n d
For detailed
theory b u t c e r t a i n l y
to a rather d e t a i l e d
These
w h i c h will be n e e d e d
to the b o o k by S c h a e f e r
latti c e s
characterizations
study real amarts,
in a B a n a c h
set functions.
lattices.
of specific
type
in the f r a m e w o r k
properties number
processes,
measure
In the final chapter,
of a m a r t
several
various
of the
to amarts may be looked at as a p o i n t
certain
processes,
and a m a r t s
additive
relation
those of the s e c o n d
in the c o n t e x t of a m a r t theory,
t h e o r y of s t o c h a s t i c
bounded
of a subset
in terms of set f u n c t i o n
only a l i m i t e d
its full variety. to study
that,
space,
lattices
theorem
the c h a r a c t e r i z a t i o n s
include
lattices
ones
operators.
and B a n a c h
of a c o n v e r g e n c e
ones,
into their own r i g h t and m a y also
Banach
spaces
summing
and can be o b t a i n e d
We shall
approach
enlightens
encompass
several
While
can be
in t h e i r proofs.
theoretic
of v i e w w h i c h cannot
involve
nature
itself.
to c o r r e s p o n d i n g
or by the v a l i d i t y
probabilistic
processes.
and w h i c h
of B a n a c h
of amarts.
theoretical
counterexamples
refer
class,
spaces and B a n a c h
processes
extend
lattice
strong p o t e n t i a l s ,
and cone a b s o l u t e l y
properties
type are t y p i c a l l y
of set f u n c t i o n
and p o s i t i v e
e i t h e r by the v a l i d i t y
of a c e r t a i n
have a m e a s u r e
come
operators
lattice
spaces and A L - s p a c e s
in terms of strong a m a r t s
summing
existing
space orl B a n a c h
space or B a n a c h Banach
characterized
for a b s o l u t e l y
relationship
in a B a n a c h
of the B a n a c h
finite d i m e n s i o n a l
in these notes.
of the t h e o r y
lattices,
we
and properties
in the a p p e n d i x
on B a n a c h
59
Let
us n o w
Let
~
their then
fix
be
a set.
union its
some
will
Let
F
be an
F(A)
mutually
A
and
in
algebra
::
~
B
be
•
~
{ B6
sets
A + B
will
on
F
used
subsets
is a s u b s e t
A
denoted
by
will
each
I B c_ A
}
finite
throughout
. If
{0,1}
F(A)
be
disjoint
. For
A
in
will
are
by
~
is a n a l g e b r a .
disjoint
which
denoted
function
F(A)
Then
If be
complement
characteristic
notation
Ac
be
set
of
or
denoted
these
~
, then
of
~A
notes.
~
,
, and
by
XA
its
.
A £ F , define
collection
{AI,A2,...,Ak}
is a p a r t i t i o n
of
A
in
of
F
if
it
satisfies k l i=I Let
P(A)
P(A)
A
=
denote
the
is d i r e c t e d
F-measurable
A
1 class
upward
simple
of
in
all
the
function
if
partitions
obvious it c a n
way.
of
A
A map
be written
in g
in
F . The
: ~ the
> ]R
class is a n
form
k g
where are
scalars.
functions. vector
which
With
lattice
AM-space
An
with
be
unit
called
[
F
be
unit X~
of
Q
in
F
and
~I'
~2'
X~
, and
its
. Following each
set
universal
{ Fn
~
sup-norm
Graves
A 6 F
vector
a stochastic
:=
is a n
I n 6~
will
U n £~ algebra
"'''
~k
]) d e n o t e t h e c l a s s of a l l F - m e a s u r a b l e simple o to t h e s u p - n o r m , t h e c l a s s D o is a n M - n o r m e d
its
[79],
completion the
map
characteristic
measure
on
F
D X
be
: F
.
}
called
basis
on
Fn
on
~
.
a stochastic
~
. Define
basis
on
is a n
function
sequence
on
F F
the
,
is a p a r t i t i o n
Let
with
:=
of a l g e b r a s
Then
~iXA. l
respect with
associates
increasing
Let
Z i=1
{ A I , A 2 ..... A k} (real)
will
=
Q
.
• • XA
, ,
80
A map
T
: ~
> ~U
{T=p}
holds
f o r all
for
[
Let
, and
T
as w e l l
it is b o u n d e d
denote with
containing
F
time
for
F
if
if the v a l u e
~(~)
. Endowed
lattice
is a s t o p p i n g
6 Fp
p 6~
sup~
is finite.
{~}
~
:=
the c l a s s
of all b o u n d e d
the p o i n t w i s e
defined
. For each bounded
{ A£
F
I A n { T = p } 6 Fp
stopping
order,
times
the c l a s s
stopping
time
for all
p6~
T
T 6 T
}
is a
, define
,
as
l~(r)
:=
{ n q]N
[ T < n )
T(T)
:=
{ (~£T
I T < c }
and .
Then F is an a l g e b r a on ~ , and T(T) T For a stopping time T £ T U {~} a n d a set FT(A) and
and
PT(A)
in the
o 6 T(~) U {~}
Almost
all
on a p a r t i c u l a r define,
n 6~
K(n)
:=
Now define
~
algebra
[0,1)
on
:=
Bn, k
k 6 K(n) standard
. The
stochastic
basis
and,
which
we
• £ T
and
PT(A)
processes
shall
,
will
construct
A 6 FT
~ Pa(A)
be b a s e d now.
First
.
for all
n 6~
is g e n e r a t e d
basis
it w i l l
basis
~
on
[0,I)
:= { F n
set.
F
n
to be
the
sets
,
I n 6~
}
will
be c a l l e d
.
a l s o be c o n v e n i e n t
on a o n e - p o i n t
, define
b y the
[ ( k - l ) 2 - n , k 2 -n)
stochastic
basis
Fa(A)
set f u n c t i o n
{I ,2,... ,2 n}
which
:=
Fr(A ) ~
~(T)
the classes
,
[0,1)
stochastic
In some c a s e s ,
concerning
stochastic
for all
have
containing
, define
T F o r all
same w a y as above.
, we then
examples
is a l a t t i c e A £ F
to c o n s i d e r
the
trivial
the
2.
Real
The theory sense,
ama
r t s .
of real amarts
the b e s t p o s s i b l e
may be r e g a r d e d solution
of all b o u n d e d m a r t i n g a l e s pointwise
convergence
to a B a n a c h
obtains.
as a s a t i s f a c t o r y
lattice
This e x t e n s i o n
of p r o c e s s e s
and it is o n l y p a r t l y
submartingales
In the f r a m e w o r k
amarts
2.3),
of set f u n c t i o n structure
(Section
2.5).
on
derivative
(finitely
we shall
properties
and q u a s i m a r t i n g a l e s
The s t r u c t u r e
martingales
to
successively of m a r t i n g a l e s
(Section
t h e o r y of g e n e r a l i z e d results
measure.
process
We shall c o n c l u d e 2.7.
will be i n t r o d u c e d
additive)
measures
the c o n s t r u c t i o n
of a b o u n d e d
probability
Section
processes,
by some a d d i t i o n a l
processes
which also c o n t a i n s
function
from
fails to be
2.4),
and
martingales
on s e m i a m a r t s
in
2.6.
Set function results
solved by g e n e r a l i z i n g
and c o n v e r g e n c e
submartingales
will be c o m p l e m e n t e d Section
for w h i c h
or q u a s i m a r t i n g a l e s .
study the b a s i c (Section
usually
in some
the class
problem originates
the fact that the c l a s s of all b o u n d e d m a r t i n g a l e s a lattice,
and,
to the p r o b l e m of e x t e n d i n g
measure
with
The g e n e r a l i z e d
will be the o b j e c t
this c h a p t e r w i t h
in Section
2.2. The n e c e s s a r y
will be d e v e l o p e d
of the g e n e r a l i z e d respect
to a
(countably
Radon-Nikodym
2.1
additive)
derivatives
of all p o i n t w i s e
some remarks
in Section
Radon-Nikodym
of a set
convergence
and c o m p l e m e n t s
theorems.
in
2.1.
The
M e a s u r e s .
principal
generalized measure This
purpose
with
respect
construction
measures, measures norm.
which
to a
on an a l g e b r a
operator
the A L - s p a c e
F
with
additive)
on the L e b e s g u e
of all
each bounded
measure
extension
~(A+B)
=
> ~
~(A)
for e a c h p a i r
fact
the b o u n d e d
to the v a r i a t i o n
in p r o v i n g
its g e n e r a l i z e d
of the c l a s s i c a l vector
measure. for b o u n d e d
that
respect
additive)
that
the
Radon-Nikodym
Radon-Nikodym
lattice
homomorphism
on
measures.
on a set
: F
probability
also crucial
continuous)
bounded
be an a l g e b r a
~
with
are
of the
(finitely
decomposition
f r o m the
f o r m an A L - s p a c e
(necessarily
A set f u n c t i o n
holds
(countably
of A L - s p a c e s
is the u n i q u e
to a
is the c o n s t r u c t i o n of a b o u n d e d
is b a s e d
The properties
derivative
section
derivative
in t u r n c a n be d e d u c e d
map associating
Let
of this
Radon-Nikodym
~
.
is a d d i t i v e
if the
identity
+ ~(B}
of d i s j o i n t
sets
A,
B £ F , and
it is b o u n d e d
if
the v a l u e
suPF is finite. Let
with
In the
a(F, ~)
ba(F, ~ )
tit(A) i sequel,
denote
denote
these
2. I. I. The c l a s s
the c l a s s
the c l a s s
the p o i n t w i s e
order,
additive
defined
classes
are
set f u n c t i o n s
of all m e a s u r e s
of all b o u n d e d addition,
ordered
will F ....~
measures
in
multiplication
vector
by
be c a l l e d , and
scalars,
Lemma. ba(F, 3R)
is a v e c t o r
lattice,
a n d the
(~v~) (A)
=
suPF(A )
(~(B) +~0(A~B))
(~t^~) (A)
=
infF(A)
(B(B) + ~0(A~B) )
identities
and
hold
for all
~, ~ £ ba(F, ~ )
I
;
I~1 (~)
and
A £ F . Moreover,
let
a(F, ~ )
spaces.
the map
measures.
. Endowed and
63 is a l a t t i c e
Proof.
Then
~
n o r m on
Consider
B, ~ 6 ba(F, 3~)
~(A)
supF(A )
: F
:= > ]R
is a d d i t i v e , BE
F(A)
ba(F, ~ )
,
A £ F
B I := A I N B ~(B)
set
and
and
+ ~(A~B)
A £ F , define
(~(B) + ~(A~B))
is a b o u n d e d
fix
. F o r all
function.
B 2 := A 2 N B
=
In o r d e r
{ A I , A 2} £ P(A)
to see t h a t
. T h e n w e have,
for all
,
~(B I) + ~(AI~B I) + ~(B 2) + ~(A2~B 2) ~ ( A I ) + ~ ( A 2)
,
hence
~(A)
Conversely,
~
~ ( A I) + ~ ( A 2)
for all
C I £ F(A I)
,
,
C 2 E F(A 2)
~(C I) + ~ ( A I ~ C I) + ~ ( C 2)
and
+ ~ ( A 2 ~ C 2)
C
:= C I + C 2 , we h a v e
=
~(C)
<
~(A)
+ ~(A~C)
,
hence
~ ( A I) + ~ ( A 2) Therefore, least
identity. -~,
bound
The
-~£ba(F,
lattice, on
we h a v e
upper
and
~(A)
~ C ba(F, ~) of
B
second ~)
<
and
~
identity
and
~^~
it o b v i o u s
, and in
it is o b v i o u s ba(F, ~ )
follows
f r o m the
= -(-~)v(-~)
that
the m a p
is the the f i r s t
one because
ba(F, ~)
l~l(~)
~
proves
first
. Thus ~ ~ >
that
. This
of
is a v e c t o r
is a l a t t i c e
norm
ba(F, m )
L e t us r e m a r k
that
the v a r i a t i o n
norm.
I~I(A) holds
for all
Lemma
2.1.1,
the n o r m
=
~.i(~)
on
More generally,
the
supp(A ) Z
~ E ba(F, ~) we have
the
and
ba(F, ~ )
is i d e n t i c a l
identity
l~(Ai) l A £ F . As an i m p r o v e m e n t
following
well-known
result:
of
with
64
2.1.2.
Theorem.
The class
ba(F, ~)
is an A L - s p a c e for the n o r m
Proof.
The n o r m
ba(F, ~)
is c o m p l e t e for the n o r m
l.I(~)
is an L-norm,
I.~(~)
and the v e c t o r lattice
l.I(~)
This result will prove to be useful in the sequel since the p r o p e r t i e s of A L - s p a c e s lead to a simple proof of the L e b e s g u e d e c o m p o s i t i o n for m e a s u r e s in
ba(F, ~)
and also serve to p r o v e that the g e n e r a l i z e d
R a d o n - N i k o d y m d e r i v a t i v e for m e a s u r e s in
ba(F, ~)
is the b e s t p o s s i b l e
g e n e r a l i z a t i o n of the c l a s s i c a l R a d o n - N i k o d y m derivative.
If
~
is a m e a s u r e in
~-continuous ~J(A) Thus,
< 6
ba(F, ~)
if for each implies
, then a m e a s u r e
e £ (0,~)
I~(A)
~ 6 ba(F, ~)
there exists
66
(0,~)
< c , and it is @ - s i n g u l a r if
is
such that I~^I~I
= 0 .
s i n g u l a r i t y of a b o u n d e d m e a s u r e w i t h respect to a n o t h e r one is
n o t h i n g else than an a s y m m e t r i c f o r m u l a t i o n of the o r t h o g o n a l i t y of these measures.
For
~ 6 ba(F, JR) , let
all ~ - c o n t i n u o u s m e a s u r e s in 2.1.3.
ba~(F, ~)
ba(F, ~)
Lemma.
{~}l holds for all
Proof.
=
ba~(F, m ) l
~Eba(F,
~)
°
We c l e a r l y have
Conversely,
consider
choose
66
choose
(0,~)
A6 F
and t h e r e f o r e
{~} c ba~(F, ~)
B E {~}I
and
such that
such that
J~l(A) + l ~ i ( ~ A )
I~I(A)
~(A)
< e , hence
, hence
~Eba~(F, < 6
+ l~l(Q~A) l~I^[~l
2.1.4.
Theorem. ~ C ba(F, JR) , the classes
and p r o j e c t i o n bands in ba~(F, JR)
ideals in
ba~(F, ~ ) I
. Fix
I~I (A) < e/2
< min {6,E/2}
= 0 , since
E
~ {~}±
e £ (0,~)
,
, and
. Then we have
was arbitrary,
~ E ba~(F, ~) I .
For each
Proof.
~)
implies
u
The n e x t result is the L e b e s @ u e d e c o m p o s i t i o n
sum of
d e n o t e the class of
and
ba(F, JR) . Moreover,
and
{~}I
ba(F, JR)
are A L - s p a c e s is the d i r e c t
{~}I .
It is easy to check that ba(F, ~)
ba~(F, ~)
for b o u n d e d measures:
. Since
ba~(F, ~)
ba(F, ~)
and
{~}I
is an AL-space,
are closed
these c l o s e d
85
ideals
are p r o j e c t i o n
complete, and
hence
bands.
Being
it is the d i r e c t
ba~(F, ~ ) ±
an AL-space,
ba(F, ~)
sum of the p r o j e c t i o n
. N o w the final a s s e r t i o n
follows
is o r d e r
bands
ba~(F, ~)
from the p r e v i o u s
lemma,
s
: F
A measure
~
E ~ ( A n) n=l holds X A
=
F
of m u t u a l l y
ca(F, ~ ) > ~
that a c o u n t a b l y
additive
if the i d e n t i t y
~( E A n) n=l
for each s e q u e n c e 6 F . Let
n measures
F
is c o u n t a b l y
denote
, and define additive
disjoint
sets
An £ F
satisfying
the class
of all c o u n t a b l y
bca(F, ~)
:= ba(F, ~) A ca(F, ~)
. Note
need not be b o u n d e d
unless
measure
F
> ~
additive
is a a-algebra.
2.1.5.
Theorem.
The class
bca(F, ~)
is an A L - s p a c e
and a p r o j e c t i o n
band
in
ba(F, ~)
We omit the easy proof.
Let
~
denote
the o - a l g e b r a
ca(F, JR)
The map a s s o c i a t i n g extension
=
by
F . Then we have
bca(F, JR)
w i t h each p o s i t i v e
to a p o s i t i v e
positively
generated
homogeneous.
measure
in
measure
ca(F, ~)
It t h e r e f o r e
in
bca(F, ~ )
is c l e a r l y
has a u n i q u e
its u n i q u e
additive
extension
and
to a p o s i t i v e
linear m a p
J
2.1.6.
Proof.
bca(F, ~)
> ca(~, ~)
Theorem.
The map onto
:
J
is an isometric
vector
lattice
isomorphism
of
bca(F, ~)
ca(T, ~)
Consider
~£bca(F,
~+(A)
(J~)+(A)
Conversely,
for
<
B £ ~(A)
IJ~i(BAC)
<
and
e
~)
and
e 6 (0,~)
A £ F . Clearly,
, choose
C £ F
we have
such that
66
[80; T h e o r e m
13.D].
(J~) (B)
Then we have
=
(J~)
(BnC)
+
(J~)
(BnC c)
=
(J~) (AnC)
-
(J~) (AnBcnc)
<
~(AnC)
+
<
~+ (A)
+ ~
+
(J~) (BnC c)
IJ~I(BAC)
,
hence
(JB) + (A)
Therefore,
(J~)+
IJ~l
which
<
means
~+ (A)
is the e x t e n s i o n
=
2(JB) + - J ~
that
J
Corollary.
For
~ E bca(F, ~)
each
vector
Proof.
First
additive.
Consider
such
that
note
I~I(C)
C E F
< 6
such
8/2
~)
of
of
=
lattice
J
to
I~I(C)
e £ (0,~) < e/2
+
ba~(F, ~) onto
<
min
since
. For
{8/2,e/2}
2.1.6,
=
IJ~I(CnA)
+
<
IJ~I(A)
IJ~I(AAC)
<
8
•
+
I J ~ I ( C n A c)
Jl~l
,
AE;
.
is an
b a J @ ( ~ , ~)
~
is c o u n t a b l y
, and choose
,
IJ~I(AAC)
=
isomorphism.
b a @ ( F , ~)
c bca(F, ~)
, fix
yields
J(2~+-~)
that
by Theorem
l~l(C)
ba~(F, ~)
implies
<
IJ~I(AAC)
T h e n we have,
that
, and this
vector
restriction
isomorphism
BEba@(F,
IJ~I(A)
choose
, the
lattice
~+
2J~ + - J~
is an i s o m e t r i c
2.1.7.
isometric
--
of
66
(0,~)
satisfying
67 hence
I~I(C)
z12
,
and t h e r e f o r e
|J~l (A)
=
~J~I(AnC)
<
l~l(C)
<
c
+ IJ~I(ANC c)
+ IJ~I(A~C)
,
as was to be shown.
For the remainder of this section,
:
be a fixed
F
>
let
[0,1]
(countably additive)
p r o b a b i l i t y measure.
By the R a d o n - N i k o d y m
theorem, the map Rl :
baJl(;, ~)
> L1(;,Jl, m)
,
w h i c h a s s o c i a t e s w i t h each J l - c o n t i n u o u s m e a s u r e on d e r i v a t i v e with respect to isomorphism.
~
F r o m this it follows, by C o r o l l a r y 2.1.7,
RI0 J :
bal(F, JR)
its R a d o n - N i k o d y m
Jl , is an isometric v e c t o r lattice that the map
> L!(;,JI, ~)
also is an isometric vector lattice isomorphism.
The map
be called the R a d o n - N i k o d y m o p e r a t o r w i t h respect to Cl :
ba(F, ~)
> bal(F, ~)
Sl :
ba(F, ~)
> {X} ±
Rl o J
will
I . Let
and
denote the band p r o j e c t i o n s g u a r a n t e e d by T h e o r e m 2.1.4, and define Dl
The map
:=
Rio J 0 C l
88
D1 :
ba(F, JR)
> L 1(;,Jl, JR)
will be called the 9 e n e r a l i z e d to
~ . For
~ 6 ba(F, ~)
be called the 9 e n e r a l i z e d to
I . The properties
exhibited
R a d o n - N i k o d y m o p e r a t o r with respect
, the random variable
DI~£LI(~,JI,
R a d o n - N i k o d y m derivative
of the g e n e r a l i z e d
in the following
of
~
Radon-Nikodym
~)
will
with respect operator are
theorem which is the main result of this
section: 2.1.8.
Theorem.
The generalized homomorphism
Radon-Nikodym
ba(F, ~)
operator
Dl
> LI(F,JI, ~)
operator
R 1 o J . Moreover,
Proof.
It is clear from the properties
D1
D1
U : which extends
projection
ba(F, ~)
C1 ,
extending
J
R1 o J
and
R~
that
and that
is
D1
> LI([,JI, ~)
R 1 o J . Then the map (R~ o j)-1 o U
band since
IR1 o j)-1 o U
of
Consider now an arbitrary vector lattice h o m o m o r p h i s m
h o m o m o r p h i s m which is continuous the kernel of
the R a d o n - N i k o d y m
is a contraction.
is a vector lattice h o m o m o r p h i s m
a contraction.
is the unique vector lattice
which extends
since
ba(F, ~)
commutes,
( ( R l o J ) -1 o U ) ~
is a vector lattice
ba(F, ~)
is complete.
is a closed ideal which actually is an AL-space.
is a band projection.
band p r o j e c t i o n s
(Rl o j)-1 0 U
Hence is a
It follows that
Using the fact that every pair of
we obtain,
for all
~ E ba(F, ~)
=
( ( R l o J ) -1 o U o C l ) ~
=
Cl~ + (S ~ o (R~ 0 J)-I o U ) ~
,
+ ( ( R l o J ) -I o U o S l ) ~ ,
hence ((Rlo j)-1 o U)~ - Cl~
Since the left hand side is in in
bal(F, ~ ) ±
and Lemma
(S ~ o (R~ o j)-I o U)~
bal(F, ~)
, both expressions
U~ = ( R l o
JoCl)~
=
of
.
while the right hand side is
must be equal to
2.1.3. Now the application ( (Rk o j)-I o U)~
yields
=
Rl o J
0 , by T h e o r e m
2.1.4
to the identity
Cl~
for all
~£ba(F,
~)
, as was to be shown,
a
2.2.
S e t
In t h i s basic
Let
f u n c t i o n
section,
we
introduce
p r o c e s s e s .
set f u n c t i o n
processes
and establish
their
properties.
F
be a stochastic
basis
on a set
~
.
A sequence
:= will
be c a l l e d
The
concepts
stochastic processes
a set f u n c t i 0 n
of s t o p p i n g
processes. and
Consider A6
{ ~n 6 ba(Fn, ~)
study
process
}
on
and conditioning
We
shall
some
a set f u n c t i o n
I n 6~
now adapt
of t h e i r
process
F
.
are
essential
these
concepts
in the
theory
of
to set f u n c t i o n
properties.
_~
and a bounded
stopping
side
extends
time
T
. For
FT , define
~T(A)
Note
that
finite
the
number
t h a t we h a v e modulus
If
~
Z p=1
~p (AA{~=p})
sum on the r i g h t of t e r m s
since
~T E ba(FT, ~)
(or v a r i a t i o n )
2.2.1. then
:=
of
hand
actually
T
is b o u n d e d .
• The
following
~T
will
be
From
this,
elementary
frequently
only
on the
in the
sequel:
stopping
time,
Lemma. is a set f u n c t i o n
the
process
and
•
is a b o u n d e d
identity
I~TI (A)
=
Z
I~pl
(An{~=p})
p=l holds
Proof.
for all
For • For
C E F P
A E
FT
p E~
and
B E F ({T=p}) T
I ~ I (B)
Cc
{T=p}
, this
, we h a v e
C 6 F
if and o n l y
yields
=
supF~(B)
(~r(C) - ~ T ( B ~ C ) )
=
sUpFp(B)
(~p(C) - ~p(B~C))
=
a
it is a l s o c l e a r lemma
used
over
i~pl (B)
if
70
and,
for
A £ F
, we d e d u c e
T
lu..c I (A)
=
)"
I]J.c I (An{'c=p})
=
)-
p=l as w a s
For
to be
~ E T
,
T E T(~) U {~}
R ~
of
RT M which will
:
we w i l l
be
and
A 6 F
, define
B(A)
in
to
F
with
~ 6 a(FT, ~)
a(F
, ~)
, and
will
be c a l l e d
linear
the
map
, ~)
each measure
interested
which
the p o s i t i v e
• a(F
the r e s t r i c t i o n
in
map
a(FT, ~)
from
in the
its r e s t r i c t i o n
a(FT, ~)
restriction
to
of
a(F
RT
, ~)
to
M
to
F
,
. In m o s t
b a ( F T , ~)
Lemma.
M £ T
and
RTX :
is a p o s i t i v e
Proof.
However,
, the
restriction
b a ( F T , JR)
> b a ( F x , ~)
~£ba(F
, ~)
T IR ~I (A)
=
suPF
(A)
(%I(B) - ~ ( A ~ B ) )
suPF
(A)
(~(B) - ~ ( A ~ B ) )
the
this
yields
restriction as
can
map
contraction.
Consider
homomorphism,
2.2.3.
T E T(~) U {~}
linear
In p a r t i c u l a r ,
be
. T h e n we have,
[RxBI (Q) <
map
RT
seen
need
from
the
for all
=
[~I (Q)
F
,
I~I(A)
, as w a s
n o t be a v e c t o r following
A6
to be
shown.
lattice
example:
Example.
Define which
,
a(FT, ZR)
associates
2.2.2. For
B
be c a l l e d
cases,
:=
is a m e a s u r e
restriction
,
shown.
T (R ~) (A)
Then
lU.pl (An{c=p})
p=l
~
:= ~
and,
is g e n e r a t e d
consists
of all
complement.
for all
b y the
subsets
Now define
of
n 6~
subsets ~
measures
, define of
which
F
n {I,2,...,n}
are
finite
~, ~ E b a ( F
, ~)
to b e the a l g e b r a . Then
of h a v e by
on
the algebra a finite
letting
F
71
:=
~(A)
]
0
,
if
A
is f i n i t e
[
I
,
if
~A
is f i n i t e
and
~(A)
TSen
~
and
:=
~
Z 2 -k k 6 A
are o r t h o g o n a l ,
are not orthogonal,
In the
sequel,
however, algebra
For
be and
we
shall
important
:=
and
is b o u n d e d .
the u p p e r
index
of
carefully
between
R T . It will, x a m e a s u r e on an
are
three ~
different
is ~ - b o u n d e d
notions
of
if the v a l u e
IUTI(n) if the n e t
}
properties
are related
as follows:
Theorem.
a set f u n c t i o n
(a)
~
is an ~ - b o u n d e d
(b)
~
is T - b o u n d e d .
Proof.
Suppose
m
:= m a x ~
process
first
and choose + B~(~)
Define
R~
if the v a l u e
For
6 T
and
l~nl (~)
suPT
I T6T
These
omit
process
it is a s e m i a m a r t
{ B~(Q)
R~B
.
there
it is T - b o u n d e d
is finite;
restrictions
to a s u b a l g e b r a .
su b
:=
their
to d i s t i n g u i s h
processes,
II ~ IIT
2.2.4.
usually
A set f u n c t i o n
il _~ I ~ is finite;
n 6~
its r e s t r i c t i o n
set f u n c t i o n
boundedness.
for all
but
A £ F
<
that such
M
BK(A)
~(~)
~
, the
following
are e q u i v a l e n t :
semiamart.
~
is an ~ - b o u n d e d
semiamart.
Consider
that
+ I
, and define
a stopping
time
u £ T
by
letting
72
=
I
M (co)
,
if
~ £A
[
m
,
if
co 6 ~ A
~(~)
Then we have +
-
1
<
~(A)
+ ~v(n~A)
By(~)
+
sup T
- ~m(~A)
IBml(n)
i~T(~) I + su b
[~nl(~)
+
hence
sup T ~T(Q)
finite.
The
we shall
is finite,
converse
is
see that the m o s t
is that of T - b o u n d e d n e s s . funct i o n
processes,
following
obvious
2.2.5.
Theorem.
and it follows
important In fact,
the m a p
~ ~ >
II. IIT .
We shall also
see that c e r t a i n lattice
Theorem
II. I ~
are ~ - b o u n d e d
2.2.4.
not
is a norm,
processes
other c l a s s e s II. IIT
between
the ~ - n o r m
if and only
in the r e s p e c t i v e
set
and we have
which
is a B a n a c h
of set f u n c t i o n
the
it c a n n o t
we t h e r e f o r e
and T - b o u n d e d n e s s
in g e n e r a l
be
norms.
is ~ - b o u n d e d
need not
For the sake of simplicity, if and only
a
if it is T - b o u n d e d
let
of this chapter,
real
by
so far as we are
For the r e m a i n d e r
be a fixed p r o b a b i l i t y
processes
and the T-norm,
if it is ~ - b o u n d e d .
>
lattice
• The T - n o r m a l s o o c c u r s
will b r i e f l y be said to be b o u n d e d
F
concepts
if they are T-bounded,
In the case of semiamarts,
process
X :
is
•
between ~-boundedness
interested
set f u n c t i o n
II ~ IIT
a n d in b o t h s i t u a t i o n s
In spite of the d i f f e r e n c e
distinguish
of t h e s e b o u n d e d n e s s
for the T - n o r m
inequality,
r e p l a c e d by the ~ - n o r m
semiamarts
IBrl(~)
on the class of all T - b o u n d e d
set f u n c t i o n
for the n o r m
form a B a n a c h
suPT
result:
The class of all T - b o u n d e d
in a m a x i m a l
that
obvious.
[0,1]
measure.
If
is a set f u n c t i o n
process,
then
73
the generalized
Radon-Nikodym
will be denoted by
DnlJ- n
for all
n EIN .
,
derivative
of
~n
with respect to
Rnl
2.3.
M a r t i n 9 a 1 e S .
A set f u n c t i o n is c o n s t a n t . be
seen
process
f r o m the
is a m a r t i n g a l e
following
characterizations
2.3•1.
~
The equivalence
of this theorem
if the n e t
definition
which
also
{ ~T(~)
with
the u s u a l
contains
Theorem• process
~
, the
following
are e q u i v a l e n t :
(a)
~
(b)
~T = R T ~ c
holds
for all
• E T
and
~ £ T(T)
(c)
~n
= RnBm
holds
for all
n 6~
and
m 6~(n)
(d)
~n
= Rn~n+1
(e)
(f)
(g)
Proof• T
further
of m a r t i n g a l e s :
F o r a set f u n c t i o n
A £ F
some
I T 6 T one can
is a m a r t i n g a l e .
There
exists
holds
for all
There
exists
holds
for all
There
exists
holds
for all
Suppose • Define
first
holds
a measure n 6~
r E T
I
.
5 £ a(F
, ~)
such
that
~n = R n ~
~ E a(F
, ~)
such
that
~T
~6 a(F
, ~)
such
that
~T(~)
= R ~
.
a measure T £ T
that
n 6~
.
a measure
a stopping
~(~)
for all
= ~(Q)
.
(a) holds.
time
v £ T
Consider
by
• C T
,
C T(T)
and
letting
T(~)
,
if
~ 6 A
~(~)
,
if
~ £ ~A
:=
Then we have
~T(A)
+ ~(~A)
=
~v(~)
:
~(~)
,
F
, there
exists
, we t h e n h a v e
A £ Fm
hence
~T (A)
Therefore, Suppose
:
~(~ (A)
(a) i m p l i e s
now that
that
A£
~m(A)
= ~n(A)
Fn
(c) holds.
holds.
~(A)
(b). For
F o r all
, hence
:=
lim ~m(A)
each
mE~(n)
A6
n£3~ and
such
}
76
exists for all
A £ F
~n(A)
=
and defines a m e a s u r e
_~
is a martingale,
~(A) A£ F
, JR)
such that
~(A)
holds for all n 61~ and A 6 F . Therefore, n The r e m a i n i n g i m p l i c a t i o n s are obvious.
If
£ a(F
:=
then the m e a s u r e
lim ~n(A)
(c) implies
~ 6 a(F
, JR)
(e).
given by
,
, will be c a l l e d the limit m e a s u r e of
~ . We shall see that
certain p r o p e r t i e s of m a r t i n g a l e s can be e x p r e s s e d by p r o p e r t i e s of their limit measures.
2.3.2. If
~
Theorem. is a m a r t i n g a l e with limit m e a s u r e
II__~ II~ In particular,
=
II__~ IIT
=
~ , then
I~I(~)
a m a r t i n g a l e is b o u n d e d if and only if its limit measure
is bounded.
Proof.
F r o m T h e o r e m 2.3.1 we obtain
II~II~ b e c a u s e of
~ E T
<
ll£11 T
and since
<
l~,l(n)
RT
Lemma 2.2.2. For each p a r t i t i o n n 61~
such that
i=1
J~(A i) ~
=
k Z i=I
for all
{ A I , A 2 , . . . , A k} £ ~ ( ~ )
{ A I , A 2 , . . . , A k} £ Pn(~)
k Z
is a contraction,
I~n(A i) I
~ 6 T , by
, there exists
. This yields
~
i~nl (~)
~
II ~ I ~
,
hence
by taking the s u p r e m u m over
P (~)
.
The close r e l a t i o n s h i p b e t w e e n b o u n d e d m a r t i n g a l e s and their limit m e a s u r e s becomes p a r t i c u l a r l y clear from the f o l l o w i n g result:
78
2.3.3.
Theorem.
The class of all b o u n d e d m a r t i n g a l e s norm
H.
l~
limitmeasure
is an i s o m e t r i c
bounded martingales
Proof.
measure
and T h e o r e m
Thus
F
structure
lattice
F
of
onto
martingale
example:
2.3.4.
Example.
On the s t a n d a r d ~
ba(F of
, ~)
ba(F
n 6~
set f u n c t i o n
stochastic
for the
of the B a n a c h
its
space of all
is u n i q u e l y follows
determined
from T h e o r e m
by a 2.3.2
on
F
it u s u a l l y
inherits
the B a n a c h
does not inherit
the
is due to the fact that the homomorphisms;
the s i t u a t i o n
becomes
see E x a m p l e clear
from the
[0,I) • d e f i n e a set f u n c t i o n
on
k £ K(n) I~I
The m i s s i n g
n E~
lattice
bounded martingales
=
if
2 -n
,
otherwise
~
is a b o u n d e d
. Then
2(I-2 -n)
k = 1
<
martingale,
but the
since
2(1-2 - (n*1))
=
l~n+ I I (~)
.
property which
is a s h o r t c o m i n g
certainly
in m a r t i n g a l e s
theorem•
such an e x t e n s i o n which
,
is not a m a r t i n g a l e
inter e s t
prope r t i e s ,
2 -n - I
:=
and
for all
Section
F
. This
basis
process
l~nl (a)
valid.
space
by letting
~n (Bn,k)
processes
on
, but
, ~)
[
for all
is a B a n a c h
, ~)
. N o w the a s s e r t i o n
For b o u n d e d m a r t i n g a l e s ,
process
ba(F
maps m a y fail to be lattice
following
holds
isomorphism
of all b o u n d e d m a r t i n g a l e s
structure
restriction 2.2.3.
on
F
w i t h each b o u n d e d m a r t i n g a l e
2.1.2.
the class
space
on
Every bounded
bounded
on
, and the map a s s o c i a t i n g
is m a i n l y
contains
should
motivates
Since
the
convergence
lead to a class of set f u n c t i o n
the b o u n d e d m a r t i n g a l e s •
to the e x t e n s i o n
has b e t t e r
convergence problem
theorem
in Section
2.5.
L e t us n o w turn to the m a r t i n g a l e
of all
its extension.
due to the p o i n t w i s e
and for w h i c h the p o i n t w i s e
We shall return
of the class
convergence
theorem.
stability remains
2.4 and in
77
A martingale exists
~
is u n i f o r m l y l - c o n t i n u o u s
6 C (0,~)
2.3.5.
such that
l(A)
< 6
if for each
implies
su b
e E (0,~)
l~n[(A)
there
<
Theorem.
For a b o u n d e d m a r t i n g a l e
~
, the following
and its limit m e a s u r e
are equivalent: (a)
~
is u n i f o r m l y l-continuous.
(b)
~
is l-continuous.
Proof.
As in the proof of T h e o r e m 2.3.2,
it can be d e d u c e d from
T h e o r e m 2.3.1 that
su b holds for all
[~nI(A) A £ F
=
[~[(A)
. From this the a s s e r t i o n follows.
This result leads us to the L e b e s ~ u e d e c p m P g s i t i o n for b o u n d e d martingales:
2.3.6.
Theorem.
Every b o u n d e d m a r t i n g a l e
is the sum of a b o u n d e d u n i f o r m l y l - c o n t i n u o u s
m a r t i n g a l e and a b o u n d e d m a r t i n g a l e w i t h l - s i n g u l a r limit measure. The d e c o m p o s i t i o n is unique.
Proof.
This is an immediate c o n s e q u e n c e of the L e b e s g u e d e c o m p o s i t i o n
for b o u n d e d m e a s u r e s b o u n d e d martingale,
(Theorem 2.1.4), a p p l i e d to the limit measure of a and c o m b i n e d with T h e o r e m 2.3.5.
In the sequel, the limit m e a s u r e of a m a r t i n g a l e
~
[] will sometimes be
denoted by
lim~ n a l t h o u g h this is a slight abuse of notation. The L e b e s g u e d e c o m p o s i t i o n for b o u n d e d m a r t i n g a l e s suggests s p l i t t i n g the proof of the m a r t i n g a l e c o n v e r g e n c e t h e o r e m into two parts:
2.3.7. If
~
Theorem. is a b o u n d e d u n i f o r m l y l - c o n t i n u o u s martingale,
lim Dn~ n
=
D
lim Bn
a.e.
then
78
Proof.
By T h e o r e m
X If
EnX
:=
2.3.5, w e h a v e
~ -- l i m ~n 6 b a l ( F ,
~)
. Define
D 5
denotes
the c o n d i t i o n a l
expectation
of
X
with respect
to
Fn
then we h a v e
Dn~ n for all
nqlq
=
EnX
. By L ~ v y ' s
lim E X n from which 2.3.8. If
_~
,
=
X
theorem
a.e.
the a s s e r t i o n
[42; T h e o r e m
,
follows,
m
Theorem. is a b o u n d e d m a r t i n g a l e
lim DnB n Proof.
--
0
Therefore,
F i r s t n o t e t h a t the A - s i n g u l a r i t y
the g e n e r a l i z e d
zero. F o r all
Xn and choose
with A-singular
of
Bn = R n ~
Radon-Nikodym
of
Dn~ n
Fn-measurable
simple
°f
IXn-Znl
d l J n R n A)
, choose
A £ F
E, 6 E (0,~)
~(A)
<
E6
functions
<
A(~A)
<
6
such that
and
k E~
°I
Z n=k
; see E x a m p l e
n C]N , d e f i n e
:=
,
such that
IXn-Znl Q
then
of the l i m i t m e a s u r e
derivatives
n~1
and choose
limit measure,
a.e.
does not imply RnA-Singularity
Fix
1.4], w e h a v e
A £ Fk
dlJnRnX)
and
<
c6
Zn
such that
2.2.3. _~
need not be
i
79
Now define Bk
:=
and, for all
n 6~(k+I)
Bn This yields,
A n { iZki > E}
::
E
Fk
,
A N {IZni > e} N ( D_k{IZpl < e } p-
for all
m£3(k)
£
Fn
_<
1St (A)
<
m
I
,
m X n=k
IXnl d(JnRnl) B
<
~ n=k
n
i~ni (B n)
m
I~I (B n)
e6
,
n=k hence
im
m
e(J l) k Z B n n=k
=
e
Z n=k
(JnRn l) (B n)
--< n--Xk B IZnl d(JnRn ~) n
<
n[k
B IZn-X nl d(JnRn ~) + nX=k n
<
2e6
,
B IXnl d(JnRn~) n
m
and therefore,
letting
(J l)\n=X k Bn>
m
tend to infinity,
<
26
This yields (J ~ ) ( { s u ~ ( k ) i Z n l which implies lim Z Furthermore,
n
from
=
0
a.e.
> e})
<
(J l)< ~ Bn> + l ( ~ A ) n k
<
36
,
co
e(J ~) ({sup~(k) IXn-Znl > e})
<
e
X n=k
<
Z
--
n=k
<
e8
I
(JnRn A) ({ IXn-Znl > e})
IX -Z ~
n
n
I d(JnRnl)
we obtain
lim Therefore,
(Xn-Z n)
=
0
a.e.
we have lim X
n
=
0
a.e.
as was to be shown. Combining
these results,
we obtain the general martingale
convergence
theorem: 2.3.9. If
~
Corollar~. is a bounded martingale,
lim Dn~ n Proof.
=
D~ lim ~n
then
a.e.
First use the Lebesgue d e c o m p o s i t i o n
(Theorem 2.3.6)
apply Theorem 2.3.7 and Theorem 2.3.8. Now the assertion the linearity of the g e n e r a l i z e d
Radon-Nikodym
operators,
and then
follows from u
2.4.
Subma
r t inca
and
~ua
1 e s
s imar
t inga
le
s
The e x t e n s i o n p r o b l e m for the class of b o u n d e d m a r t i n g a l e s n a t u r a l l y leads to set function processes which are the s u p r e m u m of two b o u n d e d martingales. v ~
If
~
and
~
are martingales,
then the set function
need not be a m a r t i n g a l e since the net
n e e d not be constant; 2.3.4 and
~
for example,
:= -~ . However,
{ ( ~ T v ~ ) (Q)
consider
the net
~
J T 6 T }
as d e f i n e d in Example
{ (~TV~T) (~)
J T £ T }
is
increasing since (~ v~T) (Q)
=
supF
(B~(A) + ~ ( Q ~ A ) ) T
sup F
(~o(A) + ~o(n~A))
=
(~ovmo)(n)
a holds for all
I £ T
and
~ £ T(T)
. This leads to the following
definition: A set f u n c t i o n process if the net
{ ~T(~)
~
is a s u b m a r t i n ~ a l e
J T 6T
}
is i n c r e a s i n g
(resp. supermartingale)
(resp. decreasing).
S u b m a r t i n g a l e s may be c h a r a c t e r i z e d as follows:
2.4.1.
Theorem.
For a set function process
~ , the f o l l o w i n g are equivalent:
(a)
~
(b)
~T ~ R T ~ a
holds for all
T £ T
and
o 6 T(T)
(c)
~n ~ Rn~m
holds for all
n 6~
and
m£~(n)
(d)
Bn ~ Rn~n+1
Proof. implies
is a submartingale.
holds for all
n 6~
.
It can be shown as in the proof of T h e o r e m 2.3.1 that (b). Obviously,
(b) implies
(c), and
(c) implies
Suppose now that
(d) holds. C o n s i d e r stopping times
and define
:= max n T(~)
m(r)
~r(n)
=
m(r) Z p=1
and
m(a)
~p({r=p})
m(T) Z
p=1
~p ({ ~=p}D {o>_p})
:= m a x ~ ~(~)
T 6 T
(a)
(d). and
o £ T(~)
. Then we have
,
82
m(x) ( -<
p:Ir ~p({r=p}N{a=p})+ Bp+1({T=p}N{o>_p+1})h/
<
re(T) re(a) X Z ~ ({r=p}N{c=q}) p=1 q=p
m(o)
q
Z q=l
Z p=l
~q ({r=p}N{~=q})
= ~a(n) Therefore,
(d) implies
(a}.
One of the m o s t e f f i c i e n t c o n c e p t s in the theory of g e n e r a l i z e d martingales
is the Riesz d e c o m p o s i t i o n of a process into a m a r t i n g a l e
and a p o t e n t i a l p a r t w h i c h c o n v e r g e s to zero in some sense. B e f o r e p r o v i n g the Riesz d e c o m p o s i t i o n for ~ - b o u n d e d
s u b m a r t i n g a l e s and
s u p e r m a r t i n g a l e s , we have to define an a p p r o p r i a t e type of potential:
A set f u n c t i o n process { ~(~)
I T £ T }
~
is a Doob p o t e n t i a l if the net
d e c r e a s e s to
0 . Clearly, e v e r y Doob p o t e n t i a l
is a
s u p e r m a r t i n g a l e w h i c h is p o s i t i v e and T-bounded.
We can now prove the Riesz d e c o m p o s i t i o n f o r ~ - b o u n d e d
submartingales
and supermartingales:
2.4.2.
Theorem.
Every ~ - b o u n d e d
s u b m a r t i n g a l e is the d i f f e r e n c e of a b o u n d e d m a r t i n g a l e
and a Doob potential. Every ~ - b o u n d e d
s u p e r m a r t i n g a l e is the sum of a b o u n d e d m a r t i n g a l e and
a Doob potential. In either case, the d e c o m p o s i t i o n is unique. Proof.
C o n s i d e r an ~ - b o u n d e d
{ ~n (A) A£ F
I n 6~
}
submartingale
is b o u n d e d and increasing,
~
. Then the sequence
hence convergent,
for all
. Therefore,
~(A) exists for all
:= A£ F
lim ~n(A) and d e f i n e s a m e a s u r e
the set f u n c t i o n p r o c e s s
~£ba(F
, ~)
. Clearly,
:=
{ Rn~
I n£~
is a b o u n d e d martingale,
}
and it is e a s i l y seen that the set f u n c t i o n
process
~
:= ~ - ~
is a Doob potential.
Hence
~
has the Riesz d e c o m p o s i t i o n
~-~
=
If
is an a r b i t r a r y Riesz d e c o m p o s i t i o n of
~ , where
m a r t i n g a l e and
then we have
~
~-_~
is a Doob potential, =
p o t e n t i a l p r o p e r t y of
for all
n 6~
~
and
- ~n(A)
and
is a b o u n d e d
~-~_
Using the m a r t i n g a l e p r o p e r t y of
~n(A)
~
A 6 F
~
and
~
as well as the Doob
~ , this y i e l d s
=
lim
(~m(A) - ~ m ( A ) )
=
lim
~m(A)
n
-
lim
~m(A)
=
0
,
. T h e r e f o r e the Riesz d e c o m p o s i t i o n is
unique.
Since every Doob p o t e n t i a l for m - b o u n d e d
2.4.3.
is T-bounded,
the Riesz d e c o m p o s i t i o n
s u b m a r t i n g a l e s yields the f o l l o w i n g result:
Corollary.
Every m - b o u n d e d
submartingale
is T-bounded.
Further results on s u b m a r t i n g a l e s and s u p e r m a r t i n g a l e s will be given in C h a p t e r 4, Section 3.
Let us now return to the e x t e n s i o n p r o b l e m for the class of b o u n d e d martingales.
F r o m the c o n s i d e r a t i o n s at the b e g i n n i n g of this section,
it is clear that such an e x t e n s i o n should at least c o m p r i s e those
84 bounded the
submartingales
infimum
function can
processes
be s e e n
2.4.4. On the
and
supermartingales
of two m a r t i n g a l e s .
from
is s t i l l
the
However,
too
following
small
which
are
the r e s u l t i n g
since
the
supremum
class
of
or
set
it n e e d n o t be l i n e a r ,
as
example:
Example. standard
processes
~
stochastic
and
~
basis
on
[0,1)
, define
set f u n c t i o n
by l e t t i n g
[
2 -n - I
,
if
k = I
2 -n
,
otherwise
2 -n - I
,
if
2 -n
,
otherwise
~
and
:=
~n (Bn,k)
and
[ ~ n (Bn,k)
for all hence
n 6~ ILl
process
and
and
I~I
A set f u n c t i o n
. Then
are b o u n d e d
is n e i t h e r
the c l a s s
supermartingales
Z
k £ K(n)
I ~ I - I~I
In p a r t i c u l a r ,
need
are bounded
submartingales,
of a l l b o u n d e d
~
~
a submartingale
not be linear
process
l~n-Rn~n+11(Q)
k = 2n
:=
and
but
nor
set
function
a supermartingale.
submartingales therefore
is a q u a s i m a r t i n @ a l e
the
martingales,
has
if the
and to b e e n l a r g e d .
series
is c o n v e r g e n t .
Quasimartingales
are
closely
related
to s u b m a r t i n g a l e s
and
supermartingales:
2.4.5.
Lemma.
Every ~-bounded
quasimartingale
is the d i f f e r e n c e
of t w o p o s i t i v e
supermartingales.
Proof.
Consider
all
,
m 6~
+ ~m Thus,
for all
an ~ - b o u n d e d
quasimartingale
)+ --<
and
A £ F
. T h e n we have,
+
(~m-Rm~m+1
n 6~
~
+
n
(Rm~m+1) , the b o u n d e d
sequence
for
85 m-1 X k=n is increasing,
hence convergent,
q0n(A) Then
~
process
:=
1;
and we may define
(~-~k+1)+(A)
supermartingale.
+ +~m(A)
Similarly,
]
the set function
given by
@n(A) n C~
m-1 Z k=n
lim
is a positive ~
+ ~m+ A( ) I m 6 ~ ( n + 1 )
(~k-~k+1)+(A)
and
A £ F ~
--
::
m-1 Z k=n
lim (
(~_Rk~+I)-(A)
, is a positive
n
~ - e
+~m(A ) h J
supermartingale,
and we have
,
D
as was to be shown. 2.4.6.
Corollary.
Every ~ - b o u n d e d Proof.
quasimartingale
Apply Corollary
More generally,
is T-bounded.
2.4.3.
every quasimartingale
is a semiamart,
as can be seen
from Theorem 2.5.1 and Lemma 2.5.2 below. Lemma 2,4.5 also leads to the Riesz d e c o m p o s i t i o n
for bounded
quasimartingales: 2.4.7.
Theorem.
Every b o u n d e d q u a s i m a r t i n g a l e the difference Proof.
and
of two Doob potentials.
First apply Lemma
supermartingale
is the sum of a b o u n d e d m a r t i n g a l e
2.4.5 and note that each positive
is ~ - b o u n d e d .
Then the assertion
follows
from Theorem
2.4.2 and Theorem 2.3.3. In the Riesz d e c o m p o s i t i o n and the difference
for a bounded quasimartingale,
of Doob p o t e n t i a l s
are unique,
the m a r t i n g a l e
but the Doob potentials
themselves need not be uniquely determined. Another consequence describes
of Lemma 2.4.5 is the following
result which
the structure of the class of all bounded quasimartingales:
86
2.4.8.
Theorem.
The c l a s s of all b o u n d e d containing
Proof.
all b o u n d e d
T h e c l a s s of all b o u n d e d
and contains vector
the b o u n d e d
space containing
us n o w p r o v e lattice. have,
quasimartingales
To this end, n 6~
+ ~n
<
(~
- R
quasimartingales
submartingales. all b o u n d e d
t h a t the c l a s s
for all
is the s m a l l e s t v e c t o r
lattice
submartingales.
Moreover,
submartingales,
of all b o u n d e d
consider
a bounded
is c l e a r l y
linear
it is the s m a l l e s t by Lemma
quasimartingales
quasimartingale
~
2.4.5.
Let
is a v e c t o r . T h e n we
, )+
+
(~n-RnBn+1
+
+ (Rn~n+1)
<
i~n-Rn~n+ I I + R n ~ n + I
hence
+ n~n+l )+
~
I~ n - Rn~n+ll
and therefore
I
+ ~n+ - Rn~n+l i
=
+ + 2 (Bn+ - Rn~n+l ) + - (~n+ - Rn~n+l )
<
2i~n-Rn~n+ll
+
This yields,
for all
m X n=l
m£~
+
+ ( R n ~ n + l - ~ n)
,
+ + l~n-Rn~n+11(~ )
~
2
m Z n=1
l~n-Rn~n+li(~)
+ + ~m+l(~)
2
X n=1
l~n-Rn~n+ I i(n)
+ 211 ~ I ~
- ~(n)
+ Letting
m
t e n d to i n f i n i t y
is c l e a r l y b o u n d e d . is a v e c t o r
In c o n t r a s t the H - n o r m
Therefore,
that
the c l a s s
is a q u a s i m a r t i n g a l e
of all b o u n d e d
which
quasimartingales s
to w h a t is k n o w n
for b o u n d e d m a r t i n g a l e s ,
a n d the T - n o r m are i d e n t i c a l
are u s u a l l y
~
lattice,
neither
quasimartingales.
2.4.9.
shows
identical
by T h e o r e m
nor even equivalent
for w h i c h
2.3.2,
these norms
for b o u n d e d
T h i s c a n be seen f r o m the f o l l o w i n g
example:
Example.
On the s t a n d a r d
stochastic
processes
,
~(m)
m£~
basis
on
, by l e t t i n g
[0,1)
, define
set f u n c t i o n
87
(m)
(Bn,k)
~n
for all ~(m)
n £~
jlT = m
Moreover,
2.4.10.
,
if
I
0
,
otherwise
k 6 K(n)
. Then
n ~ m
each
lattice
for the ~ - n o r m
of all b o u n d e d
as w e l l
for all
. On an a r b i t r a r y
processes
_B(m)
,
n £~
:=
and
is a b o u n d e d
potentials
and
In the B a n a c h
[~
processes
= 1
and
quasimartingales
fails
as c a n be
to
seen
stochastic
m C~
F
basis
on
~
, define
set
, by letting
t
(-1) n n
l
0
- m
-I
,
if
~ 6 A
,
otherwise
and
n < m
A £ F . T h e n e a c h of the set f u n c t i o n p r o c e s s e s n quasimartingale w h i c h is the d i f f e r e n c e of t w o D o o b
for w h i c h
the ~ - n o r m
lattice
of all
{ _~(m)
[
}
to the
set
and
T-bounded
the T - n o r m
are
set f u n c t i o n
identical.
processes
on
F
,
sequence
converges
m£•
function
i
for all
n 6~
Z n=1
and
lack
A E F
2n+1 nln+l)
set f u n c t i o n
In p a r t i c u l a r ,
process
~
which
is d e f i n e d
(-1)n n-1
'
if
0
,
otherwise
by
letting
~6A
:=
Bn (A)
The
[i ~(m)
as f o r the T - n o r m ,
-I
the
set f u n c t i o n
satisfying
example:
,,(m) (A) ~n
the
of the
k = 2
Example. ~ 6 ~
function
~(m)
and
.
the f o l l o w i n g
Choose
I
quasimartingale
the v e c t o r
be c o m p l e t e from
and
is a b o u n d e d
II ~(m)
:=
[
n
=
process ~
cannot
of c o m p l e t e n e s s
. Due
Z
to the
I~ n
identity
Rn~n+ll(~)
n=l ~
fails
to be a q u a s i m a r t i n g a l e .
be the d i f f e r e n c e
of t h e c l a s s
of two D o o b
of all b o u n d e d
potentials.
quasimartingales
88
for b o t h t h e ~ - n o r m extending lattice
and the T - n o r m may be r e g a r d e d
this class
theoretical
this e x t e n s i o n convincing.
2.4.11.
of set f u n c t i o n point of view,
which
concerns
processes
there
is also a n o t h e r
Doob p o t e n t i a l s
Let us first prove
as a m o t i v a t i o n
to a larger one.
the f o l l o w i n g
for
F r o m the
argument
for
and may appear to be more result:
Theorem.
The class of all set f u n c t i o n Doob p o t e n t i a l s
processes
is the s m a l l e s t
vector
w h i c h are the d i f f e r e n c e lattice
containing
of two
all Doob
potentials.
Proof.
The class
difference
set f u n c t i o n
of two Doob p o t e n t i a l s
Doob potentials. Doob p o t e n t i a l
If
it follows
and
=
(~+
~)
~
-
that the m o d u l u s
a difference
This v e c t o r
~
is c l e a r l y
lattice
in w h i c h
of a d i f f e r e n c e
complete
desired
it can be shown
for b o u n d e d
~
processes;
again
is
see Example
is not a d i f f e r e n c e
This
of
and a l t h o u g h
section.
processes
will
By the Riesz convergence
of the g e n e r a l i z e d
process
process
that a p o i n t w i s e processes
a.e.
Furthermore,
converge
to
is m a j o r i z e d
convergence
in the ideal
generated
is
for p o t e n t i a l s w i t h the above
lattice
which
that the ideal g e n e r a t e d
of all T - b o u n d e d
turn out to be c o m p l e t e
a.e.
by the
theorem
let us also remark
0
by a Doob
theorem
In c o n n e c t i o n
in the B a n a c h
it
that the g e n e r a l i z e d
of a set f u n c t i o n
of c o m p l e t e n e s s ,
by the Doob p o t e n t i a l s
0
Radon-Nikodym
and the m a r t i n g a l e
quasimartingale.
true)
is the c o n v e r g e n c e
in the next
to
convergence
of a b o u n d e d
for all set f u n c t i o n
Doob potentials.
that the g e n e r a l i z e d
of the set f u n c t i o n
But this means
will be g i v e n
function
process
converge
(and a c t u a l l y
derivatives
the m o d u l u s
discussion
is a
for the T - n o r m nor is it an
set f u n c t i o n
quasimartingales
the p o i n t w i s e
derivatives
to e x p e c t
Radon-Nikodym
potential.
of Doob p o t e n t i a l s
it is the limit of such p r o c e s s e s
of a Doob p o t e n t i a l
Radon-Nikodym
whenever
the
~ ^~
D
the set f u n c t i o n
this yields
is n a t u r a l
then
is a Doob potential.
decomposition theorem,
linear and contains
2(~A~)
is n e i t h e r
although
On the o t h e r hand, derivatives
are the
of Doob potentials.
Doob potentials, its m o d u l u s
which
are Doob p o t e n t i a l s ,
in the class of all T - b o u n d e d
2.4.10,
processes
too, and from the i d e n t i t y
I~-~1
ideal
of all
set
for the T-norm.
2.5.
A m a r t s .
If m a r t i n g a l e s and b o u n d e d s u b m a r t i n g a l e s are c o n s i d e r e d as set function processes w h i c h are r e s t r i c t i o n s of a limit m e a s u r e or m o n o t o n i c a l l y increase to a limit measure,
it is natural to go one step further and to
study set function p r o c e s s e s w h i c h c o n v e r g e to a limit measure.
A set f u n c t i o n process net
{ ~(~)
I T 6 T }
~
is an amart
(or a s y m p t o t i c martingale)
if the
is convergent. The class of all amarts is clearly
linear and contains all m a r t i n g a l e s and Doob potentials;
it then follows
from the c o r r e s p o n d i n g Riesz d e c o m p o s i t i o n theorems that the class of all amarts also c o n t a i n s the b o u n d e d submartingales, and q u a s i m a r t i n g a l e s .
For q u a s i m a r t i n g a l e s ,
supermartingales,
the b o u n d e d n e s s c o n d i t i o n
can be omitted:
2.5.1.
Theorem.
Every q u a s i m a r t i n g a l e is an amart.
Proof. k 6~
Consider a quasimartingale
~ . Fix
E 6 (0,~)
and choose
such that
T
n=k
T h e n we have,
<
I~n - Rn]~n+ I I (~) for all
r6T(k)
E
and
m£1~(T)
,
m-1 ' ~ ( ~ ) - ~ m (~) '
=
I p__Z k
(~p-Rp~m)({r=p})
I
m-1 m-1 :
p=k n=p m-1
n
<
l~n-Rn~n+11 ({~=P}) n=k p=k
This yields,
for all
<
X n=k
<
E
~, • 6 T(k)
l~n-Rn~n+11(~)
,
I
90
lu..~(n) -u.a(n) Therefore•
The
{ ~(~)
following
2.5.2. Every
I
<
J r 6 T
result
Proof.
Consider
for all
an a m a r t
o 6 T(k)
to be
but will
hence
convergent.
be u s e f u l :
~
~
. Choose
k £~
such
that
I
. Then
we have,
for all
5_
I~..~vk(n)
!
I +
k Z p=1
l~p({rAk=p}) I
_<
1 +
k X p=1
l~pl(S)
• £ T
,
- ~.~k(n) 1 + I~.~^k(n) I
,
shown.
In p a r t i c u l a r , quasimartingale
it f o l l o w s
from Theorem
2.5.1
and Lemma
the
2.5.3.
Theorem.
The c l a s s
of all b o u n d e d
Proof.
Consider
structure
amarts
a bounded
of the c l a s s
is a B a n a c h
amart
~
. Fix
of all b o u n d e d
lattice
that every
~, ~ E T ( k )
suPT(k)
~+(Q)
. By L e m m a
<
suPT(k)
2.5.2,
l~
J (~)
e 6 (0,~)
we h a v e
<
amarts:
for the n o r m
that
for all
2.5.2
is a s e m i a m a r t .
we c a n n o w d e s c r i b e
holds
net,
is a s e m i a m a r t .
lu._~(~) I
such
is a C a u c h y
is e l e m e n t a r y
l~a(n) - ~k(n) I
as w a s
}
Lemma. amart
holds
2£:
H _~ IIT
and choose
H.
IIT
k q~
.
91 hence there exists
T
holds
(n)
for all
• £ T(k)
+ ~(~)
<
T 6 T(k)
<
~(a)
For
M £ T(k)
such that
+
2£;
. Choose
,,.I. (A) +
A £ F
£:
, define a stopping
=
~) (co)
such t h a t
M
time
v 6 T(k)
I
T (co)
,
if
o~ £ A
[
(~)
,
if
0~ 6 ~ A
by letting
T h e n we h a v e +
~){(~)
<
~R (~ ) - BR ( ~ A )
=
~x(~)
- BV(~)
<
+ ]~c(O)
+
2e
+
e
+ ~T(A)
+
e
,
hence
I-¢.~(a) This yields,
+ -~(Q)
for all
I
_<
2~
o, T 6 T(k)
I
.
,
_<
+ Therefore, follows
~
is an a m a r t w h i c h
is c l e a r l y b o u n d e d .
t h a t the c l a s s of all b o u n d e d
amarts
In o r d e r to see t h a t the c l a s s of all b o u n d e d the n o r m
II. IIT , c o n s i d e r { ~(m)
of b o u n d e d
amarts
all T - b o u n d e d and
m E~
a Cauchy
From this
is a v e c t o r amarts
it
lattice.
is c o m p l e t e
for
sequence
I m 61~ }
a n d let
~
denote
set f u n c t i o n p r o c e s s e s .
its l i m i t in the B a n a c h T h e n w e have,
f o r all
l a t t i c e of o, T £ T
, l~o(n)-~T(~) I _< II ~_~(m)
iiT + l'~o(m)(n)_u(m)_r (n) I + II ~ ( m ) _ ~ iiT
Fix
e E (0,~)
, choose
m 63~
such that
<
e
,
(m) (n) I (Q) - ~x
<
II ~ - ~ ( m )
and choose
~ £ T
(m)
~o holds for all
i[T
such that
o,
T C T(~)
I~olS) - ~T(n) [
for all
o,
the vector
However,
T £ T(K) lattice
Alternatively, can be proven method
Amarts their
can be
lattice
amarts
property
the Riesz
may be characterized
amarts
the ~-norm
false;
amart,
is c o m p l e t e
of all bounded
see C h a p t e r
of t h e c l a s s
decomposition
2.5.4.
would
have
2.4.9.
of all bounded
for amarts.
For
amarts this
4.
in a s i m i l a r
process
~
, the
way
as m a r t i n g a l e s
in t e r m s
following
are equivalent:
is a n a m a r t .
(b)
There
(c)
There
lim
exists
Proof.
exists
, ~)
~6 a(F
, ~)
such that
.
holds
a measure
first
, and choose
I ~ (0) for all
~C a(F
for all 56 a(F
A6 , ~)
such that F such that
= l i m BT(n)
Suppose
e £ (0,~)
= 0
a measure
= l i m Br(A)
There ~(Q)
a measure
I~r-RT~I(~)
~(A)
(d)
exists
- ~(n)
that ~ C T
I
v, ~ E T(x)
is a n a m a r t . such that
o
need not be complete
Theorem.
(a)
that
for the T-norm.
and the T-norm
see E x a m p l e
4, S e c t i o n
a n d it f o l l o w s
limit measure:
For a set function
holds
is a b o u n d e d
if it w e r e ,
by using
of p r o o f ,
3e
~
lattice
which
the
<_
. Hence
since
to be equivalent
. Then we have
of a l l b o u n d e d
the vector
for the ~-norm
e
A C F
Consider and
A E F
, fix
of
For
o,
T 6 T(~)
, define
v(~)
stopping
times
v, ~ 6 T(M)
0 (~)
,
if
¢06 A
(OVT) (~)
,
if
CO 6 Q ~ A
T(~)
,
if
~ £ A
(ovT) (~)
,
if
~ £ Q~A
by letting
:=
and
~(~)
:=
Then we have
Therefore,
{ ~r(A)
~(A)
:=
exists
for all
A£
Again,
fix
[ r 6 T
F
and defines
e £ (0,~)
for all
Consider 6 T(T)
net,
hence
and choose
I
a measure M £ T
56a(F
, ~)
such that
<_
~, n £ T(K)
r £ T(~) by
is a C a u c h y
l i m ~T(A)
l~v(n) - ~ ( n ) holds
}
,
A6
F
and
T
n £ T(T)
. Define
a stopping
time
letting
r (~)
~(~)
,
if
¢0 6 A
:=
Then we have
for all
T 6 T(~)
,
A6
F
and
T
infinity,
l~r(A)
- ~(A) I
~
S
n £ T(T)
, hence,
letting
~
tend to
g4
This
yields
sup F
I~T(A) - 5 ( A ) I
~
T Therefore,
(a) i m p l i e s
As a corollary, measure
we obtain
is n o t
involved.
2.5.5.
Corollary.
F o r a set
function
(a)
~
(b)
lim suPT(T )
Proof. Fix
remaining
implications
a characterization This
process
of a m a r t s
is the d i f f e r e n c e
~
, the f o l l o w i n g
are obvious,
in w h i c h
property
are
a
the
limit
for a m a r t s :
equivalent:
is an amart.
Suppose
E £ (0,~)
for all
I~-RT~aI(~)
first
that
and choose
I~ v - R holds
(b). T h e
51(n) v 6 T(~)
= 0 .
~
is an a m a r t
~ £ T
~
such
limit
measure
~
.
¢
. T h e n we have,
I~ z-RT~al(~)
with
that
~
z £ T(~)
and
o 6 T(T)
I~z-R$~l(n) + IRr(Ra~-~a) l(~) I~ z - R
<
for all
2e
~l(n)
+ IRa~-~al(S)
,
hence
suPT(r) for all
z E T(~)
As a n o t h e r amart
this
this
. The converse
consequence
is b o u n d e d
deduce
I ~ T - RT~C[(~)
2~
implication
of T h e o r e m
if a n d o n l y
result
~
2.5.4,
if its l i m i t
f r o m the c o r r e s p o n d i n g
is o b v i o u s .
it is e a s y measure one
to prove
that
is b o u n d e d .
for m a r t i n g a l e s
an
We shall later
in
section.
The m a i n
interest
the
that
fact
similar
in the
it l e a d s
to the Riesz
quasimartingales.
limit
measure
to a R i e s z
decompositions
Since
amarts
of an a m a r t
decomposition for b o u n d e d
generalize
actually
for a m a r t s
comes which
submartingales
quasimartingales,
the
from is
and set
,
g5 funct i o n
processes
decomposition potentials.
and only
in the p o t e n t i a l
to
set f u n c t i o n
process
~
0 . Clearly•
if its m o d u l u s
Lemma.
If
J~J
~
{ I~TI(S)
is a p o t e n t i a l
J~J
is a potential;
lattice.
of every p o t e n t i a l
is a potential•
Doob
if the net
and every p o t e n t i a l
2.5.6.
of)
process
potential,
result:
(the d i f f e r e n c e s
will be studied now.
is a p o t e n t i a l
is a v e c t o r
following
processes
a set f u n c t i o n
potentials
The m o d u l u s
part of the Riesz
for amarts have to g e n e r a l i z e
These
A set f u n c t i o n converges
occuring
Moreover,
hence
every
~
I T £ T }
the class of all
Doob p o t e n t i a l
is a
is a b o u n d e d
amart.
is m a j o r i z e d
by a Doob potential.
then the s m a l l e s t
if
We also have
Doob p o t e n t i a l
2'
the
majorizing
satisfies
B~(A) for all
n 6~
Proof.
=
suPT(n ) JBrJ(A)
and
A £ F
Consider
~n(A)
n
a potential
~
suPT(n )
J~ J(A)
:=
,
. For
n 6~
and
A £ F
n
• define
Then each of the set f u n c t i o n s
ddn
is c l e a r l y
:
bounded
superadditive, stopp i n g
times
~n(A) and d e f i n e
Fn
> ]R
and subadditive.
consider
v, o £ T(n)
+
~n(B)
a stopping
T(~)
disjoint
:=
In o r d e r
sets
. Fix
such that
~
IBvI(A)
time
T E T(n)
+ IUoI(B)
•
if
~ £A
oho)
•
if
~CB
•
otherwise
(co)
+ ~
by letting
v(~)
(vva)
to see that
A, B 6 F n
•
~n
is also
e £ (0,~)
, choose
98 T h e n we h a v e
Cn(A) by L e m m a
2.2.1.
it f o l l o w s process
~
since
hence
~n
: 0
=
is the
smallest
following
But
~n
'
n 6~
this
means
+ ~
hence
, define
,
additive,
and
a set f u n c t i o n
satisfying
that
. Finally,
if
, t h e n we have,
suPT(n)
result
Cn(A+B)
supermartingale
ILl
ILl
~n(A)
<
,
majorizes
majorizing
+ ~
is s u p e r a d d i t i v e ,
is a p o s i t i v e
is a p o t e n t i a l .
~
I]I [(A+B)
set f u n c t i o n s
~n(~)
obviously
potential
The
the
which
L
<_
Therefore,
that
in~
which
+ Cn(B)
I~ I(A)
Doob
~
the
is a D o o b
for all
n 6~
majorizing
structure
potential
is an a r b i t r a r y
s u P T ( n ) ~T(A)
potential
describes
~ ~
and
= ILl
of the c l a s s
Doob
A £ Fn
,
~n(A) .
of all
potentials:
2.5.7.
Theorem.
The c l a s s
of all p o t e n t i a l s
Moreover,
it is a s u b l a t t i c e
amarts the
and,
smallest
Proof.
in the ideal
remarks
for the T - n o r m . following
We can n o w p r o v e
2.5.8.
the
of all
all D o o b
that
The
lattice
set
function
lattice
assertions
of a p o t e n t i a l
decomposition
II. II T of all b o u n d e d
for the n o r m
processes,
it is
of all p o t e n t i a l s
are
and
immediate
from Lemma
is the
If the a m a r t
Proof.
sum of a m a r t i n g a l e
for a m a r t s :
Consider
and a potential.
is unique.
is b o u n d e d ,
then
an a m a r t
I]~T(A) - ~(A) I
<
so is the m a r t i n g a l e .
L
that
I
with
limit
measure
5
. Choose
is
f r o m the 2.5.6.
Theorem. amart
"
potentials.
the v e c t o r
remaining
definition
the R i e s z
The decomposition
such
lattice
seen
lattice
in the B a n a c h
containing
It is e a s i l y
complete
Every
vector
is a B a n a c h
M 6T
[]
97
holds
for all
T 6 T(M)
and
I~T - RT~I(~)
hence
R ~ 6 ba(FT, ~)
because
of
for all
n 6~
set f u n c t i o n
T 6T(~)
~
!
IRnv~l(n)
the set f u n c t i o n s
which
Rn~
is a m a r t i n g a l e .
Rn~£ba(Fn,
~)
,
,
n 6~
, define
It is then c l e a r
a
that
process
Therefore,
~
has the Riesz d e c o m p o s i t i o n
Riesz
decomposition
is a potential,
sup T
l~T-~l(n)
implies
~ = ~
of
~
, where
then the set f u n c t i o n
is at the same time a s u b m a r t i n g a l e
which
. This y i e l d s
if
is an a r b i t r a r y ~
,
IRn(Rnv~)i(~)
. Therefore,
is a potential.
. Then we have
and
=
process
the set f u n c t i o n
Moreover,
2
, for all
nvM £ T(x)
IRnSl(n)
and
~
A 6 F
and
=
lim
~ = ~
~
and a potential.
l~T-~l(n)
. Therefore,
is a m a r t i n g a l e
process
=
lim
This yields
I~T-~TI(Q)
the Riesz
=
0
decomposition
,
is
unique. Finally, bounded
2.5.9.
if the a m a r t
~
is bounded,
since e v e r y p o t e n t i a l
then the m a r t i n g a l e
is bounded.
~ = ~- ~
is Q
Corollary.
For an a m a r t
~
and its limit m e a s u r e
(a)
~
is bounded.
(b)
~
is bounded.
~
, the f o l l o w i n g
are equivalent:
98 Proof.
By the Riesz decomposition,
if its m a r t i n g a l e part is bounded.
an amart is b o u n d e d if and only
Since the limit m e a s u r e of an amart
is identical w i t h the limit m e a s u r e of its m a r t i n g a l e part, the a s s e r t i o n follows f r o m T h e o r e m 2.3.2.
u
It should be noted, however, e x t e n d to amarts. the T-norm;
that not all a s s e r t i o n s of T h e o r e m 2.3.2
Even for q u a s i m a r t i n g a l e s ,
see Example 2.4.9. F u r t h e r m o r e ,
the~-norm
may d i f f e r from
the T - n o r m of a n o n t r i v i a l
p o t e n t i a l c e r t a i n l y differs from the n o r m of its limit m e a s u r e w h i c h is equal to zero.
Before p a s s i n g to the amart c o n v e r g e n c e theorem, we include a n o t h e r c o n s e q u e n c e of the Riesz d e c o m p o s i t i o n w h i c h shows that the usual d e f i n i t i o n of an a m a r t is e q u i v a l e n t to C h a t t e r j i ' s condition:
2.5.10.
Corollary.
For a set f u n c t i o n p r o c e s s
2 , the f o l l o w i n g are equivalent:
(a)
~
is an amart.
(b)
There exists a m a r t i n g a l e satisfying
Proof.
If
2
is an amart, define
by the limit m e a s u r e of
if
2
Doob p o t e n t i a l potential.
2'
Hence
= (~-~) + ~
and a Doob p o t e n t i a l
~
2 , and define
m a j o r i z i n g the p o t e n t i a l Conversely,
~
2,
I ~ - ~I < ~'
I~- ~
to be the m a r t i n g a l e induced 2'
to be a Doob p o t e n t i a l
I.
is such that there exists a m a r t i n g a l e satisfying 2- ~
I~- ~I < ~'
is a potential,
, then
1 2 - ~I
~
and a is a
and it follows that
is an amart.
It is r e m a r k a b l e that the a b o v e c h a r a c t e r i z a t i o n y i e l d s an e q u i v a l e n t d e f i n i t i o n of amarts in w h i c h s t o p p i n g times n e e d not be used. This is due to the fact that m a r t i n g a l e s and Doob p o t e n t i a l s may be d e f i n e d w i t h o u t u s i n g s t o p p i n g times;
see T h e o r e m 2.3.1 and T h e o r e m 2.4.1.
Let us now prove the a m a r t c o n v e r g e n c e theorem.
Due to the Riesz
d e c o m p o s i t i o n for amarts and the m a r t i n g a l e c o n v e r g e n c e theorem,
it is
s u f f i c i e n t to prove the c o n v e r g e n c e of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a p o t e n t i a l in order to prove the c o n v e r g e n c e of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a b o u n d e d amart.
The c o n v e r g e n c e of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a p o t e n t i a l can be o b t a i n e d f r o m a m a x i m a l i n e q u a l i t y w h i c h we shall
99
prove now.
For all
n E~
, let
Cn :
~_n
d e n o t e the o - a l g e b r a g e n e r a t e d by
b a ( F n, ~)
Fn , let
> b a R n l ( F n , ~)
denote the b a n d p r o j e c t i o n g u a r a n t e e d by T h e o r e m 2.1.4, and let
Jn
:
bca(Fn' ~)
~ ca(Fn' m)
denote the e x t e n s i o n o p e r a t o r given by T h e o r e m 2.1.6. For each set function process
~
on
[
, we then o b t a i n a set f u n c t i o n
process
:=
{ JnCn~n
on the stochastic basis Let
T
I n E~
}
[ := { ~n
I n E~
} .
denote the class of all b o u n d e d stopping times for
~
.
The proof of the a n n o u n c e d m a x i m a l i n e q u a l i t y will be b a s e d on the f o l l o w i n g t e c h n i c a l lemma:
2.5.11. If
~
Lemma. is a set f u n c t i o n process on
sUP~T(k ) holds for all
kE~
I~I(S)
!
suPT(k)
[
, then
I~TI(n)
.
Proof.
W i t h o u t loss of generality, we m a y and do a s s u m e
Consider
~ ET
and d e f i n e
m
N o w define
B1
:=
GI
:=
and
choose
A 1E F1
{~=1}
,
such that
IP.j I ( G I ~ 1 )
<_ Cm-2
:= m a x ~ {(~)
. Fix
e E (0,=)
k = I
100
holds
for all
j E {1•2, .... m}
DI
:=
BI n AI
=
, and define
AI
T h e n we h a v e
{~=1}
S
(GIAA I ) U D I
,
hence
I~11({~=1}) For A
P
E F
pE {2,3,...,m} and
P
D
B
E F
P
~
cm-1 + lUlI(D 1)
, define
inductively
, as f o l l o w s :
P p-1
:=
U
P
j=1
sets
B p E Fp
Define
Ac ]
and
Gp choose
A
::
E F P
{~:p} N Bp such that
P l~j I (GpAAp)
holds
for all
D
G
:=
Therefore•
em -2
P
we o b t a i n
G
AD P
c _
, and define
NA
B
P
c B p -- p Gp
<
j E {p,p+1,...,m}
P From
•
c G AA , hence P -- p P
(GpAAp) U D P
we h a v e {~:p}
:
p-1 X j:l
c
( p;1
--
j:1
c
-
{~:p} N Bj h A . + {~:p} 0 B 3 P
1
GCND
(P
U
j=1
]
G .~ A A .
J
]
U G P
)
U Dp
'
,
GpEFp
101
hence l~pl({~=p})
~
em -I + l~pl(Dp)
Since the sets D , p £ {I,2,...,m} , are mutually disjoint, P define a stopping time T 6 T by letting
x(~)
p
,
if
~ E Dp
m
,
if
0~6~
I~I(~)
~
e + IUTI(~)
From this the assertion
~
~ D p=1 p
yields
~
e + suPT
IUT](~)
follows.
We can now prove the maximal
If
m
:=
Summing up the above inequalities
2.5.12.
we can
inequality:
Lemma. is a set function process,
then
~(J I) ( { s u ~ ( k ) IDn~nl > E}) holds for all Proof.
e £ (0,~)
k 6~
suPT(k ) I ~ I (Q)
.
Define
Ak and, for all
An For each
and
~
=
n 6~(k+I)
:=
m £~(k)
m
This yields
{Ivk~l>E}
(~)
C
;k
,
{ lDn~ nl > ~} n
n 1{ ) ~ IDp~pl < E } p=k
, define a stopping
time
n
,
if
~£ An
m
,
if
~6~
:=
6
{m 6 T(k)
and m Z A n=k n
Fn by letting
k < n < m
102
(~_)
m
¢(J l)
An
=
c
Z n=k
n-k
(JnRn l) (An )
m I
--< nZ--k A IDn~nl d(JnRnl) n m :
I~nl (A n)
~-
n=k
m
suPT(k) by Lemma 2.5.11. Letting
m
l~Ti(n)
,
tend to infinity, we obtain
~(J A ) ( { s u b ( k ) , D n ~ n l
> ¢})
=
E(J l)( ~
An)
\n=k
suPT(k)
IBTI(Q)
,
as was to be shown. From this maximal inequality, easily deduced: 2.5.13. If
~
theorem is
Theorem. is a potential, lim Dn~ n
Proof.
the potential convergence
Fix
0
a.e.
e, 6 £ (0,~)
suPT(k) Now the maximal
=
then
IUTI(~)
and choose ~
k £~
such that
e6
inequality yields
(J A) ( { s u b ( k } IDn~nl > E})
<
6
,
from which the assertion follows. By the Riesz decomposition
for amarts and the linearity of the
103
generalized
Radon-Nikodym
theorem is a consequence potential
convergence
2.5.14
Corollary.
If
~
By Corollary
2.5.15. ~
=
2.5.10,
in the following
the following
of the m a r t i n g a l e
amart c 0 n v e r ~ e n c e
convergence
theorem and the
theorem:
is a bounded amart,
lim Dn~ n
If
operators,
then
D~ lim ~n
a.e.
the amart convergence
theorem may also be stated
form:
Corollary. is an ~ - b o u n d e d
martingale
~
set function process
and a Doob potential
lim DnB n
=
D
lim ~n
~'
such that there exists a
satisfying
I~- ~I < ~'
, then
a.e.
In f a c t , this is the earliest form in which the amart convergence theorem was stated;
see Chatterji
[35; Theorem 2]. A l t h o u g h no stopping
times at all are needed in the formulation of C o r o l l a r y remarks following Corollary its proof.
It may be interesting
of Corollary Proof.
Xn and choose
lim ~n
=
0
lim
~
has to be bounded.
lim Dn5 n = D (~n-~n)
lim 5n
:=
a.e.
~, 6 6 (0,~)
simple functions
IXn-Znl d(JnRnl) and choose
k 6~
Z
n
< such that
Since
sufficient to prove
D n ( B n - ~ n)
F -measurable n
Now the
a.e.
= 0 . Thus we obtain
" It is therefore
, define
n~1 ~Q Fix
we have
lim ~n = D
lim D n ( ~ n - ~ n) n C~
in
to include the following direct proof
theorem yields
is a Doob potential,
For all
(see the
2.5.15.
convergence
lim Dn5 n = D
2.5.15
they seem to be indispensable
First note that the m a r t i n g a l e
martingale ~'
2.5.10),
such that
104
~.~.(s)
<_ ~8
and Z n=k It follows
IXn-Zn I d (JnRnl)
<
e6
~
exactly lim
as in the proof of Theorem
(Xn-Z n)
=
0
2.3.8 that
a.e.
Now define
:= and,
for all
{IZk[ > e}
n £~(k+I)
q
Fk
,
/n-1 An For all
:=
m £~(k)
Tm(~)
Thisyields,
>
{[Znl > e } NkpZ__k{[Zp[_<e} , define a stopping
time
,
if
~EA n
and
m
,
if
~ E ~~
m Z An n=k
E (J A) < n L
mq~(k)
An>
=
C
m~-
k ~ n ~ m
(JnRnl) (An)
n=k
mf IZnl nZ--k A
°I <
by letting
,
=
<
Fn
r m 6 T(k)
n :=
for all
6
d (JnRnA)
n
n--Zk A
ml ~Zn-Xnl
IXnl
d(JnRn ~) + nZ__k A
n
n
m
_<
~8 +
Z n=k
IBnl (An )
~6 + I~ml(~)
<
e8
+
~.~'
(n) m
_<
~6
+
~(~)
<
2,~6
,
d (JnRnl)
105
by T h e o r e m
2.4.1.
Letting
m
(J l ) ( { s u ~ ( k ) , Z n , which
t e n d to i n f i n i t y ,
>e})
=
we obtain
(J l)( ~ A n ) n=k
<
26
,
implies
lim Z
Therefore,
n
=
0
a.e.
=
0
a.e.
we h a v e
lira X
n
as was to be shown.
,
[]
2.6.
S e m i a m a r t s .
Semiamarts
were
encountered
chapter,
in p a r t i c u l a r
function
processes.
own
sake,
and
potentials.
Riesz
Semiamarts
times
section,
properties
The m a i n
(non-unique)
2.6.1.
In this
their
characterization
several
in the c o n t e x t
result
are
semiamarts
compared
are
for
sections
studied
those
de S n e l l
semiamarts
of t h i s
properties
are
with
is the e n v e l o p p e
decomposition
of a m a r t s
in e a r l i e r
of b o u n d e d n e s s
of set
for
their
of a m a r t s
and
from which
as w e l l
a
as a f u r t h e r
deduced.
may be characterized
as f o l l o w s :
Theorem.
F o r a set f u n c t i o n (a)
~
(b)
For
process
~
, the
following
are equivalent:
is a s e m i a m a r t . all
M £ T
, the
family
that
~
{ ~T(A)
I ~ £ T(~)
and
A6
F
is b o u n d e d .
Proof. For
Suppose
• E T(M)
first
and
A £ F
is a s e m i a m a r t .
, define
a stopping
Consider
time
v 6 T(~)
T £ T by
. letting
M
=
I
~ (~)
'
if
~ £A
(~)
,
if
~ £ ~A
[ T h e n we h a v e
I~T(A) I <_ I~(~)I for all
r £ T(x)
and
+ I~(~A) I
<
suPT
l~v(~)l + lull(n)
A C F M
The converse
Similar
is o b v i o u s .
to a m a r t s ,
2.6.2.
we have
(a)
~
(b)
The
and
A £ FT
property
for
semiamarts:
Theorem.
F o r a set f u n c t i o n
Proof.
a difference
process
~
, the
following
are
equivalent:
is a s e m i a m a r t . value
Suppose
suPT
first
, define
suPT(r )
that
stopping
~
[~T-R~a[(~)
is a s e m i a m a r t .
times
9, ~ C T(T)
is finite.
For by
• E T
letting
,
aET(~)
107
v(~)
T (co)
,
if
~ 6 A
O(c0)
,
if
co 6 ~ A
o(~)
,
if
~ £ A
T(~)
,
if
~ £ ~kA
:=
and
~(~)
:=
Then we have
(~ -R ~o) (A) - (~T-RT~o) (n~A)
=
~V(Q)
- ~n(~)
2 suPT for all
T £ T ,
o C T(T)
and
I~ T - RT~OI(n) for all
T £ T
and
~
A E F
T
2 suPT
o £ T(T)
l~(n) I
<
~
2 SuPT
l~x(n) I
<
if the v a l u e
sup T suPT(T) is finite,
, hence
, which yields
suPT s u P T ( T ) I~T-RT~oI(n) Conversely,
IBx(Q) I
I~T-RT~oI(~)
t h e n we h a v e
IB 1(n) - B~(Q) I
<_
SuPT(1)
I~I-RI~cl (~)
<
~
IlLI(~) l + suPT(1 ) IBI-RIBol (~)
<
,
hence
IBM(~) I for all
_<
x £ T = T(1)
. Hence
~
is a s e m i a m a r t .
The m a i n r e s u l t of this s e c t i o n
is the e n v e l o p p e
w h i c h we s h a l l p r o v e now:
de S n e l l
for s e m i a m a r t s
108 2.6.3.
Theorem.
Every semiamart is m a j o r i z e d by a s u p e r m a r t i n g a l e and m i n o r i z e d by a submartingale. If
~
is a semiamart,
then the smallest s u p e r m a r t i n g a l e
_~ m a j o r i z i n g
satisfies
~n(A)
=
suPT(n ) ~T(A)
and the largest s u b m a r t i n g a l e
~n(A) for all
n EI~
= and
~
satisfies
,
A E F
n bounded,
if
Proof.
Consider a semiamart
:=
minorizing
Br (A)
Moreover,
~n(A)
is
infT(n)
~
suPT(n)
then
so are
~ . For
_~
and
n E~
_~ .
and
A E Fn , d e f i n e
BT (A)
By T h e o r e m 2.6.1, each of the set functions
~n
is bounded,
and it is
also additive, w h i c h can be seen by using a stopping time a r g u m e n t as in the proof of Lemma 2.5.6. Therefore, deffne a set function process . Again,
~
the set f u n c t i o n s
~n
'
n 6~
,
w h i c h is a s u p e r m a r t i n g a l e m a j o r i z i n g
it can be seen as in the proof of Lemma 2.5.6 that
the smallest s u p e r m a r t i n g a l e m a j o r i z i n g
~
is
~ .
In order to c o n s t r u c t the largest s u b m a r t i n g a l e m i n o r i z i n g
~
, it is
s u f f i c i e n t to c o n s i d e r the smallest s u p e r m a r t i n g a l e m a j o r i z i n g The final a s s e r t i o n c o n c e r n i n g b o u n d e d n e s s
-~ .
is obvious.
[]
The f o l l o w i n g r e s u l t d e s c r i b e s the s t r u c t u r e of the class of all b o u n d e d semiamarts:
2.6.4.
Theorem.
The class of all b o u n d e d s e m i a m a r t s is a Banach lattice for the n o r m [[. [IT . Moreover,
in the v e c t o r lattice of all set f u n c t i o n processes,
it is the smallest ideal c o n t a i n i n g all b o u n d e d submartingales. Proof.
A p p l y T h e o r e m 2.2.4, T h e o r e m 2.2.5, and T h e o r e m 2.6.3.
The ideal p r o p e r t y of the class of all b o u n d e d s e m i a m a r t s T - b o u n d e d set f u n c t i o n processes)
(or: all
should be c o m p a r e d w i t h the ideal
p r o p e r t y of the class of all potentials.
109 The enveloppe de Snell for semiamarts
yields another c h a r a c t e r i z a t i o n
of amarts: 2.6.5. If
Corollary.
~
is a semiamart with smallest m a j o r i z i n g
largest minorizing
submartingale
(a)
~
(b)
lim ~n = lim ~n "
Proof.
Suppose first that
supermartingale
and
is an amart.
~
is an amart. By Corollary
there exists a m a r t i n g a l e
~
IB- ~I < B' . Then
is a supermartingale
- B'
~
~ , then the following are equivalent:
~ + B'
and a Doob potential
is a submartingale m i n o r i z i n g
lim ~n
=
lim
~'
2.5.10•
satisfying
majorizing
~ , and
~ . This yields
(~n - Bn)
lim ~n lim ~n lim
(~n + ~n)
=
lim ~n
'
hence
lim ~n Conversely•
=
lim ~n
suppose that
lim ~n = lim ~n
holds.
For all
A q F
define
~(A) Then
~
lim ~n(A)
is an additive A ~n
hence
:=
<-- Rnfi <
Rn~£ba(Fn,
function process
~) ~
positive potentials• majorizing a potential•
i~-~i
=
lim ~n(A)
set function. V ~n
Moreover,
n 6~
, we have
'
. Therefore•
the restrictions
of
hence
is a positive potential
. Now it follows
and
define a set
Then
(~- 5) v ( ~ - ~ )
~ - ~
~
which is a martingale.
from Theorem 2.5.7 that
from which it follows that
as was to be shown,
for all
~ = (B - ~) + ~
~ - ~ ~- ~
are is
is an amart, o
110 The
enveloppe
2.6.6. Every
de S n e l l
yields
a Riesz
decomposition
for
is the
Consider
supermartingale
sum of a m a r t i n g a l e
and a bounded
semiamart.
~
a semiamart
~
~(A)
. F o r all
~
2.6.5
in
, ~)
of
~
martingale.
Then
the
set f u n c t i o n
a n d we h a v e
to s h o w t h a t
Fn
, choose
~(A)
~
, and
Rn~£ ba(Fn, ~)
the r e s t r i c t i o n s
A£
smallest
define
2"
in can b e s e e n
as in the p r o o f
holds
n 6~
a set f u n c t i o n process
is b o u n d e d .
~, ~ E T(n)
~(A)
majorizing
, define
.
a(F
that
with
A £ F
v l i m ~n(A)
:=
is a m e a s u r e
of C o r o l l a r y
and
semiamarts:
Theorem. semiamart
Proof.
Then
also
such
for all process
2"
:= ~ - ~
To this
end,
~
. Therefore,
which
is a
is a s e m i a m a r t , consider
n C~
that
+ I
and
~(n~A)
and define
~
stopping
~(~A)
+ I
times
g,
n a(~)
,
T E T(n)
by
,
if
c06A
(co)
,
if
c0 £ Q ~ A
~ (co)
,
if
~ £ A
n
,
if
co £ ~ A
letting
:=
and
z(~)
:--
T h e n we h a v e
(~n-Rn~) (A) -
(~n-Rn~) ( ~ A )
<
~g(~)
2 suPT
for all
n E~
and
AC
Fn
• Therefore,
2"
- ~T(~)
+ 2
[~(~)
[ + 2
is b o u n d e d ,
,
a
111
Instead of using the smallest m a j o r i z i n g
s u p e r m a r t i n g a l e for g e n e r a t i n g
the m a r t i n g a l e of a Riesz d e c o m p o s i t i o n for a semiamart, use its largest m i n o r i z i n g submartingale. least two
(and thus i n f i n i t e l y many)
Therefore,
one can also
a semiamart has at
d i f f e r e n t Riesz d e c o m p o s i t i o n s
w h e n e v e r the limits of the smallest m a j o r i z i n g s u p e r m a r t i n g a l e and the largest m i n o r i z i n g s u b m a r t i n g a l e are d i f f e r e n t from each other. By C o r o l l a r y 2.6.5, this is the case for every s e m i a m a r t w h i c h is not an amart.
F r o m the Riesz d e c o m p o s i t i o n for semiamarts, we o b t a i n a n o t h e r c h a r a c t e r i z a t i o n of s e m i a m a r t s w h i c h is similar to a c h a r a c t e r i z a t i o n of amarts
(Corollary 2.5.10):
2.6.7.
Corollary.
For a set function process
~
, the f o l l o w i n g are equivalent:
(a)
~
is a semiamart.
(b)
There exists a m a r t i n g a l e satisfying
Proof.
semiamart.
and a s u p e r m a r t i n g a l e
i~ - ~i < ~'
Suppose first that
decomposition,
~
is a semiamart.
choose a m a r t i n g a l e
~
such that
By the Riesz ~-
By T h e o r e m 2.6.4, the set function process
b o u n d e d semiamart. Now apply T h e o r e m 2.6.3 and define smallest s u p e r m a r t i n g a l e m a j o r i z i n g
i~- 5l
The c o n v e r s e is obvious from T h e o r e m 2.6.4.
is a b o u n d e d I~- ~ ~'
is
a
to be the
2.7.
R e m a r k s .
F r o m the lattice t h e o r e t i c a l point of view,
it is c o n v e n i e n t to define
certain p r o p e r t i e s of b o u n d e d m e a s u r e s in terms of their variations. The a d v a n t a g e in doing this is due to the fact that the v a r i a t i o n of a b o u n d e d m e a s u r e is identical w i t h its modulus and also is a m e a s u r e w h i c h is positive.
In this way, we have d e f i n e d ~ - c o n t i n u i t y and
~ - s i n g u l a r i t y of a b o u n d e d measure. Also, by T h e o r e m 2.1.5, c o u n t a b l e a d d i t i v i t y of a b o u n d e d m e a s u r e may be d e f i n e d in terms of its variation. A n o t h e r example will be given below.
Our d e f i n i t i o n s of ~ - c o n t i n u i t y and ~ - s i n g u l a r i t y are e q u i v a l e n t to those given by B o c h n e r and P h i l l i p s
[23] and Darst
[45], who p r o v e d the
L e b e s g u e d e c o m p o s i t i o n for b o u n d e d measures. While B o c h n e r and Phillips u s e d lattice t h e o r e t i c a l arguments,
Darst's proof is a purely m e a s u r e
theoretic one. For c o u n t a b l y a d d i t i v e measures,
a lattice t h e o r e t i c a l
proof of the L e b e s g u e d e c o m p o s i t i o n was given by Yosida
[133].
A measure
~ = 0
~ 6 ba(F, ~)
for each m e a s u r e
is purely f i n i t e l y additive if
~ £ b c a ( F , ~)
satisfying
l~l ~
holds
i~i . It is not hard
to see that the class of all purely f i n i t e l y a d d i t i v e m e a s u r e s on is identical w i t h
bca(F, ~ ) ±
2.1.4, it can be p r o v e n that the class a p r o j e c t i o n b a n d in of
bca(F, ~)
and
ba(F, ~) bca(F, ~ ) ±
F
° A l o n g the lines of the proof of T h e o r e m bca(F, ~ ) ±
, and that
is an A L - s p a c e and
ba(F, ~)
is the direct sum
. This is the Y o s i d a - H e w i t t d e c o m p o s i t i o n
w h i c h was first stated by W o o d b u r y [130].
W o o d b u r y also gave a sketch of
a proof, while the first c o m p l e t e proof was given by Y o s i d a and Hewitt [134].
The Y o s i d a - H e w i t t d e c o m p o s i t i o n was u s e d by C h a t t e r j i
[35,36]
in order
to c o n s t r u c t the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e of a b o u n d e d a d d i t i v e set function.
In C h a t t e r j i ' s construction,
the Y o s i d a - H e w i t t
d e c o m p o s i t i o n serves to reduce the p r o b l e m to the c o u n t a b l y a d d i t i v e case in w h i c h the c l a s s i c a l L e b e s g u e d e c o m p o s i t i o n can be applied. Due to the L e b e s g u e d e c o m p o s i t i o n for b o u n d e d a d d i t i v e set functions, the Y o s i d a - H e w i t t d e c o m p o s i t i o n is not n e e d e d in the c o n s t r u c t i o n of the g e n e r a l i z e d R a d o n - N i k o d y m derivative.
The lattice t h e o r e t i c a l and more
direct c o n s t r u c t i o n given in Section 2.1 also reveals the c h a r a c t e r i s t i c p r o p e r t i e s of the g e n e r a l i z e d R a d o n - N i k o d y m o p e r a t o r w h i c h are important but not obvious from C h a t t e r j i ' s approach.
113 In the c o u n t a b l y a d d i t i v e case, the idea of c o n s t r u c t i n g a g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e of a m e a s u r e goes back to A n d e r s e n and Jessen [1,2]. Using g e n e r a l i z e d R a d o n - N i k o d y m derivatives,
they proved the
c o n v e r g e n c e of a real m a r t i n g a l e and also o b t a i n e d the i d e n t i f i c a t i o n of its limit. The results of A n d e r s e n and J e s s e n w e r e e x t e n d e d by J o h a n s e n and K a r u s h
[85,86], Rao
[107], L a m b
[90], and others. C h a t t e r j i
[35,36]
was the first to study the f i n i t e l y a d d i t i v e case. For further c o m m e n t s on the g e n e r a l i z e d R a d o n - N i k o d y m derivative,
see C h a p t e r I.
The m o s t i n t e r e s t i n g set f u n c t i o n p r o c e s s e s are c e r t a i n l y those w h i c h arise from i n t e g r a t i n g a stochastic process. F r o m this, it is easy to u n d e r s t a n d the d e f i n i t i o n s and p r o p e r t i e s of a stopped set function process and the r e s t r i c t i o n of a m e a s u r e as b e i n g the c o u n t e r p a r t s , on the level of expectations,
of a stopped s t o c h a s t i c process and the
c o n d i t i o n a l e x p e c t a t i o n of a r a n d o m variable. W h i l e those set f u n c t i o n processes, w h i c h arise from i n t e g r a t i n g a stochastic process, c o n s i s t of m e a s u r e s which are c o u n t a b l y a d d i t i v e and even a b s o l u t e l y c o n t i n u o u s with respect to the u n d e r l y i n g p r o b a b i l i t y measure, for general set function processes.
this is not r e q u i r e d
For this reason, certain p r o p e r t i e s
of set function p r o c e s s e s are easier to obtain than the c o r r e s p o n d i n g ones for stochastic processes. A typical example of this kind is the Riesz d e c o m p o s i t i o n for amarts.
Roughly speaking,
structure p r o p e r t i e s
of set function p r o c e s s e s are easier to prove than those of stochastic processes, whereas
some more effort is n e e d e d for p r o v i n g c o n v e r g e n c e
theorems.
A l t h o u g h the d e f i n i t i o n of a m a r t i n g a l e as given in Section 2.3 is not the c l a s s i c a l one,
it is the a p p r o p r i a t e one in the c o n t e x t of amart
theory, and it also seems to be the m o s t c o m p r e h e n s i b l e one if m a r t i n g a l e s are looked at as b e i n g fair games. The gain of g e n e r a l i t y in m a r t i n g a l e theory, w h e n c o m p a r e d w i t h the theory of sequences of c o n d i t i o n a l e x p e c t a t i o n s , b e c o m e s clear from the L e b e s g u e d e c o m p o s i t i o n for b o u n d e d martingales. given by C h a t t e r j i
This L e b e s g u e d e c o m p o s i t i o n was i m p l i c i t l y
[35,36]; see also B a e z - D u a r t e
[4] for the L e b e s g u e
d e c o m p o s i t i o n for m a r t i n g a l e s w i t h a c o u n t a b l y a d d i t i v e limit measure, and N e v e u
[100] for a c o m p o n e n t w i s e L e b e s g u e d e c o m p o s i t i o n .
The latter
result a l s o indicates that the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a m a r t i n g a l e of m e a s u r e s n e e d not form a m a r t i n g a l e of r a n d o m variables. The proof of the m a r t i n g a l e c o n v e r g e n c e t h e o r e m given here is due to Chatterji
[35,36].
In the real case,
s u b m a r t i n g a l e s and s u p e r m a r t i n g a l e s are the m o s t
114
obvious g e n e r a l i z a t i o n s of martingales.
In a sense, the step from b o u n d e d
m a r t i n g a l e s to b o u n d e d s u b m a r t i n g a l e s is the essential one in e x t e n d i n g the class of all b o u n d e d m a r t i n g a l e s :
The b o u n d e d q u a s i m a r t i n g a l e s are
then o b t a i n e d as the set f u n c t i o n p r o c e s s e s in the linear hull of the b o u n d e d submartingales,
the p o t e n t i a l s are o b t a i n e d as the set f u n c t i o n
p r o c e s s e s in the ideal g e n e r a t e d by the D o o b potentials,
and the class
of all b o u n d e d amarts is identical w i t h the v e c t o r lattice g e n e r a t e d by the b o u n d e d m a r t i n g a l e s and potentials. Also,
the ideal g e n e r a t e d by the
b o u n d e d s u b m a r t i n g a l e s c o i n c i d e s with the class of all b o u n d e d semiamarts (or T - b o u n d e d set f u n c t i o n processes).
F u r t h e r results on s u b m a r t i n g a l e s
may be found in Chapter 4, Section 3.
Q u a s i m a r t i n g a l e s were i n t r o d u c e d by Fisk Orey
[102] and Rao
[70]. They were also studied by
[106]. The d e c o m p o s i t i o n of a b o u n d e d q u a s i m a r t i n g a l e
into the d i f f e r e n c e of two p o s i t i v e s u p e r m a r t i n g a l e s is due to Rao see also S t r i c k e r
[106];
[122]. The lattice p r o p e r t y of the class of all b o u n d e d
q u a s i m a r t i n g a l e s was first p r o v e n by Y o e u r p
[132] and later by J e u l i n
[84], who u s e d a lattice t h e o r e t i c a l argument. for q u a s i m a r t i n g a l e s ,
In the Riesz d e c o m p o s i t i o n
the b o u n d e d n e s s c o n d i t i o n can be omitted. We thus
have the f o l l o w i n g Riesz d e c o m p o s i t i o n for general q u a s i m a r t i n g a l e s : 2.7.1.
Theorem.
Every q u a s i m a r t i n g a l e is the sum of a m a r t i n g a l e and the d i f f e r e n c e of two Doob potentials.
Proof.
Since every q u a s i m a r t i n g a l e is an amart,
m a r t i n g a l e and a potential, quasimartingale.
it is the sum of a
by T h e o r e m 2.5.8. But this p o t e n t i a l is a
Since every p o t e n t i a l is bounded,
the Riesz d e c o m p o s i t i o n
for b o u n d e d q u a s i m a r t i n g a l e s can be a p p l i e d to the potential. F r o m this the a s s e r t i o n follows,
a
A m a r t t h e o r y e m e r g e d from a n u m b e r of papers c o n c e r n i n g e x t e n s i o n s of the m a r t i n g a l e c o n v e r g e n c e theorem. The first results on amarts of random v a r i a b l e s c o n c e r n e d amarts w i t h an integrable supremum; Mertens
[92], and Baxter
v a r i a b l e s started w i t h the w o r k of Austin, and Chacon [56,57];
see M e y e r
[93],
[5]. The general theory of amarts of r a n d o m Edgar, and Ionescu Tulcea
[3]
[32]. The t e r m "amart" was i n t r o d u c e d by Edgar and S u c h e s t o n
see also S u c h e s t o n
[123]. As r e m a r k e d in C h a p t e r I, the general
theory of amarts of m e a s u r e s started e a r l i e r than the general theory of amarts of r a n d o m variables. [35,36] and Lamb
It dates b a c k to the p a p e r s by C h a t t e r j i
[90], a l t h o u g h these authors did not work e x p l i c i t l y
w i t h b o u n d e d s t o p p i n g times.
In the f o r m of C o r o l l a r y 2.5.15, the a m a r t
115
c o n v e r g e n c e t h e o r e m is due to C h a t t e r j i
[35,36] who also gave a sketch of
its proof. Our direct proof of C o r o l l a r y 2.5.15 is b a s e d on C h a t t e r j i ' s arguments.
In C h a t t e r j i ' s proof, however,
the stopping time a r g u m e n t
is missing. His proof is thus incomplete, but not really incorrect, as r e g r e t t a b l y stated in
[113,117]. C o r o l l a r y 2.5.10, w h i c h yields the
e q u i v a l e n c e of C h a t t e r j i ' s result and the amart c o n v e r g e n c e theorem, is due to G h o u s s o u b and S u c h e s t o n o b s e r v e d in
[75]. This e q u i v a l e n c e was first
[113,117]. The proof of the a m a r t c o n v e r g e n c e t h e o r e m given
here is taken from
[117].
For amarts of r a n d o m variables,
the c o n v e r g e n c e t h e o r e m was p r o v e n by
Austin, Edgar, and Ionescu Tulcea
[3] and Chacon
[32]. For further
results and proofs related to the a m a r t c o n v e r g e n c e theorem, Baxter
[6], B e l l o w
and D v o r e t z k y
[13,14], B e l l o w and D v o r e t z k y
[16], Chen
see also [38,39,40],
[52,53]. The first e x p o s i t o r y paper on amarts is that of
Edgar and S u c h e s t o n
[57]; see also
[56,123].
S t a b i l i t y p r o p e r t i e s of amarts were a l r e a d y d i s c u s s e d by Austin, Edgar, and Ionescu Tulcea
[3], Edgar and S u c h e s t o n
[56,57], and B e l l o w
[8,9].
The proof of the lattice p r o p e r t y of the class of all b o u n d e d amarts given in Section 2.5 does not rely on the Riesz d e c o m p o s i t i o n . on the Riesz decomposition,
however,
see C h a p t e r 4, Section 4. B e l l o w
Based
a more elegant proof can be given;
[8,9] also gave c o n d i t i o n s under w h i c h
a c o n t i n u o u s f u n c t i o n t r a n s f o r m s b o u n d e d amarts into b o u n d e d amarts.
A different, but t y p i c a l l y stochastic type of s t a b i l i t y is the stability of amarts u n d e r o p t i o n a l s t o p p i n g and o p t i o n a l sampling.
The f o l l o w i n g
result is the o p t i o n a l stopping t h e o r e m for amarts:
2.7.2. If
~
Theorem. is an amart and
function process basis Proof.
[*
2"
:= { Fn^ O Let
T*
I nE~
times
I n E~
}
is an amart on the stochastic
} .
F* . Fix
~, n q T(k)
v, T E T(k)
is an a r b i t r a r y stopping time, then the set
d e n o t e the class of all b o u n d e d stopping times for
the stochastic basis
holds for all
a
:= { ~ n ^ o
. For
by letting
c £ (0,~)
and choose
v*, T* £ T*(k^u)
k E~
such that
, define stopping
116
v(~)
:=
p
,
if
~ 6 {v*=pAO}
T(~)
:=
p
,
if
~E
and
Then we
we
have
~*
= ~AO
and
{r*=pAO}
r*
= ~AO
<
~)* (CO)
. Moreover,
for
all
~ £ {k>a}
have
O(CO)
:
(kAo) (CO)
:
(VAO) (CO)
by
letting
<
O(CO)
,
hence
(VAO)(~) and
=
C(CO)
=
c(~o)
,
similarly
( T ^ o ) (co)
Now
define
stopping
x(~)
times
M,
n £ T(k)
(VAO) (CO)
,
if
CO £ {k
k
,
if
~ £ {k>o}
(~AO) (~)
,
if
~ E {k
k
,
if
~ E {k>a}
:=
and
~(~)
Then
we
:=
have
l~v.(n) -uT.(Q) I
<_
I~VAo({k
<
as
was
to
be
shown.
~
,
--~I Ao({k<_o}) I
[~^o({k>o})
- ~TAo({k>c})
[
,
117 We also h a v e the f o l l o w i n g o p t i o n a l samplin 9 t h e o r e m for amarts:
2.7.3. If
~
Theorem. is an amart and
{ ~n E T(n)
I n 6~
}
is an increasing sequence
of b o u n d e d stopping times, then the set f u n c t i o n p r o c e s s ~*
:= { ~T
F* := { F n -T n Proof.
I n E~
}
I n E~
} .
Let
T*
denote the class of all b o u n d e d stopping times for
the stochastic basis
I~.
~* £ T*(T k)
holds for all
E £ (0,~)
and c h o o s e
k E~
such that
_<
~, n C T(k)
{~*=p}
, we h a v e
=
~* E T(k)
Z {v*=Tn}n{rn= p} n=k
p E~
I~o.(~) for all
F* . Fix
(~) - ~.Tc(~) I
holds for all For all
is an amart on the s t o c h a s t i c basis
since
6
F P
. Therefore, we have
-l~T.(n)
I
<_
~
,
a*, T* £ T*(T k)
The g r o w t h c o n d i t i o n
T
£ T(n) , for all n E ~ , can be o m i t t e d in the n case of amarts of r a n d o m variables, but a p p a r e n t l y it c a n n o t be r e m o v e d for amarts of measures. N e v e r t h e l e s s , the Riesz d e c o m p o s i t i o n ,
c o m b i n e d w i t h a very weak form of
this w e a k form of the o p t i o n a l s a m p l i n g t h e o r e m
is still s u f f i c i e n t l y strong for p r o v i n g another o p t i m a l i t y result for amarts. Let
(C) be an arbitrary, but fixed p r o p e r t y of set f u n c t i o n
processes.
2.7.4.
Theorem.
Suppose that (i)
e v e r y set f u n c t i o n p r o c e s s of a m a r t i n g a l e lim ~n(~)
(ii}
~
exists,
~
having property
and a set f u n c t i o n process
~
for w h i c h
and that
for each set f u n c t i o n process each i n c r e a s i n g sequence
~
having property
{ T n E T(n)
I n 6~
s t o p p i n g times, the set f u n c t i o n process property
(C) is the sum
}
{ ~n
(C) and for
of b o u n d e d I n E~
(C).
Then every set f u n c t i o n process h a v i n g p r o p e r t y
(C) is an amart.
}
has
118
Proof.
Consider
a exists, If
process
~
having property
(C). T h e n
l i m ~n(~)
(i) and T h e o r e m
{ i n E T(n)
times, by
by
:=
a set f u n c t i o n
I n E~
}
2.3.1.
is any i n c r e a s i n g
t h e n the set f u n c t i o n
(ii), h e n c e
l i m ~r
(~)
process
sequence
{ ~I
exists,
I n 6~
of b o u n d e d }
stopping
has p r o p e r t y
(C),
n
n Now select a sequence for all
k 6~
For each
{ n k 6~
I k E~
, define
a stopping
nk m
satisfying
n k < Tnk < nk+ 1 ,
.
m£~
o
}
(~)
time
O m E T(m)
,
if
m
=
2k - I
,
if
m
=
2k
by letting
:=
T
(~)
nk Then times,
{ O m 6 T(m) hence
Therefore,
I m 6~
}
is an i n c r e a s i n g
the set f u n c t i o n lim Bo
(~)
process
exists,
sequence
{ ~Om
I m E~
of b o u n d e d }
stopping
has p r o p e r t y
(C).
a n d we h a v e
m l i m ~n(~)
=
l i m ~nk (Q)
=
l i m PO2k_l (~)
=
l i m ~ O m (~)
lim Ba2k(~)
=
lim Bo
as w e l l as
! i m ~T
(~)
=
l i m ~r
n
(~)
=
nk
(Q) m
which yields
a
=
l i m ~n(~)
=
lim ~r
(~) n
Finally,
suppose
that
s E (0,~)
such that,
T n £ T(n)
satisfying
e
<
lim
is n o t an amart.
for e a c h
n 6~
Then there exists
, there exists
a stopping
some time
la-I~T. (n) I n
S i n c e the s t o p p i n g subsequence
~
t i m e s are b o u n d e d ,
{ Tnk 6 T ( n k) la - ~T
(~) I
I k E~ =
we m a y s e l e c t an i n c r e a s i n g
} . F o r this s u b s e q u e n c e ,
0
nk B u t this c o n t r a d i c t s
the a s s u m p t i o n ,
hence
_~
is an amart.
we h a v e
'
119
This r e s u l t means, to a larger class form of e i t h e r This r e s u l t the limit;
that
it is i m p o s s i b l e
of set f u n c t i o n
see C h a p t e r
counterpart
of T h e o r e m
and S u c h e s t o n
2.7.4
of r a n d o m variables.
any r e a s o n a b l e
sampling
theorem.
to m a r t i n g a l e s
in
of these processes.
processes
the a n s w e r
to c o m p a r e
For a r b i t r a r y
or not t h e i r g e n e r a l i z e d
form an a m a r t of r a n d o m variables.
2.7.5.
loosing
respect
discussion
for s t o c h a s t i c
it seems to be s u i t a b l e
is not c l e a r w h e t h e r
however,
without
or the o p t i o n a l
interesting.with
5 for a b r i e f
the class of amarts
The
is due to E d g a r
[61].
At this point, with amarts
processes
the Riesz d e c o m p o s i t i o n
is p a r t i c u l a r l y
to e x t e n d
amarts
amarts
of m e a s u r e s
of measures,
Radon-Nikodym
For b o u n d e d
amarts
it
derivatives
of measures,
is positive:
Theorem.
The g e n e r a l i z e d
Radon-Nikodym
derivatives
of a b o u n d e d
a m a r t of m e a s u r e s
form an a m a r t of r a n d o m variables.
Proof.
F i r s t note
a potential Riesz
that the g e n e r a l i z e d
of m e a s u r e s
decomposition
form a p o t e n t i a l
for amarts,
to a b o u n d e d m a r t i n g a l e Jordan
decomposition
construction
~
[100; P r o p o s i t i o n
III-2-7] of
Radon-Nikodym
- ~
. It then
Radon-Nikodym
form a p o s i t i v e The
derivatives
, ~)
By the attention
follows
from the
derivatives
and from
Radon-Nikodym
apply
to
5-
of
. N o w the
supermartingale,
same a r g u m e n t s
form an a m a r t of r a n d o m v a r i a b l e s
to r e s t r i c t ~£ba(F
that the g e n e r a l i z e d
~+
amart of r a n d o m variables. generalized
~ = ~
of the g e n e r a l i z e d
of the r e s t r i c t i o n s
limit m e a s u r e .+
derivatives
of r a n d o m variables.
it is s u f f i c i e n t
with
yields
Radon-Nikodym
derivatives h e n c e an
. Thus,
the
of the b o u n d e d m a r t i n g a l e
(although
they need not form a
martingale),
Semiamarts
were
u
introduced
by K r e n g e l
Krengel
[87]. T h e s e a u t h o r s
bounded
semiamarts
amarts
also
introduced
with an a d d i t i o n a l
in the c l a s s of s e m i a m a r t s
enveloppe
de Snell goes b a c k
semiamarts
of m e a s u r e s ,
On the trivial and o n l y
de Snell,
stochastic
basis,
if its g e n e r a l i z e d
On an a r b i t r a r y
to G h o u s s o u b
stochastic
a
The
see also Snell
(bounded)
property
w h i c h are of
of the
[75]. F o r
first c o n s t r u c t e d [121] a n d N e v e u
semiamart
derivatives
this e q u i v a l e n c e
see also
identification
and S u c h e s t o n
de Snell was
Radon-Nikodym basis,
[88,89];
semipotentials
property.
by a p a r t i c u l a r
the e n v e l o p p e
[112]. F o r the e n v e l o p p e
and S u c h e s t o n
is an a m a r t converge
in
[100].
if
a.e.
need not be true.
120
By the f o l l o w i n g Tulcea
example,
although
their g e n e r a l i z e d
2.7.6.
process
~
for all
n 6~
and
lim D n ~ n
1
is taken
Closely
:=
I
k £ K(n)
=
0
[50]
Edgar,
and I o n e s c u
fail to be an a m a r t converge:
denote
,
otherwise
. Then
[88,89]
the class
and
~
is odd and
k = I
is a b o u n d e d
semiamart
which
we have
,
measure.
are the f o l l o w i n g
by C h o w
well
problems
processes, [41]
and Schmidt
which were
by D u b i n s
and
for s u b m a r t i n g a l e s ,
[110]
for semiamarts.
suited to the m e a s u r e of set f u n c t i o n
of all s t o p p i n g
a set f u n c t i o n A 6 F
and by These
theoretic
approach.
processes,
we h a v e
times
process
~
for the s t o c h a s t i c
. For an a r b i t r a r y
basis
stopping
, the series
~p(AN{a=p})
n e e d not converge.
Thus,
in the u s u a l way. it can be p r o v e n
Therefore,
0
, d e f i n e a set f u n c t i o n
the notation:
Z p=1
then
n
t h e m in the f r a m e w o r k
, and c o n s i d e r
~o
if
of s t o c h a s t i c
are p a r t i c u l a r l y
stating
~ £ S
,
for m a r t i n g a l e s ,
and S u c h e s t o n
to e x p l a i n
I
to s e m i a m a r t s
in the f r a m e w o r k
problems
[0,1)
to be the L e b e s g u e
related
Krengel
on
a.e.
studied,
time
which
derivatives
basis
Moreover,
Freedman
S
Radon-Nikodym
stochastic
fails to be an amart.
Let
semiamarts
by letting
~n"(Bn,k)
Before
is due to Austin,
Example.
On the s t a n d a r d
if
which
[3], there e x i s t b o u n d e d
we define,
I~I(A)
it m a y be i m p o s s i b l e
If the a b o v e
series
that the i d e n t i t y for all
:=
N o w the a b o v e m e n t i o n e d
Z p=l
~6 S
and
to d e f i n e
converges
of Lemma A£ F
a set f u n c t i o n
for all
2.2.1
A E F
,
obtains.
,
I~pI(AN{~=p}]
problems
can be stated
in the f o l l o w i n g
way:
121
When
-
-
is the v a l u e
SUPs
I~oI(~)
finite?
When
is the value
sup s
I~I(~)
attained
time
in
for some
stopping
S ?
It can be shown that the first p r o b l e m
reduces
to b o u n d e d
stopping
times,
due to the i d e n t i t y
sup S
I~oI (~)
It then follows only
if
~
out that
suPT
from T h e o r e m
is a b o u n d e d SUPs
is either process
=
I~oI(~)
be attained.
fails
these
it reveals
that
T-boundedness theory.
By T h e o r e m
-
-
Example
n £~
the sets
stopping
on
of a m a r t
times.
This
In our opinion,
boundedness
property
is
in a m a r t
the v e c t o r
by the p o s i t i v e
lattice
of all T - b o u n d e d
of all
set f u n c t i o n
set f u n c t i o n
lattice but
processes.
of all b o u n d e d
amarts
it need not be c o m p l e t e
since these n o r m s may
inequality
(Lemma
set f u n c t i o n
in the m a x i m a l
be r e p l a c e d
2.7.7.
some c o m m e n t s merits
fail to be equivalent,
for by
2.4.9.
T-bounded
For
important
the ideal g e n e r a t e d
for the T-norm,
The m a x i m a l
that
if
set f u n c t i o n
properties.
in the v e c t o r
2.5.3,
is c o m p l e t e
-
important
consists
the ~ - n o r m
S
this c l a i m by some examples: 2.6.4,
By T h e o r e m
it turns
in
the s u p r e m u m m a y not
on real amarts by g i v i n g
to b o u n d e d n e s s
supermartingales processes
time
if and
[114].
one of the m o s t
is the really
is finite
or an ~ - b o u n d e d
otherwise,
the role of the b o u n d e d
respect
We i l l u s t r a t e
I~oI(~)
stopping
is not ~ - b o u n d e d
remarks
theory
true w i t h
for some
see Schmidt
It is c e r t a i n l y
SUPs
As to the second problem,
to be a semiamart;
the T-norm.
also
that
semiamart.
For details,
Let us c o m p l e t e
2.2.4
is a t t a i n e d
a semiamart which
which
[~TI (~)
2.5.12)
processes.
inequality
makes
sense only
The f o l l o w i n g
the T - n o r m
cannot
example
for shows
in general
by the ~ - n o r m :
Example. and Bn, p
k £ K(n) ,
p 6K(n)
, let
F2n+k_2
, and define
denote
the algebra
a measure
by letting
~ 2 n + k _ 2 (B n , p )
2 -n
t
if
k = p
0
,
otherwise
:=
~2n+k_2
generated on
by
F2n+k_2
122
Then the set f u n c t i o n p r o c e s s
su~
]~n[(n)
=
2-I
sup T
l~xl(S)
=
1
_~
is a b o u n d e d
semiamart,
and we have
and
Moreover,
if
l
is taken to be the L e b e s g u e measure,
(J X) ( { s u ~ I D n ~ n l > E}) for all
e £ (0,1)
sup~ for all
=
then we have
I
, hence
l~nl(n)
<
e ( J l ) ( { s u ~ I D n ~ n l > e})
<
suPT
l~xl(~)
e £ (2-1,1)
Due to T h e o r e m 2.2.4, T-boundedness
the d i s t i n c t i o n b e t w e e n ~ - b o u n d e d n e s s
and
is s o m e w h a t o b s c u r e d in the real case, but it w i l l turn
out to be substantial and i n d i s p e n s a b l e in the v e c t o r - v a l u e d case.
3.
Amar
t s
To a large extent,
in
the p r o p e r t i e s
space can be a t t r i b u t e d turn,
depend
differ
a
on the s t r u c t u r e
Lebesgue
measure,
which
Radon-Nikodym
be T-bounded, an infinite
this
Radon-Nikodym
unless
and even a T - b o u n d e d
dimensional
Banach
amarts
and the v a r i a t i o n
to the e x i s t e n c e
space has the
Banach
strong a m a r t
space may fail to
the R a d o n - N i k o d y m
connected the
with
These
the fact that,
semivariation
n o r m are not equivalent,
in
property
st r o n g l y b u t only weakly.
variation,
By the
amart of r a n d o m v a r i a b l e s
space h a v i n g
are c l o s e l y
of b o u n d e d
if and only
part of the limit
an ~ - b o u n d e d
dimensional strong
in
an ~ - b o u n d e d
derivative.
reduces
the B a n a c h
example,
dual n e e d not c o n v e r g e
of strong
For example,
space c o n v e r g e s
of the A - c o n t i n u o u s
As a n o t h e r
which,
space and m a y c o n s i d e r a b l y
latter c o n d i t i o n
in an infinite
for v e c t o r m e a s u r e s sup-norm)
in a Banach
in a B a n a c h
is not g u a r a n t e e d property.
and a s e p a r a b l e properties
processes
of v e c t o r m e a s u r e s
of real measures.
derivative
of r a n d o m v a r i a b l e s
of stochastic
has a g e n e r a l i z e d
decomposition,
of a R a d o n - N i k o d y m
.
of the B a n a c h
of r a n d o m v a r i a b l e s
if its limit m e a s u r e
space
to the p r o p e r t i e s
from the p r o p e r t i e s
martingale
B a n a c h
unless
norm
(or
the B a n a c h
space has finite dimension.
The a b o v e - m e n t i o n e d dimensional
Banach
deficiencies
u n i f o r m weak amarts: property whereas
Uniform
amarts
and form an e n t i r e l y u n i f o r m w e a k amarts
the weak c o n v e r g e n c e Banach
space
of strong a m a r t s
space lead to the n o t i o n s are
theorem remains
is thus much
strong
satisfactory
generalize
richer
than
in an i n f i n i t e
of u n i f o r m amarts and amarts with an a d d i t i o n a l
counterpart
strong amarts
valid.
to real amarts,
in such a way that
The t h e o r y of a m a r t s
the theory of real amarts.
in a
124
We shall start w i t h a rather d e t a i l e d d i s c u s s i o n of vector m e a s u r e s and their r e p r e s e n t i n g linear o p e r a t o r s
(Section 3.1). A f t e r giving some
basic results on v e c t o r - v a l u e d set f u n c t i o n p r o c e s s e s we shall study m a r t i n g a l e s u n i f o r m amarts
this c h a p t e r with some remarks
T h r o u g h o u t this chapter,
its dual.
If
U(~) ~
(Section 3.4),
(Section 3.5), as well as w e a k amarts, u n i f o r m weak
amarts, and weak sequential amarts
II. II , let
(Section 3.2),
(Section 3.3), strong a m a r t s
let
~
(Section 3.6)~ We shall c o n c l u d e
(Section 3.7).
be a
(real) Banach space with n o r m
denote its closed unit ball, and let and
~
are Banach spaces,
all b o u n d e d linear o p e r a t o r s
~
> ~
~'
denote
then the B a n a c h space of
will be d e n o t e d by* ~ ( ~ , ~)
3.1.
V e c t o r
This
section
consists
bounded
vector
on the
sup-norm
These
results
out
result
shed
to b e u s e f u l
lattice.
In t h e
for vector
in t h e
For detailed
Let
be an algebra
and uniform
investigation
variation derivative
space with
of all
measures
amarts.
measures,
bounded on t h e
also turn
in a B a n a c h
and used
of a v e c t o r
variation.
and also
They will
measures
is g i v e n
functions.
between
the Lebesgue
the Radon-Nikodym
on v e c t o r
simple
existing
that
operators
of b o u n d e d
variation,
of vector
section,
i t is s h o w n linear
decomposition to construct
measure
of b o u n d e d
property.
we refer
to t h e b o o k b y
[49].
A set function
~
~(A+B)
holds
lattice
of b o u n d e d
part of this
information
and Uhl
F
amarts
Radon-Nikodym
Diestel
first part, by bounded
for vector
measures
of bounded
in a B a n a c h
In t h e
light on the difference
strong
measures
the generalized variation
some
second
.
of the vector
is p r o v e n
and vector
between
e s
may be represented
completion
measures
difference
a s ur
of two parts.
measures
A corresponding
vector
me
on a set
: F
=
> ~
~(A)
~
.
is a d d i t i v e
if t h e
identity
+ B(B)
f o r e a c h p a i r of d i s j o i n t
sets
A, B 6 F , a n d
it is b o u n d e d
if t h e
value
sup F
is f i n i t e . measures. and
lJ ~(A)
In t h e Let
b a ( F , ]E)
a(F, ~ )
. Endowed
scalars,
these
i
is a n o r m although
on
sequel,
a(F, ~ )
let
by
with
sup F
ba(F, ~)
measure
set functions
the class
the class
are vector
Jl ~(A)
be called
of all vector
measures
vector
vector F
measures
> ~ in
defined
addition
and multiplication
spaces.
Clearly,
the map
H
, but we
norm
will
of all b o u n d e d
the pointwise
classes
>
additive
denote
denote
equivalent
For a vector
11
on this
~ 6 a(F, ~ )
shall
see that there
space.
, define
a map
is a m o r e
natural
,
126
III ~ III
:
,.I-
by l e t t i n g
III ~ Ill(A)
for all
:=
A E F , where
sup
II Z ~i~(Ai)
II
,
the s u p r e m u m
is t a k e n o v e r all p a r t i t i o n s
and scalars
~I' ~2 . . . . .
{ A I , A 2 ..... ~ }
E P(A)
Then
is a s u b a d d i t i v e
set f u n c t i o n w h i c h w i l l be c a l l e d
of
~ . F o r all
A E F , we h a v e
li ~(B) II
III ~ III(A)
III ~ III
semivariation
supF(A) [49; P r o p o s i t i o n
I.I.11].
<_
Consequently,
a n d o n l y if its s e m i v a r i a t i o n
<
~
sup F II ~(A) II
I
>
III ~ III(S)
[-1,1] the
2 s u P F ( A ) II ~(B) II
a vector measure
is b o u n d e d ,
~
eke
is b o u n d e d
if
a n d the m a p s
and
are equivalent
n o r m s on
the s e m i v a r i a t i o n vector measures
ba(F, ~ )
. The above-mentioned
preference
n o r m w i l l be j u s t i f i e d b y the r e p r e s e n t a t i o n
by bounded
linear operators
for
of b o u n d e d
w h i c h w e a r e g o i n g to d i s c u s s
now.
For a vector measure
~ E a(F, ~ )
T O : ]DO
, define a linear operator
> ]E
by letting
To(Z~i~A
1
)
:=
for e a c h s i m p l e f u n c t i o n operator
]Do
(A) for all
> ~
=
Z ~i~(Ai) Z ~iXAi
£ ]Do " T h e n
T
o
satisfying
ToX A
A q F . Furthermore,
SUPu (]Do)
,
we have
TO( X ~iXA'I )
=
III ~ III(Q)
,
is the u n i q u e
linear
127
w h i c h means case,
To
is b o u n d e d if and only if B is bounded. o has a u n i q u e e x t e n s i o n to a b o u n d e d linear o p e r a t o r that
the s u p - n o r m c o m p l e t i o n representin~
denote
X
T0X
Do
" This e x t e n s i o n
of the b o u n d e d in
on
~
the
, and it
satisfying
,
vector measure
X
> ~ ( m , ~)
:
T
will be c a l l e d
vector measure
~ ( D , ~)
is the u n i v e r s a l
ba(F, ~)
the map a s s o c i a t i n g
its r e p r e s e n t i n g
3.1.1.
of
linear operator
=
where
D
linear o p e r a t o r
is the u n i q u e
In this
T
on
w i t h each b o u n d e d
linear operator.
F . Let
vector measure
F
>
T h e n we have:
Theorem.
The class the map
ba(F, ~) X
Proof.
is an isometric
The m a p
E b a ( F , ~) III U I]1 ( f i )
is a B a n a c h
and
=
X
for the n o r m
isomorphism
is c l e a r l y
T£~(
space
D , ~)
of
III. lll(fi) , and
ba(F, ~)
onto
linear and b i j e c t i v e .
satisfying
~ = T 0X
~ ( D , ~)
Moreover,
for
, we have D
I] T I[
L e t us n e x t c o n s i d e r
vector measures
For a v e c t o r m e a s u r e
II~II
"
6 a(F, ~)
of b o u n d e d v a r i a t i o n .
, define
a map
> ~+
F
by letting
II ~ II ( A ) for all called
: =
A6 F . Then the v a r i a t i o n
variation
is bounded,
finite dimension; vector measures
sup?(A ) Z II ~ II
of
is an a d d i t i v e . Clearly,
Let
In or d e r to c h a r a c t e r i z e
set f u n c t i o n
in
vector measures
linear operators,
w h i c h w i l l be of b o u n d e d
n e e d not be true unless
bva(F, ~)
of b o u n d e d v a r i a t i o n
,
each vector measure
b u t the c o n v e r s e
see below.
of their r e p r e s e n t i n g defin i t i o n :
~
II ~ ( A i) II
denote
~
has
the class of all
a~F, ~)
of b o u n d e d v a r i a t i o n let us recall
in terms
the f o l l o w i n g
128
If
~
linear
is a B a n a c h
lattice
operator
........>......~..
~
and
~
is a B a n a c h
is c o n e
absolutely
summable
sequences
in the p o s i t i v e
summable
sequences
in
absolutely such
summing
cone
. A linear
if a n d o n l y
of
~
into
operator
if t h e r e
then a bounded if it m a p s
the a b s o l u t e l y
S 6 ~ ( ~ , ~)
exists
the
is c o n e
a constant
p 6~+
that
Z holds
II Sx i II
for e a c h
absolutely smallest
where
constant
the
is a n o r m
collection
=
is t a k e n
~
[109;
over
c~+
, let
. For a cone
II S II1
inequality.
denote
the
T h e n we h a v e
,
all
finite
collections
II Z x i Hi = I . M o r e o v e r ,
the m a p
II S II1
> ~
Section
3.1.2.
> ~
the a b o v e
satisfying
>
If' (x i) I
{Xl,X2,...,Xk} :~
II S x i II
sup Z
on the c l a s s
operators
S
satisfying
c~+
I
P SUPu ( IF ' ) Z
operator
supremum
{Xl,X2,...,Xk}
S
_<
finite
summing
II S II1
see
~
space,
summing
~ i ( ~ , ~)
, which
of all
is a B a n a c h
cone
space
absolutely
for
this
summing
norm.
For details,
IV.3].
Theorem.
The c l a s s the m a p
bva(P, 2) X
Proof.
In o r d e r
a vector
is a B a n a c h
is a n i s o m e t r i c
measure
to p r o v e
space
isomorphism
the t h e o r e m ,
~ 6 ba (P, ~.)
and
for
the n o r m
of
bva(F, 2)
II . II (Q) onto
it is s u f f i c i e n t
its r e p r e s e n t i n g
, and
11.1( • , ]E)
to c o n s i d e r
linear
operator
T 6"IT.( ]19, ~.) Suppose
first
collection
that
~
has bounded
{x I ,x2,... ,x k}
II X x i II = I . C h o o s e ~ij 6JR+
such
variation.
of p o s i t i v e
a partition
Consider
simple
{B I , B 2 , . . . , B m} £ F(~)
that m
xi holds
for all k X i=1
=
X j=1
~ijXB.3
i £ {I,2, .... ,k}
ti Tx i il
=
k Z i=I
. Then we have m
li Z j=1
~ij~(Bj)
II
a finite
functions
satisfying and
scalars
129 m
k
z j=1
x i=I
~ij
<
sup j
k x i=I
~ ~ij
=
II ~ x i II II ~ ll(n)
=
II ~ II (n)
<
hence
T
is c o n e
absolutely
II T II1
<
Conversely,
if
partition
T
II ~(B.) 3
II
II ~ ll(n)
,
summing
a n d we h a v e
II ~ll(s) is c o n e
{ A I , A 2 ..... % }
5- II ~(A i) II
absolutely 6 P(~)
=
z
summing,
t h e n w e have,
for e a c h
II ~ XA. II
II T II1
,
II TXA.
II
<
II T II1
l
hence
~
has bounded
II B II (~) From
this
the
variation
<
In o r d e r
to g i v e
of b o u n d e d
and we have
li T [iI
assertion
measures
=
l
follows.
a variant
of the p r e c e d i n g
variation,
let us a l s o
characterization recall
the
of v e c t o r
following
definition:
If
~
and
~
are Banach
is a b s o l u t e l y
summin~
absolutely
summable
absolutely
summing
spaces,
if it m a p s
sequences if a n d o n l y
in
then a bounded the
summable
sequences
I{ . A l i n e a r
if t h e r e
linear
operator in
operator
exists
a constant
~
~
>
into
the
S 6~( ~, ~) p£]R+
is
such
that
I holds
Jl Sx iil
for e a c h
summing constant
<
finite
operator
S
satisfying
P SUPu(IF') collection
:~
> ~
the a b o v e
Z
If' (x i)
{xl,x2,...,XkJ
, let
it S il
inequality.
as Then
c~
denote
. F o r an a b s o l u t e l y the
the m a p
smallest
130
S
I
•
II S llas
is a n o r m on the class operators see
~
[83] and
3.1.3.
~ a s ( ~ , ~)
of all a b s o l u t e l y summing
, w h i c h is a B a n a c h space for this norm. For details,
[104].
Corollary.
The map
X
Proof. and
> ~
is an isometric i s o m o r p h i s m of
Since
D
~ a s ( 9 , ~)
is an AM-space,
are identical;
see
bva(F, ~)
the B a n a c h spaces
onto
~ a s ( 9 , ~)
~ i ( D , ~)
[109; Section IV.5].
A p a r t i c u l a r situation arises in the case w h e r e
~
has finite dimension.
Let us first c h a r a c t e r i z e a b s o l u t e l y summing o p e r a t o r s in terms of v e c t o r measures.
This c h a r a c t e r i z a t i o n will then lead to a c h a r a c t e r i z a t i o n of
finite d i m e n s i o n a l Banach spaces.
3.1.4.
Theorem.
Suppose
~
and
~
linear operator.
are B a n a c h spaces and
S :~
•
is a b o u n d e d
Then the f o l l o w i n g are equivalent:
(a)
S
(b)
There exists a c o n s t a n t
is a b s o l u t e l y summing. p E]R+
such that
II S~ IJ(Q) < p III ~ III(~) holds for each algebra each v e c t o r m e a s u r e (c)
S~ £ bva(F, ~) measure
Moreover, with
if
S
~£ba(F,
and for
~)
holds for each algebra
~£ba(F,
F
F
and for each v e c t o r
~)
is a b s o l u t e l y summing,
then the i n e q u a l i t y in
(b) holds
p = II S llas
Proof.
Suppose first that
vector measure T 6 ~ ( D , ~)
~6ba(F,
. Then
~)
S 0 T
S
is a b s o l u t e l y summing. C o n s i d e r a
w i t h r e p r e s e n t i n g linear o p e r a t o r
is a b s o l u t e l y summing, and it is the
r e p r e s e n t i n g linear o p e r a t o r of the v e c t o r m e a s u r e b o u n d e d varia{ion,
Therefore,
=
II S 0 T llas
<
II S llas II T II
(a) implies
(b) implies
3.1.5 that
S~
by C o r o l l a r y 3.1.3, and we have
II S~ II (n)
Clearly,
S~ . Hence
=
II S llas
III l~ III (n)
(b).
(c), and it follows f r o m the s u b s e q u e n t Example
(c) implies
(a).
has
131
3.1.5.
Example:
Suppose
~
and
~
are Banach
linear operator which
Then there exists a summable sequence Consider Bn, 2 :=
{ Sx n
I n E~
}
the a l g e b r a [2-n,2 -n+1)
spaces and
is n o t a b s o l u t e l y
{ xn E~
is n o t a b s o l u t e l y
F ,
sequence
S :~
on
[0,1)
n 6~
--~
is a b o u n d e d
summing.
which
, and define
I n 6~
}
such t h a t the
summable.
is g e n e r a t e d
b y the sets
a vector measure
~ 6 a(F, ~)
by l e t t i n g
~(Bn, 2) for all
n 6~
:=
. T h e n the v e c t o r m e a s u r e
{ ZH x n is b o u n d e d
3.1.6.
I H c~
[83; T h e o r e m
have bounded
,
xn
finite 14.6.1],
~
b u t the v e c t o r m e a s u r e
S~
does not
are equivalent: has finite dimension.
[[I B [il(~) = il ~ li(n)
(b)
vector measure (c)
bva(F, ~)
Proof.
By the t h e o r e m
the B a n a c h
space
~
=
H i~
N o w the a s s e r t i o n
holds
for each algebra
F
a n d for e a c h
for e a c h a l g e b r a
F .
~ 6 ba(F, ~)
= ba(F, ~ )
holds
of D v o r e t z k y - R o g e r s
has finite dimension
is a b s o l u t e l y
I
Recall
}
Corollar[~
(a)
i~
s i n c e the f a m i l y
variation.
The following
map
is b o u n d e d
summing,
iS =
follows
19.6.9],
a n d in t h i s c a s e w e h a v e
il i ~
llas
from Theorem
t h a t the s e m i v a r i a t i o n
[83; T h e o r e m
if a n d o n l y if the i d e n t i t y
3.1.4.
of a v e c t o r m e a s u r e
~ E ba(F, ~ )
satisfies
the i d e n t i t y
HI ~ iiI (A) for all
A£ F
vector measures
=
SUPu(]E')
[49; P r o p o s i t i o n of b o u n d e d
ie' 0 I/I (A)
I.I.11].
variation
,
A related
characterization
is the f o l l o w i n g :
of
132
3.1.7.
Lemma.
For a vector measure
~ 6 ba(F, ~ )
(a)
~
(b)
The f a m i l y
, the f o l l o w i n g
are equivalent:
has bounded variation. { le' 0 ~]
I e' 6 U ( ~ ' )
}
has a s u p r e m u m
in
ba(F, ~ ) Moreover,
if
~
Proof.
If
B
all
e' £ U ( ~ ' )
AL-space
variation,
then
has bounded
variation,
then
, hence
SUPu ( ~ , )
le' o ~i
H ~ i~ = S U P u ( ~ , )
]e' o ~] < exists
H ~ II
since
le' o ~i
holds
for
ba(F, ~)
as an
is o r d e r c o m p l e t e .
Conversely, Fix
has b o u n d e d
suppose
e £ (0,~)
e~, e½ . . . . .
that
. For
~
AE F
e~ 6 U ( ~ ' )
Z
H ~ ( A i) H
:= S U P u ( ~ , a n d for
) Ie' 0 ~]
exists
{ A I , A 2 , . . . , A k} 6 P(A)
in
b a ( F , JR)
, choose
satisfying
<
Z
Hi B if] (A i)
<
Z
le! 0 ~ l ( A i) + e
<
~ (A) + C
--
1
This yields
H B H (A)
for a l l
If
~
<
Q0(A)
,
A £ F .
is a m e a s u r e
in
ba(F, JR) , t h e n a v e c t o r m e a s u r e
w i l l be s a i d to be ~ - c o n t i n u o u s ~-continuous,
if its v a r i a t i o n
~6bva(F,
II ~ II 6 b a ( F ,
a n d it w i l l be said to be ~ - s i n g u l a r
~)
~) is
if i t s v a r i a t i o n
is ~ - s i n g u l a r . 3.1.8.
Lemma.
For a measure following
~6ba(F,
(a)
~
(b)
e' o ~
Proof.
If
II ~II
and a vector measure
is ~ - c o n t i n u o u s
~£bva(F,
~)
, the
JR)
band
= SUPu(~,)
~-continuous.
(resp. ~ - s i n g u l a r ) .
is ~ - c o n t i n u o u s
e' o B
Je' o ~i 6 b a ~ ( F , projection
~)
are equivalent:
in
(resp. ~ - s i n g u l a r )
is q0-continuous f o r e a c h
e' £ U ( ~ E ' )
, for all
e' EU(IR.,)
. Since
ba(F, JR)
, by T h e o r e m
2.1.4,
]e' o ~] 6baq°(F, JR) , by L e m m a
The c o n v e r s e
is o b v i o u s .
for e a c h
e' 6 U ( ~ ' )
, then we have
baQ°(F, JR)
is a
we h a v e 3.1.7,
hence
~
is D
133
The next result is the L e b e s ~ u e d e c o m p o s i t i o n for v e c t o r m e a s u r e s of b o u n d e d variation:
3.1.9.
Theorem.
For each m e a s u r e
~ 6 b a ( F , ~)
and for each vector m e a s u r e
there exists a ~ - c o n t i n u o u s v e c t o r m e a s u r e ~-singular vector measure
~s
£ b v a ( F , ~)
~£bva(F,
~@c E bva(F, ~) satisfying
~)
and a
~ = ~c
+ ~s
The d e c o m p o s i t i o n is unique. Moreover, ~I ~ I~
H ~ ~I = II ~ c
with r e s p e c t to
H + II ~ s
H
is the L e b e s g u e d e c o m p o s i t i o n of
~ .
The proof of the L e b e s g u e d e c o m p o s i t i o n for v e c t o r m e a s u r e s of b o u n d e d v a r i a t i o n p r o c e e d s via a Stone space argument;
see
[49; T h e o r e m 1.5.9]
and use Lemma 3.1.8.
For
~ 6 ba(F, ~)
vector measures
, let in
C@ :
bva@(F, ~)
bva(F, ~)
bva(F, ~)
denote the class of all ~ - c o n t i n u o u s
, and let
> bva~(F, ~)
denote the p r o j e c t i o n given by T h e o r e m 3.1.9.
3.1.10.
Corollary.
For each m e a s u r e
~6ba(F,
~)
is a c o n t r a c t i v e projection, and
(i~-
C~) (bva(F, ~))
c o m p l e m e n t e d subspaces of
A vector m e a s u r e
~ : F
Z B(A n) n=1
=
, the map
C ~ : bva(F, ~)
and the classes
£ F . Let
are B a n a c h spaces for the n o r m
> ~
for the n o r m
II. li(~)
and
is c o u n t a b l y a d d i t i v e if the identity
~( Z A n) n=1
bvca(F, ~)
n v e c t o r m e a s u r e s in
= C@(bva(F, ~))
bva(F, ~)
holds for each sequence of m u t u a l l y d i s j o i n t sets Z A
> bva~(F, ~)
bva@(F, ~)
An 6 F
satisfying
denote the class of all c o u n t a b l y a d d i t i v e
bva(F, ~)
. The class
bvca(F, ~)
Jl. ll(~) , and a v e c t o r m e a s u r e in
is a B a n a c h space
bva(F, ~)
is c o u n t a b l y
a d d i t i v e if and only if its v a r i a t i o n is c o u n t a b l y additive. Let denote the ~ - a l g e b r a g e n e r a t e d by bvca(F, ~)
F . Then each v e c t o r m e a s u r e in
has a unique e x t e n s i o n to a v e c t o r m e a s u r e in
and the map
J :
bvca(F, 7R)
> bvca([, TR.)
,
bvca(;, ~)
,
134
w h i c h a s s o c i a t e s with each v e c t o r m e a s u r e in to a v e c t o r m e a s u r e in bvca(F, ~)
onto
onto
J
to
bva ~(F, ]E)
bvaJ~(T, ~)
its e x t e n s i o n
for a m e a s u r e
~£bca(F,
JR) ,
is an isometric i s o m o r p h i s m of
For details,
For the r e m a i n d e r of this section,
bvca(F, ]E)
, is an isometric i s o m o r p h i s m of
bvca(~, 7R.) . Moreover,
the r e s t r i c t i o n of bva~(F, ~)
bvca(~, ~)
see
suppose that
[49; C h a p t e r I].
]E
has the R a d o n - N i k o d y m
p r o p e r t y and that
A :
F
:~
[0,1]
is a fixed p r o b a b i l i t y measure. RA :
bvaJA(T, I~.)
By the R a d o n - N i k o d y m property, > L 1 (F,JA, ]E)
,
w h i c h a s s o c i a t e s with each J A - c o n t i n u o u s v e c t o r m e a s u r e in its R a d o n - N i k o d y m d e r i v a t i v e w i t h r e s p e c t to isomorphism.
the map
JA
bva(~, IR)
, is an i s o m e t r i c
It then follows that the map
RA O J :
bvaA(F, ~E)
> L I(~,JA, ]E)
also is an isometric isomorphism. The map P ~ d o n - N i k o d y m o p e r a t o r w i t h respect to CA :
bva(F, ~)
RA 0 J
will be c a l l e d the
A . Let
> bvaA(F, ~R)
d e n o t e the c o n t r a c t i v e p r o j e c t i o n given by T h e o r e m 3.1.9 and define DA
:=
RA o J o CA
The map DA :
bva(F, ~)
> LI(~,JA, ~)
will be c a l l e d the @ e n e r a l i z e d R a d o n - N i k o d y m o p e r a t o r with r e s p e c t to
i . For
~6bva(F,
~)
, the r a n d o m v a r i a b l e
DAb£LI(~,JA,
be c a l l e d the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e of to
B
~)
will
with respect
A .
The f o l l o w i n g e l e m e n t a r y result w i l l be u s e f u l in p r o v i n g w e a k c o n v e r g e n c e theorems for the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a set f u n c t i o n process:
3.1.11. Suppose
Theorem. ~
has the Radon-Nikodym property.
Then e' (DAb) holds for all Proof.
=
DA(e'~)
~ £ bva(F, ~)
and
e' 6~'
The Lebesgue decomposition of e'~
=
e'~ Ac + e'~ As
By Lemma 3.1.8,
=
yields
,
and the Lebesgue decomposition of e'~
~
e'~
is given by
(e'~) Ac . (e'~) As
e'~ Ac
is A-continuous and
e'~ As
By Theorem 2.1.4, the Lebesgue decomposition of
is A-singular.
e'~
is unique.
Therefore, we have e,~ lc
=
(e,~) xc
hence e,(j~ lc)
=
Jle,~ lc)
=
j((e,~) Ac)
This yields [ e'(DA~) d(JA) JA for all
=
e' I DAb d(Jl) A
=
A 6 F , from which the assertion follows.
;
DA(e'~) d(JA) A
,
3.2.
Set
Let
F
f u n c t i o n
be a s t o c h a s t i c
p r o c e s s e s
b a s i s on a set
~ .
A sequence
:=
{ ~n6bva(Fn
, ~)
w i l l be c a l l e d a v e c t o r - v a l u e d
Consider For
Then
~r
~
and a bounded
on
[
stopping
.
time
T .
:=
Z Up(A{T=p}) p=1
is a v e c t o r m e a s u r e
in
b v a ( F r , ~)
a n d we h a v e the f o l l o w i n g
lemma:
Lemma.
3.2.1.
If
~
process
, define
uT(A)
elementary
}
set f u n c t i o n
a set f u n c t i o n p r o c e s s
AE F
I n6m
is a set f u n c t i o n
process
and
•
is a b o u n d e d
stopping
time,
then the i d e n t i t y ao
II u T II (A) holds
for all
Proof.
For
A £ F
p £~
11 U~ If(B)
For
=
Z p=l
II Up II (AN{T=p})
T
and
B C FT({r=p})
, we h a v e
FT(B)
= Pp(B)
k i=IZ II u T ( B i) II
=
s u p p T(B)
=
k s U p P p (B) i=IZ II ~ p ( B i) II
=
II ~p II (B)
A E F T , we d e d u c e
II u T II (A)
=
Z p=l
[[ ~
[I(AN{~=P})
=
Z p=l
l[ U p [I(AN{T=p})
as was to be shown.
For
, hence
MET
,
~CT(x)
U {~}
,
u£a(Fr,
~.)
and
AC F
, define
,
137
(Rx~~ ) (A)
::
~(A)
T
Then
R~_
is a v e c t o r
restriction
of
R M~ which to
~
:
, will
a(F
the
> a(F
we w i l l
be
which
will
be c a l l e d
the
map
, ~)
each vector the
, ~)
linear
measure
restriction
interested
in
map
in the
a(F
, ~)
from
its r e s t r i c t i o n
a(FT, ~ )
restriction
of
to
a(F
RT M
to
, ~)
Lemma.
x 6 T
and
is a l i n e a r
Proof.
r £ T(x) U {~}
:
R~
In p a r t i c u l a r ,
(A)
map
, ~)
(A) E
II
~ ( A i)
11
<
SUpp
(A) E
]]
~ ( A i)
II
yields
II R U T II(~) !
omit
~
is
it is T - b o u n d e d
and
the u p p e r
~-bounded
sup T
is n o r m b o u n d e d .
I TCT
=
for all
II 1.L [I
A £
F
,
(A)
II ~ II(~)
index
of
o
R~
if the v a l u e
if the v a l u e
II ~T II (Q)
it is a s e m i a m a r t
{ ~(0)
. T h e n we have,
SUpp
process
:=
, ~)
=
usually
II ~ H T is finite;
> bva(F
~6bva(F
this
shall
A set f u n c t i o n
is finite;
restriction
contraction.
Consider
we
, the
b v a ( F T , ~)
[I R~P, II
Again,
in
, and
~)
3.2.2. For
with
be c a l l e d
In m o s t c a s e s , bva(F,
F
a ( F T, ~ )
associates
F
measure
to
if the n e t
}
In c o n t r a s t
to w h a t
is k n o w n
to b e t r u e
in t h e real
138 case
(Theorem
2.2.4),
we shall
infinite
dimensional
Instead,
we have the f o l l o w i n g
3.2.3.
semiamart
space need not be T - b o u n d e d
in an
(Theorem
3.2.4).
result:
Lemma.
For a set f u n c t i o n
process
(a)
~
(b)
The value
Proof. su~
Banach
see that an T - b o u n d e d
is a s e m i a m a r t
Suppose I]~ ~n ~il(~)
m £~(M)
~
and the value
Ill ~
sup T
li~(~)
first that is finite.
and define
(~)
, the f o l l o w i n g
~
are equivalent:
ili ~n iH(~)
su~
is a s e m i a m a r t
Consider
a stopping
is finite.
is finite.
K £ T
time
~ £ T
i
M(~)
,
if
~ 6A
I
m
,
if
~ £ ~A
such that the v a l u e
and
A 6 F
. Choose
by letting
:=
Then we have
II bE(A)
II <
II ~v(s) II + II ~m(~A)
< F r o m this
it follows
The c o n v e r s e
The main
II ~T(~)
that the v a l u e
of this
~
and
~
linear operator.
section
is finite.
is the following:
are B a n a c h
Then
spaces
the f o l l o w i n g
(a)
S
is a b s o l u t e l y
(b)
S
m a p s the ~ - b o u n d e d
stochastic in
~
Suppose
first
Lemma
hence
is a T - b o u n d e d
The c o n v e r s e
S
:~
> ~
is a b o u n d e d
are equivalent:
summing.
basis)
semiamart
3.2.3,
and
semiamarts
in
~
into the T - b o u n d e d
(on an a r b i t r a r y
set f u n c t i o n
processes
.
an ~ - b o u n d e d
S~
Ill ~T lll(~)
sup T
Theorem.
Suppose
Proof.
lli ~n lli(n)
II + s u ~
is obvious.
result
3.2.4.
sup T
II
suPT
that ~
S
in
~
is a b s o l u t e l y . Then
~I S~ T If(Q)
set f u n c t i o n
summing.
sup T
is finite,
process
in
can be seen from the s u b s e q u e n t
~
Consider
H~ ~T Hi(~)
is finite,
by T h e o r e m
3.1.4,
by
hence
. Example
3.2.5.
m
139
3.2.5.
Example.
Suppose
~
and
linear operator
~
Then there exists sequence
{ Sx n
a constant HC~
are B a n a c h
which
[83; T h e o r e m
process
~
S :~
> ~
sequence { x £~ I n 6~ n is n o t a b s o l u t e l y s u m m a b l e ,
}
such that 14.6.1].
stochastic
lJ Z H x n II ~ P Let
is a b o u n d e d
summing.
a summable
I n £~
p 6~+
On the s t a n d a r d
spaces and
is n o t a b s o l u t e l y
holds
}
such t h a t the
and t h e r e e x i s t s
for e a c h f i n i t e
set
x := X x
basis
n [0,1)
on
, d e f i n e a set f u n c t i o n
by l e t t i n g n
X-
:=
Bn(Bn,k )
X
Z Xj j=1
n
0
First,
for all
n 6~
,
if
k
=
I
•
if
k
=
2
,
otherwise
, we h a v e n
=
II ~n II ( n )
II x-
:E j=l
II x II
<
+
x,
II x II
the set f u n c t i o n
consider
n 6~
,
~p(AN{T=p}) Define For all
m
ZK(p)
p£ {n,n+1,...,m-1}
II x n II
~
and
is ~ - b o u n d e d . A £ F
. For all
p 6~(n)
, we h a v e
~p(An{T=p}NBp,k)
p = m
, we t h e n h a v e
0 6 ~{~=p}
, which yields
= @ ' which yields
~p(AN{T=p}) , we have
~m(AN{T=m}) F o r all
=
+
2p
process
T £ T(n)
II 3
:= T(0)
{~=p}DBp•I
For
+
II x n II
x.
j=l
<
Therefore,
+
n T
II
--
Next•
It 3
pE~(m+1)
=
~ p ( A N { T = p } N B p , 2)
Bm, I c_ {~=m}
=
x-
, we have
~p (AN{T=p})
=
0
, which yields
m Z xj + B m ( A N { ~ = m } N B m , 2 ) 9=I {T=p}NBp, I = {T=p}NBp, 2 = ~
, which yields
140
T h u s we h a v e ~o
~T(A)
=
X ~p (An{r=p}) p=n m m ~Z x. + Z ~ p ( A N { x = p } N B p , 2) j=1 ] p=n
=
Let
H
denote
the
AN{T=p}NBp, 2 ~ @ Then
the a b o v e
set of all
, which
identity
p £ {n,n+1 ,... ,m}
is e q u i v a l e n t
to
satisfying
A N { T = p } N B p , 2 = Bp, 2 .
yields m
~T(A)
From
this
=
identity,
II ~..~ (s) for all
n 6~
Therefore, For
choose
II
<
II x II
+ 2p
,
, hence
set f u n c t i o n
process
which
n E~
x. + Z H X p 3
T 6 T(n)
reference, ~
Z j=1
we o b t a i n
and
the
later
process
x -
we also
~
exhibit
can be proven
such
for all
r 6 T
.
is a s e m i a m a r t .
another
b y the a b o v e
property
of t h e
argument.
Fix
set f u n c t i o n E 6 (0,~)
and
that
m
llxholds
for all
and
A6
the
last
F
Z
x
j=1
II +
II z H x
3
II
P
mqlW(n)
and
for e a c h
, choose mq~N(n) T i d e n t i t y for ~T(A)
<
-
and
finite
set
Hcl~(n)
Hcl~(n)
as a b o v e .
. For
T 6 T(n)
Then we have,
by
, m
~T(A)
II
This
yields,
II
<
for all
III ~,.~ III (n) Therefore, which
the
means
II
x-
x 6 T(n)
Xj
II + II Z H
Xp
II
<_
e
,
<__ 2E
set f u n c t i o n
that
Z
j=1
process
it is a s t r o n g
~
satisfies
potential
lim
(see S e c t i o n
Ill ~T III(~) = 0 , 3.4
for the
by
letting
definition). Finally,
for all
~ n (~)
n £~
, define
a stopping
time
p
,
if
~ £ Bp, 2 N B c n,1
n
,
if
~0 E Bn, I
:=
Tn £ T
141
Then we have n
n
z
II sx
p=1
II P
<
z
--
p=1
=
II sl~P ll(Bp, 2) + II SU n 11(Bn, I)
I1 S~ t
II (~) n
Letting
n
tend to infinity yields
Therefore,
the set function process
su b S~
II S ~ n
II(~) = ~
is not T-bounded.
We thus o b t a i n a first c h a r a c t e r i z a t i o n of finite d i m e n s i o n a l B a n a c h spaces in terms of set function processes:
3.2.6.
Corollary.
The f o l l o w i n g are equivalent: (a) (b)
~
has finite dimension.
Every ~ - b o u n d e d
s e m i a m a r t is T-bounded.
We also record the f o l l o w i n g obvious result:
3.2.7.
Theorem.
The class of all T - b o u n d e d set f u n c t i o n p r o c e s s e s is a B a n a c h space for the n o r m
II. II T
For the r e m a i n d e r of this chapter,
i :
F
>
let
[0,1]
be a fixed p r o b a b i l i t y measure.
If
~
is a set f u n c t i o n process,
the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e of will be d e n o t e d by
Dn~ n
for all
n 61~ .
,
~n
w i t h respect to
then Rnl
3.3.
M a r t i n @ a 1 e s .
The
results
of t h i s
martingales mostly
given
without
the proofs
proof
given
A set f u n c t i o n
section
are essentially
in S e c t i o n since,
They
are
apart
from
lattice
in t h e r e a l
process
~
the
2.3.
case carry
s a m e as t h o s e
stated
over
is a m a r t i n g a l e
on real
for reference
theoretical
and
arguments,
to t h e v e c t o r - v a l u e d
if the n e t
{ ~ T (Q)
case.
I T E T }
is c o n s t a n t .
3.3.1.
Theorem.
For a set function
process
~
, the following
are equivalent:
(a)
~
is a m a r t i n g a l e .
(b)
~
= RT~ O
holds
for all
• £ T
and
o 6 T(~)
holds
for all
n 6~
and
m £ ~ (n)
(c)
~n = Rn~m
(d)
~n
(e)
There
= Rn~n+1 exists
~n = Rn~ (f)
There ~T
(g)
If
~
= RT5
~r(~)
= ~(~)
is a m a r t i n g a l e ,
, will
measure
3.3.2. If
of
5C a(F
T C T
the
a martingale
~
, ~)
such that
, ~)
such that
.
measure
~E a(F • £ T
then the vector
be called
such that
.
for all
l i m ~n(A)
is a m a r t i n g a l e
process
II ~ II
sup~
Proof.
:=
n 6~
, ~)
.
measure
~E a(F
, ~)
given
,
limit measure
will
of
sometimes
~
. Again,
be denoted
the
by
limit
lim
~n
"
Theorem.
~
holds
holds
.
5£ a(F
measure
for all
a vector
n E~
measure
for all
a vector
holds
exists
for all
a vector
holds
exists
There
5(A)
A £ F
holds
for all
with
is a r e a l
II ~ n II (A)
limit measure
submartingale,
=
sup T
~
, then
and the
II IJ..~ II (A)
=
the
set function
identity
II ~. II (A)
A 6 F
For all
II ~.~ l l ( n )
zET
=
and
oET(z)
II R.~ a l l ( n )
, we have,
<
II u.a l l ( n )
by Lemma
,
3.2.2,
by
143
hence
II ~ H
is a real submartingale.
su~ A6 F
H ~ n H(A)
=
sup T
The identity
II Br H (A)
H ~ If(A)
,
, can be p r o v e n as in the real case.
3.3.3.
Corollary.
For a m a r t i n g a l e equivalent: (a)
, the f o l l o w i n g are
and its limit m e a s u r e
is ~ - b o u n d e d .
(b)
is T-bounded.
(c)
has b o u n d e d variation.
Moreover,
if
3.3.4.
Theorem.
is ~ - b o u n d e d ,
then
H _~ i~
The class of all ~ - b o u n d e d m a r t i n g a l e s on norm
=
11. i ~
=
If _~ fiT =
F
it ~ il (2)
is a Banach space for the
, and the map a s s o c i a t i n g w i t h each ~ - b o u n d e d m a r t i n g a l e
its limit m e a s u r e is an isometric i s o m o r p h i s m of the Banach space of all ~ - b o u n d e d m a r t i n g a l e s on
A martingale exists
6q
~
such that
a martingale
the real s u b m a r t i n g a l e
3.3.5.
onto
bva(F
, ~)
is u n i f o r m l y A - c o n t i n u o u s if for each
(0,~)
other words,
F
A(A) ~
< 6
implies
SU~N
c 6 (0,~)
II ~ n H(A)
there
< ~ . In
is u n i f o r m l y A - c o n t i n u o u s if and only if
it ~ II
is u n i f o r m l y A - c o n t i n u o u s .
Theorem.
For an ~ - b o u n d e d m a r t i n g a l e
~
and its limit m e a s u r e
, the following
are equivalent: (a)
~
is u n i f o r m l y A - c o n t i n u o u s .
(b)
~
is A - c o n t i n u o u s .
We thus have the L e b e s ~ u e d e c o m p o s i t i o n for
3.3.6.
~-bounded
martingales:
Theorem.
Every ~ - b o u n d e d m a r t i n g a l e is the sum of an ~ - b o u n d e d
uniformly
A - c o n t i n u o u s m a r t i n g a l e and an ~ - b o u n d e d m a r t i n g a l e w i t h A - s i n g u l a r limit measure. The d e c o m p o s i t i o n
is unique.
As in the real case,
the L e b e s g u e d e c o m p o s i t i o n can be used to split
the proof of the m a r t i n g a l e c o n v e r g e n c e t h e o r e m into two parts:
144
3.3.7.
Theorem.
Suppose If
~
~
has the R a d o n - N i k o d y m
is an ~ - b o u n d e d
lim Dn~ n 3.3.8. ~
=
D
lim ~n
martingale,
then
a.e.
Theorem.
Suppose If
property.
uniformly A-continuous
~
has the R a d o n - N i k o d y m property.
is an ~ - b o u n d e d
lim Dn~ n Combining
martingale with A-singular
=
0
these results,
limit measure,
then
a.e. we obtain the general martingale
convergence
theorem: 3.3.9. Suppose If
~
Corollary. ~
has the R a d o n - N i k o d y m
is an ~ - b o u n d e d
lim Dn~ n
martingale,
=
D
lim ~n
property. then
a.e.
It should be noted that, apart from the definition of the g e n e r a l i z e d Radon-Nikodym
derivatives,
the R a d o n - N i k o d y m property
the proof of the convergence martingale.
Thus,
if
~
is needed only in
of the uniformly A-continuous
is an ~ - b o u n d e d
part of the
m a r t i n g a l e with A-singular
limit measure which results from integrating a m a r t i n g a l e of random variables,
then
lim Dn~ n = 0
not have the R a d o n - N i k o d y m
a.e. holds even if the Banach space does
property;
see also
[49; T h e o r e m V.2.9].
3.4.
S t r o n ~
a m a r t s .
Strong amarts
form the i m m e d i a t e
vector-valued
case.
turns out that strong remarked
amarts have
at the b e g i n n i n g
not be T-bounded,
generalization
and even in the case of a T - b o u n d e d
Radon-Nikodym
property
derivatives
A set f u n c t i o n
process
~
dual,
need not converge
As a l r e a d y
becomes
in a B a n a c h
useless,
space with
the g e n e r a l i z e d
strongly
is a stron9 a m a r t
it
strong a m a r t s n e e d
inequality
strong a m a r t
and a s e p a r a b l e
to the
since
rather weak properties.
of this chapter, ~ - b o u n d e d
in w h i c h case the m a x i m a l
the R a d o n - N i k o d y m
of real amarts
Their name may be somewhat m i s l e a d i n g
b u t only weakly.
if the net
{ ~T(~)
I T E T }
is n o r m c o n v e r g e n t .
3.4.1. Every
Lemma. strong a m a r t
The proof This
of this
is a semiamart.
lemma
is the same as in the real case
is also true for the proof
strong amarts
3.4.2.
in terms
of the f o l l o w i n g
of their limit m e a s u r e
2.5.2).
(see T h e o r e m
of
2.5.4):
Theorem.
For a set f u n c t i o n p r o c e s s , the f o l l o w i n g (a) is a strong amart.
(b)
There exists
(C)
There
Ill ~ T - R T ~
lim
~(A) (d)
exists
Ill(Q)
= 0
are equivalent:
~£a(F
, ~)
such that
~ E a(F
, ~)
such that
, ~)
such that
.
a vector measure holds
for all
a vector measure
A£ F ~ C a(F
= lim ~ ( ~ )
The next r e s u l t
is the d i f f e r e n c e
the limit m e a s u r e
3.4.3.
a vector measure
= lim ~T(A)
There e x i s t s ~(~)
property
for strong amarts,
in w h i c h
is not involved:
Corollary.
F o r a set f u n c t i o n
process
(a)
~
(b)
lim suPT(T )
Again,
(Lemma
characterization
, the f o l l o w i n g
are e q u i v a l e n t :
is a strong amart.
the proof
provided
~
III ~ -R ~
lll(n) = 0
is the same as in the real case
the v a r i a t i o n
n o r m is r e p l a c e d
(Corollary
by the s e m i v a r i a t i o n
2.5.5), norm.
146
As a first and important difference between real amarts and strong amarts,
we remark that the semivariation
norm appearing
in the preceding
results cannot in general be replaced by the v a r i a t i o n norm. This is also true for the potential part occuring for Y - b o u n d e d
strong amarts.
A set function process { Ill ~T III(~) I T E T }
~
is a stron9 potential
converges
to
an ~ - b o u n d e d
strong amarts:
strong amart is the sum of an ~ - b o u n d e d
Consider an ~ - b o u n d e d
e 6 (0,~)
then so is the strong potential.
strong amart
. For each partition
{AI,A 2 .... , ~ } 6 Pn(Q)
k Z i=I
II ~(A i) II
~
with limit measure
{AI,A2,...,~) and
E P (~)
Z II ~(A i) - ~n(Ai)
, choose
licit(n) and it follows
_~
_<
_<
k Z (II ~(A i) - B n ( A i) II + II ~n(Ai) IIh/ i=I
<
E + l:~n ll(S)
:=
{ Rn5
~
and Corollary
3.3.3 that
I n 61%1 }
martingale,
is a strong potential, if
,
from Theorem 3.3.1
is an ~ - b o u n d e d
Moreover,
ll~tL~
hence
by Theorem 3.4.2, and it is also ~ - b o u n d e d .
has a Riesz decomposition. ~
is an arbitrary
lim ~n = 0 . Thus,
if
~
strong potential,
is an ~ - b o u n d e d
then we have
martingale
and
~ . n 6~
II ~ e . Then
This yields
Therefore,
and
is unique.
If the strong amart is T-bounded,
satisfying we have
martingale
strong potential.
The d e c o m p o s i t i o n
Proof.
for ~ - b o u n d e d
Theorem.
Every ~ - b o u n d e d
Fix
if the net
0 .
We can now prove the Riesz d e c o m p o s i t i o n 3.4.4.
in the Riesz d e c o m p o s i t i o n
~
is
147
an~-bounded
strong p o t e n t i a l s a t i s f y i n g
lim ~n hence
~ = ~
=
lim ~n
=
~
~ = @ + ~ , then we h a v e
'
, by T h e o r e m 3.3.1. Therefore,
the Riesz d e c o m p o s i t i o n is
unique. The final a s s e r t i o n c o n c e r n i n g T - b o u n d e d n e s s
is obvious since
every ~ - b o u n d e d m a r t i n g a l e is T-bounded, by C o r o l l a r y 3.3.3.
F r o m the f o l l o w i n g c h a r a c t e r i z a t i o n of a b s o l u t e l y summing operators,
it
can be seen that the b o u n d e d n e s s p r o p e r t i e s of strong amarts d e p e n d on the d i m e n s i o n of the B a n a c h space.
3.4.5.
Theorem.
Suppose
~
and
~
are Banach spaces and
S :~
is a b o u n d e d
>
linear operator. Then the following are equivalent: (a) S is a b s o l u t e l y summing. S
(b) (c)
maps the strong p o t e n t i a l s in
~
into the T - b o u n d e d set
f u n c t i o n p r o c e s s e s in
~
S
strong p o t e n t i a l s in
maps the ~ - b o u n d e d
.
T - b o u n d e d set f u n c t i o n p r o c e s s e s in (d)
S
maps the ~ - b o u n d e d
~
Proof.
Suppose first that
strong p o t e n t i a l in
~
II S~T I{ (~) hence
II S~ li
(c). Moreover,
then there exists an ~ - b o u n d e d S~
3~4.6.
(c) and
~
is a
,
(a) implies if
S
(b).
is not a b s o l u t e l y summing,
strong p o t e n t i a l
~
is not T-bounded, by Example 3.2.5. Therefore,
Finally,
If
~ £ T ,
and it follows that the value
is finite. Therefore,
(b) implies
for all
II S llas III ~T III (Q)
<
.
is a b s o l u t e l y summing.
, then we have,
is a real potential,
sup T li S~ T li(~) Obviously,
S
~
into the
into the
strong amarts in
T - b o u n d e d set function p r o c e s s e s in
~
.
in
~
such that
(c) implies
(d) are e q u i v a l e n t by the Riesz d e c o m p o s i t i o n ,
Corollary.
The following are equivalent: (a)
~
(b)
Every strong potential
has finite dimension.
(c)
Every ~ - b o u n d e d
strong p o t e n t i a l
(d)
Every ~ - b o u n d e d
strong a m a r t is T-bounded.
is T-bounded. is T-bounded.
(a). o
148
Another
deficiency
poor b o u n d e d n e s s theorem.
Even
a separable
this
3.6.7
amarts,
which
is the a b s e n s e
space h a v i n g
since
severe
of a T - b o u n d e d
than their
of a strong c o n v e r g e n c e
obtains
property
and
for the g e n e r a l i z e d
strong amart.
it is a c o n s e q u e n c e
and its corollaries.
is more
the R a d o n - N i k o d y m
only w e a k c o n v e r g e n c e
derivatives
result here
Theorem
properties,
in a B a n a c h
dual,
Radon-Nikodym
of strong
We do n o t p r o v e
of a more g e n e r a l
one;
see
3.5.
U n i f o r m
Although
strong
boundedness than
are
defined
and convergence
those
of real
essentially
the
f r o m the R i e s z due
amarts
a m a r t s
to t h e i r
decomposition
which
to e x p e c t
and actually
for t h o s e
strong
satisfying
II ~T I1(~)
= 0
lim
process
limit measure
converges
Without
which
to
3.5.1.
strong
evident
amarts
are
in t e r m s
by variations,
it
results
a potential
part
condition
to the
limit
that
amart
the n e t
if it is a s t r o n g
{ II ~ T - R T ~
JJ(Q)
amart
I T E T
}
.
measure,
difference
uniform
amarts
following
are
m a y be c h a r a c t e r i z e d
property:
Theorem.
F o r a set f u n c t i o n (a)
~
(b)
lim
Proof.
Suppose
process
is a u n i f o r m suPT(r )
. T h e n we have,
~
, the
first
that
for all
equivalent:
amart.
II ~ T - R T ~ o
II t~.~-RTI~ a II(s)
II(~) = 0 .
~
T £ T
5_
is a u n i f o r m
amart
and
,
a £ T(T)
I1 1.<~-R1;~- I1(~)
with
yields
Conversely,
lim suPT(T ) suppose
lim Fix
e E (0,~)
suPT(~)
II ~ - R ~ B a Jl(~) = 0 .
that
suPT(T )
II ~ -
Jl ~ r - R T ~ o Jl(~) and choose
R T ~ o II (n)
~ ET
<
=
0 such
limit measure
+ II RI:(Rag.-1.L a) II(n)
<-- JJ ~r- RT5 JJ (n) + II Ro~-~a
holds.
have
is t h e n
satisfactory
have
of the w e a k e r
is a u n i f o r m
such
0
appeal
~
~
by the f o l l o w i n g
which
satisfactory
are defined
be replaced
that more
instead
of
their
III (~) = 0 .
A set f u n c t i o n with
true amarts
It
potentials
in g e n e r a l
can be obtained
lira III ~
strong
less
amarts,
martingales
martingales.
the d e f i c i e n c i e s
Since
cannot
in g e n e r a l
vector-valued
as real
that
part.
same w a y as real
are
Contrarily,
properties
potential
of s e m i v a r i a t i o n s is n a t u r a l
properties
amarts°
same
in the
that
jj (Q)
'
IS0 holds
for all
T 6 T(x)
II ~ ( ~ ) for all limit
-~o(n)
T, o 6 T(M)
measure
oqT(~)
such k Z i=I
. This
of
II
<
, hence
~
. For
that
yields
e ~
is a s t r o n g
T E T(x)
and
II ~ ( A i) - ~ o ( A i )
X
li ~T(Ai)-~(A i) II ~
k Z i=I
amart.
Let
~
{ A I , A 2 .... , ~ }
denote
6 ~T(~)
the
, choose
il ~ ~ . T h e n w e h a v e
II ~T(Ai)-~o(Ai)
k Z II ~ o ( A i ) - ~ ( A i ) i=I
il +
il ~r - R r ~ o If(n) + e
SUPT(T ) This yields
II ~T - RT5 II(n) for all
r £ T(~)
A set f u n c t i o n {
, hence
process
II ~T II(~) I ~ £ T } is a u n i f o r m
~
potential
for e a c h u n i f o r m ~'
is a u n i f o r m
Therefore,
3.5.2.
Theorem.
The c l a s s
of all
amart.
potential
0 . Clearly,
every
potential
if the
set f u n c t i o n
uniform ~
if the n e t a set f u n c t i o n
potential
, there
exists
process
process
II ~ II
is T - b o u n d e d . a real
Doob
II ~ II ~ ~' , by L e m m a 2.5.6. We a l s o h a v e
satisfying obvious
to
if a n d o n l y
Moreover,
following
~
,
is a u n i f o r m
potential.
the
2~
converges
is a r e a l
potential
~
result:
uniform
potentials
is a B a n a c h
space
for t h e n o r m
II. IIT From
the d e f i n i t i o n s
Riesz
decomposition
3.5.3. Every
of u n i f o r m
amarts
and uniform
for u n i f o r m
amarts
is a l m o s t
potentials,
the
evident:
Theorem. uniform
amart
The decomposition If the u n i f o r m
Proof.
amart
Consider that
and
a uniform
potential.
is ~ - b o u n d e d ,
amart
then
_~
so is the m a r t i n g a l e .
with
limit measure
~
. Choose
II ~n-Rn~ II (~) < I h o l d s f o r a l l n 61~(k) . T h e n we II ~n ll(n) + I , h e n c e R n ~ £ b v a ( F n , ]E) , for all
such
have
II R n ~ II(~) < , and
sum of a m a r t i n g a l e
a uniform
k £I~
n £3W(k)
is the
is u n i q u e .
thus
for all
n 6~
since
R
n
is a c o n t r a c t i o n .
II
151 Therefore,
the limit m e a s u r e
:= which
{ Rn~ I n 6 ~
is a m a r t i n g a l e ,
is a u n i f o r m Therefore,
~
defines
a set f u n c t i o n
process
}
and it is c l e a r
that the set f u n c t i o n
process
potential. ~
has a Riesz
The u n i q u e n e s s
decomposition
of the Riesz
~ = ~ + ~
decomposition
case or as in the proof
of the Riesz d e c o m p o s i t i o n
Finally,
amart
if the u n i f o r m
the m a r t i n g a l e ,
since
amarts.
In p a r t i c u l a r ,
3.5.4.
Corollary.
For a u n i f o r m a m a r t
is ~ - b o u n d e d ,
plays a c e n t r a l
role
we h a v e the f o l l o w i n g
~
as in the real
for strong amarts.
then the same is true for
every u n i f o r m p o t e n t i a l
The Riesz d e c o m p o s i t i o n
.
can be p r o v e n
is T-bounded.
u
in the theory of u n i f o r m corollaries:
, the f o l l o w i n g
and its limit m e a s u r e
are
equivalent: (a)
~
(b)
~
is T-bounded.
(c)
~
has b o u n d e d variation.
Proof.
Since u n i f o r m p o t e n t i a l s
equivalent
is ~ - b o u n d e d .
by the Riesz
are T-bounded,
decomposition
u n i f o r m a m a r t has the same limit m e a s u r e Riesz d e c o m p o s i t i o n ,
3.5.5. If
~
(a) and
as the m a r t i n g a l e
(c) are e q u i v a l e n t
process
uniform II ~ II
lira II ~T II (A)
Proof.
(b) are
3.3.3.
Since a of its
by C o r o l l a r y
3.3.3.
Corollary. is an ~ - b o u n d e d
set f u n c t i o n
holds
(a) and
and C o r o l l a r y
for all
is a b o u n d e d
limit m e a s u r e real amart,
~ , then the
and the i d e n t i t y
II 5 II (A)
A 6 F
Consider
lim
=
amart with
A 6 I: . By the Riesz
III ~T II (A) - I{ R T ~ II (A) I
<
decomposition,
lim
II ~ - R T ~
we have
II (A)
=
0
,
152
and Theorem
3.3.2
I II R T ~ II (A) - II ~ II (A) I
lim By
the t r i a n g l e
lim for all
As
inequality,
real
in the
. In p a r t i c u l a r ,
obtain
the
the R i e s z
of u n i f o r m
=
0
,
set f u n c t i o n
is a
II ~ II
process
decomposition
also yields
the
following
amarts:
Corollary.
F o r a set f u n c t i o n (a)
~
(b)
There
process
is a u n i f o r m exists
Proof.
Suppose
martingale
~
potential, majorized
hence
~
result
T-bounded
that
exists ~
II
, then
amart
Then real
~- 5
the
is a u n i f o r m
potential
, by L e m m a ~
and consider
~ - ~
which
is
2.5.6.
a n d a real
Doob
is a u n i f o r m
potential
potential,
the
o
structure
of the c l a s s
of all ~ - b o u n d e d ,
uniform
amarts
is a B a n a c h
space
for
the
.
Consider amarts
and
a Cauchy let
T-bounded
set f u n c t i o n
and
,
m E~
potential
amarts:
of all ~ - b o u n d e d
uniform
~'
a martingale
< ~'
describes uniform
Theorem.
Proof.
Doob
amart,
3.5.7.
II. IIT
is a u n i f o r m
potential
The c l a s s norm
~
is a p o s i t i v e
II
- ~
Doob
II ~ -
are equivalent:
a n d a real
II ~
is a u n i f o r m
The n e x t
~
decomposition.
if t h e r e
satisfying
hence
following
its R i e s z
by a real
Conversely,
, the
II ~ - ~ il < ~'
first
of
~
amart.
a martingale
satisfying
hence
0
Q
case,
characterization
~'
=
amart.
real
3.5.6.
we then
III ~T If(A) -II 5 II(A) I
A E F
bounded
yields
~
sequence denote
processes.
II ~ x - R : ~ o II (n) ~
{ ~(m)
its
limit
I m E~
of K - b o u n d e d
in the B a n a c h
Then we have,
II ~-~(m)
}
for all
space r E T
,
of all a 6 T(T)
iiT + II ~:(m) -R:~o(m) II (S) + II _B(m)-~_ iiT
15S Fix
e E (0,~)
, choose
II u -
and choose
~ £ T
such
~T(m)
II holds
U (m) IIT
for all
m E~
such
<_
,
E
that
that
- R x ~ o(m) ll(n)
T E T(~)
and
<
E
o 6 T(T)
, by the d i f f e r e n c e
property.
T h e n we h a v e
II ~ T - RTU o t l ( n )
~
3e
for all
T £ T(~)
and
property
that
is a u n i f o r m
Before
~
studying
include
of s t r o n g
characterization
3.5.8. ~
and
~
operator.
the d i f f e r e n c e
uniform
of a b s o l u t e l y amarts.
dimensional
are B a n a c h
Then
the
S
is a b s o l u t e l y
(b)
S
maps
uniform (c)
S
Proof.
maps
Banach
This
amart,
summing
will
lead
let us
operators to a n o t h e r
spaces.
potentials
in
Suppose
first
that
S
potential
in
in
~
since
II S~ T II (~) for all if
T £T ~
<
~
> ~
is a b o u n d e d
potentials
in
~
into
the
. strong
amarts
in
~
into
the
is a b s o l u t e l y ~
, then
summing.
S~
is a u n i f o r m
potential
S~
is a u n i f o r m
amart
Hl ~T Ill (~)
.
is a s t r o n g
T E T
:~
.
II S llas
II S ( ~ r - R r B O) II(~) for all
S
equivalent:
strong ~
the ( ~ - b o u n d e d ) in
is a s t r o n g
and
are
summing.
the ( ~ - b o u n d e d )
amarts
~
Similarly,
spaces
following
uniform
If
holds
of an ~ - b o u n d e d
and uniform
of f i n i t e
from
amart.
characterization
amarts
(a)
holds
it f o l l o w s
Theorem.
Suppose linear
. Now
the c o n v e r g e n c e
the following
in t e r m s
o E T(T)
,
and
amart,
~
o £ T(r)
then
since
II S llas III ( B ~ - R T ~ O) III(~) , and by the d i f f e r e n c e
properties
154 for strong amarts and u n i f o r m amarts. Conversely,
if
an ~ - b o u n d e d
S
is not a b s o l u t e l y summing, then there exists
strong p o t e n t i a l
~
Since the set f u n c t i o n process
in S~
~
such that
S~
is not T-bounded.
is clearly ~ - b o u n d e d ,
a u n i f o r m amart, by C o r o l l a r y 3.5.4.
In particular,
S~
it c a n n o t be cannot be a
u n i f o r m potential,
3.5.9.
u
Corollary.
The f o l l o w i n g are equivalent: (a)
~
(b)
Every (~-bounded)
has finite dimension. strong p o t e n t i a l is a u n i f o r m potential.
(c)
Every (~-bounded)
strong a m a r t is a u n i f o r m amart.
Let us now study the c o n v e r g e n c e of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of an P - b o u n d e d
u n i f o r m amart. E x c e p t for the r e q u i r e m e n t
that the B a n a c h space have the R a d o n - N i k o d y m property, w h i c h is even n e c e s s a r y in the case of m a r t i n g a l e s ,
the results o b t a i n e d for real
amarts e x t e n d to u n i f o r m amarts. Moreover,
the p r o o f s g i v e n in the real
case c a r r y over to the v e c t o r - v a l u e d case where v e c t o r m e a s u r e s of b o u n d e d v a r i a t i o n take the place of b o u n d e d real measures. First,
let
us note that L e m m a 2.5.11 extends to the v e c t o r - v a l u e d case. We thus obtain the f o l l o w i n g m a x i m a l inequality:
3.5.10. Suppose If
~
Lemma. ~
has the R a d o n - N i k o d y m property.
is a set f u n c t i o n process,
then
e(J A) ( { s u B ( k ) II Dn~ n II > E}) holds for all
~ E (0,~)
As in the real case,
and
k 6~
the m a x i m a l
<
suPT(k ) II ~T II (n)
.
inequality leads to the c o n v e r g e n c e of
the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of a u n i f o r m potential.
This
is the u n i f o r m p o t e n t i a l c o n v e r g e n c e theorem:
3.5.11. Suppose If
~
Theorem. ~
has the R a d o n - N i k o d y m property.
is a u n i f o r m potential,
lim DnB n
=
0
then
a.e.
We remark that in the m a x i m a l i n e q u a l i t y and in the u n i f o r m p o t e n t i a l
155 c o n v e r g e n c e t h e o r e m the R a d o n - N i k o d y m p r o p e r t y only enters in the d e f i n i t i o n of the g e n e r a l i z e d R a d o n - N i k o d y m derivatives.
Thus,
if
is a u n i f o r m p o t e n t i a l w h i c h r e s u l t s from i n t e g r a t i n g a u n i f o r m p o t e n t i a l of r a n d o m variables,
then
lim D n ~ n = 0
a.e. holds even if
the Banach space does not h a v e the R a d o n - N i k o d y m property; remark f o l l o w i n g the m a r t i n g a l e c o n v e r g e n c e t h e o r e m
see a l s o the
(Corollary 3.3.9).
A l s o as in the real case, by the Riesz d e c o m p o s i t i o n for u n i f o r m amarts, the u n i f o r m amart c o n v e r g e n c e t h e o r e m is a c o n s e q u e n c e of the m a r t i n g a l e c o n v e r g e n c e t h e o r e m and the u n i f o r m p o t e n t i a l c o n v e r g e n c e theorem:
3:5.12. Suppose If
~
Corollary. ~
has the R a d o n - N i k o d y m property.
is an N - b o u n d e d
lim DnB n
=
u n i f o r m amart, then
D
lim ~n
a.e.
By C o r o l l a r y 3.5.6, the u n i f o r m amart c o n v e r g e n c e t h e o r e m may also be stated in the f o l l o w i n g form:
3.5.13. Suppose If
~
Corollary. ~
has the R a d o n - N i k o d y m property.
is an N - b o u n d e d
martingale
~
set f u n c t i o n process such that there exists a
and a real Doob p o t e n t i a l
~'
satisfying
II ~ - ~ ~I < ~'
then
lim Dn~ n
=
D
lim ~n
a.e.
We c o n c l u d e this section on u n i f o r m amarts with a result c o n c e r n i n g quasimartingales.
A set function process Z II ~ n - R n ~ n + 1 H (~) 3.5.14.
~
is a ~ u a s i m a r t i n g a l e
if the series
is convergent.
Theorem.
Every q u a s i m a r t i n g a l e is a u n i f o r m amart.
Proof. k £I~
Consider a quasimartingale such that
~-
n=k
il ~.n-Rn~n+l
II (~q)
<
~
. Fix
E 6 (0,~)
and c h o o s e
156 T h e n w e have, 1 X i=I
f o r all
TET(k)
,
{ A I , A 2 , . . . , A I} 6 Pt(~)
li ~T(Ai) -%Im(A i) I[
<
1 Z Z i=I n=k
and
II ~ n - R n ~ n + 1
mE~l(T)
11 (A i)
oo
where
the f i r s t i n e q u a l i t y
<_
X n=k
<
~
tl lln-Rnlln+ 1 II (n)
,
is P r O v e n as in the real c a s e
( T h e o r e m 2.5.1).
This yields
11 ~tfor all
T E T(k)
RT~m I1(~) and
~
m£~(r)
II Rt1-~0 - R.r]/m I1 (n) for all
T ET(k)
,
for all
• 6 T(k)
and
property.
, . In p a r t i c u l a r ,
_<
o E T(T)
II ~T - R r B o II(~)
difference
c
II 11o - Rol"~m II (n)
and
~
o E T(~)
2e
mE~(o)
we o b t a i n
<
E
,
. This yields
,
, hence
~
is a u n i f o r m
amart,
b y the Q
3.6.
Weak
amar
t s ,
u n i f o r m and
weak
The intention processes,
sequential
of g e n e r a l i z i n g
amarts.
for weak
for strong obtains,
strong amarts
r t s
to a larger class of
obtains
led to the n o t i o n s
under
the same c o n d i t i o n s
of w e a k a m a r t s
just as the strong c o n v e r g e n c e
For weak
and we shall
sequential
see that this
which generalize
weak
In what
we shall w r i t e
follows,
ama
and w e a k
It turned out that the weak c o n v e r g e n c e
amarts,
amarts.
r t s ,
s e q u e n t i a l
for w h i c h w e a k c o n v e r g e n c e
as for strong amarts,
fail
ama
weak
sequential
amarts,
t h e o r e m may
t h e o r e m may fail
however,
weak c o n v e r g e n c e
is also true for u n i f o r m weak amarts,
amarts.
w-lim
for the weak
limit of a w e a k l y
A set f u n c t i o n is w e a k l y
process
convergent.
~
for all
A £ F
The a b o v e - m e n t i o n e d
and this weakest
, although
{ ~T(A)
if the net deficiency
I ~ £ T }
it is e a s i l y
reasonable
a set f u n c t i o n to
~
for all
A E F
process
such that the sequence
the net
. This
I n 6~
comes
weakly
the
into a m a r t i n g a l e
and
{ ~T(A)
leads
}
to a limit measure,
need not p o s s e s s
I ~ 6T }
to the f o l l o w i n g
is a u n i f o r m weak a m a r t { ~n (A)
of weak a m a r t s
weakly
decomposition
for w h i c h
I r 6 T }
seen to be a w e a k C a u c h y net.
that weak amarts
form of a Riesz
process 0
A set f u n c t i o n
to saying
{ ~T (~)
n e e d not c o n v e r g e
that w e a k amarts n e e d not c o n v e r g e
is e q u i v a l e n t
converges
net or sequence.
is a weak a m a r t
from the fact that the net
This m e a n s
convergent
is w e a k l y
weakly definition:
if it is a w e a k a m a r t convergent
for all
A £ F The d i f f e r e n c e
b e t w e e n weak
by the f o l l o w i n g
amarts
characterization
and u n i f o r m w e a k of w e a k l y
spaces:
3.6.1.
Theorem.
The f o l l o w i n g (a) (b)
~
are equivalent: is w e a k l y
sequentially
E v e r y weak a m a r t
amarts
sequentially
complete.
is a u n i f o r m w e a k amart.
is c l a r i f i e d
complete
Banach
158
Proof.
Suppose
consider For
a weak
first
amart
n, m 6 ~ ( k )
that _B . F o r
, define
a(~)
~
is w e a k l y AC
F~
sequentially
, choose
stopping
times
~,
n
,
if
~EA
nvm
,
if
0~ £ ~ A
m
,
if
~ £A
nvm
,
if
~ £ ~A
k 6~
~ £ T(k)
complete
such
and
that
A£
Fk
by letting
:=
and
(~)
:--
Then we have
Bn(A) Therefore, convergent uniform
- ~m(A)
{ ~n (A) since
amart.
The c o n v e r s e
c a n be
3.6.2. Suppose Then
}
is a w e a k
is w e a k l y
seen
- BT(~)
from
Cauchy
sequentially
the
sequence,
complete.
subsequent
Example
hence
Thus,
~
weakly is a
3.6.2.
Example. ~
there
not weakly F o r all
~o(Q)
I n 6~
~
weak
=
is n o t w e a k l y exists
stochastic
Cauchy
complete.
sequence
{ x n 6~
I n 6~
}
which
is
convergent.
n C~
generated
a weak
sequentially
by
, let the
F
sets
basis
~
be a c o p y
n
B1, 1 :=
:= { F n
of the a l g e b r a
[0,2 -I )
I n 6~
and
} , define
on
B1, 2 :=
[0,1) [2-1,1)
a set f u n c t i o n
which
is
. On the process
by letting
:=
~n(B1,k )
Then
the net
sequence Therefore,
{ ~T(~)
{ ~n(B1,1 ) the
to be a u n i f o r m
As a c o n s e q u e n c e
set
i
xn
,
if
k = I
[
-x n
,
if
k = 2
I ~ E T I n £~
function
weak
} }
is w e a k l y
convergent
is n o t w e a k l y
process
~
to
0 , b u t the
convergent.
is a w e a k
amart,
but
it f a i l s
amart.
of T h e o r e m
3.6.1,
we g e t t h e
following
result:
.
15g
Corollary.
3.6.3. Suppose
has
(a)
a separable
dual.
Then
the
following
are equivalent:
is r e f l e x i v e .
(b)
Every
Proof.
Each
and a weakly
weak
amart
reflexive
is a u n i f o r m
Banach
sequentially
space
complete
weak
amart.
is w e a k l y
Banach
space
sequentially having
complete,
a separable
dual
is r e f l e x i v e .
Similar
to s t r o n g
characterized
amarts
in t e r m s
3.6.4.
Theorem.
F o r a set
function
(a)
~
(b)
There
(c)
There
and uniform
of t h e i r
process
~
is a u n i f o r m
lim
exists
exists = w-lim
Proof.
Suppose
Consider
A £ F
and choose
m £~(T)
and define
(~)
first
following
measure
= 0
a vector ~T(A)
are
~ C a(F
holds
weak
amarts
m a y be
equivalent:
~
~ C a(F
for all
such
a stopping
that time
I
T(~)
,
if
~ 6A
I
m
,
if
~ £ ~A
such
that
e' £ ~ ' , ~)
such
that
A E F
is a u n i f o r m
~ £ T
, ~)
for all
measure
holds
that
uniform
amart.
a vector
le'(~r-Rr~)l(g)
~(A)
choose
, the
weak
amarts,
limit measure:
weak A £ F
amart. M
. For
• £ T(~)
~ C T(r)
by l e t t i n g
is w e a k l y
convergent.
:=
Then we have
~T(A) It f o l l o w s define
=
~v(~)
- ~m(~A)
that
the n e t
{ ~T(A)
a vector
measure
~ 6 a(F
~(A)
for all
A 6 F
Consider
now
:=
e' 61R'
w-lim
by
}
letting
~T(A)
. Fix
l e ' (~"u ( ~ ) - ~ ' ( D ) )
I T £ T , ~)
I
~ 6 (0,~)
<
and choose
M E T
such
that
Now
160 holds
for all
v 6 T(M)
. For
T £ T(~)
and
A6
F
T
, choose
m6~(T)
such that
le'(~m(~A)-5(~A))
and define
a stopping
(~)
i
time
~
e
,
v £ T(r)
by
letting
~(w)
,
if
~ 6 A
m
,
if
~ 6 ~A
:=
Then we have
le'(Bz(A)-~(A))I
It f o l l o w s
for all
T 6 T(~)
The remaining
3.6.5.
~
<
4s
(a)
implies
(b)
are obvious.
~
is w e a k l y
sequentially
is a set f u n c t i o n ~
(b)
There
Proof.
exists = w-lim
Combine
A set function
for all
A E F
process
~
3.6.4
~
the
measure
following
~6 a(F
and Theorem
is a u n i f o r m
sequence
. I t is c l e a r
We can now prove
the
are equivalent:
, ~)
such that
~r(~)
process
e' 6 ~ '
complete. then
amart.
a vector
is a u n i f o r m
for all
weak
Theorem
such that the
amarts:
process,
is a u n i f o r m
~(n)
holds
2s
. Therefore,
assertions
(a)
amart
<
+ le'(~m(Q~A)-5(Q~A))I
Corollary.
Suppose If
le'(~v(~)-~(~))l
that
le' (~r - RT~) I (n)
holds
~
weak
{ ~ n (A)
weak I n 6~
from Theorem potential
3.6.1.
D
potential
if it is a w e a k
}
converges
3.6.4
weakly
to
if a n d o n l y
if
lim
[e'~Tl(O)
.
Riesz
decomposition
0
that a set function
for H-bounded
uniform
weak
= 0
161
3.6.6.
Theorem.
Every ~ - b o u n d e d
u n i f o r m weak amart is the sum of an ~ - b o u n d e d
and an ~ - b o u n d e d
martingale
u n i f o r m weak potential.
The decomposition
is unique.
If the u n i f o r m weak amart is T-bounded,
then so is the u n i f o r m weak
potential. Proof. measure choose
Consider an ~ - b o u n d e d ~ . Fix
e E (0,~)
e~, e~ . . . . .
holds for all
e~ 6 U ( ~ ' )
i £ {I,2,...,k}
{AI,A 2 ..... A k} £ Pn(~) k X i=1
uniform weak amart
. For each partition
and
[[ ~(A i) H
such that , and choose
~
II ~(A i) II = n E~
with limit
{A1,A2,..., ~ }
6 P (~)
ei~(A i)
satisfying
Z iei~(Ai)-elBn(Ai) [ ~ e . Then we have
=
k X el~(A i) i=1
<
E +
<
e +
<
e + I[ ~n II (~)
k
Z i=1
le:~ln(Ai) [
k
Z i=1
t[ ~n II (A i)
This yields
II ~ II(n)
_<
il_~
I1~
,
and it follows from T h e o r e m 3.3. I and C o r o l l a r y
_~
:=
is an ~ - b o u n d e d
_~
=
is an ~ - b o u n d e d
{ Rn~
I n£~
martingale,
3.3.3 that
} hence the set function process
_~-_~ u n i f o r m weak potential.
The remaining assertions
can be proven as usual;
see the proof of
Theorem 3.4.4. Let us now study the convergence derivatives
of the g e n e r a l i z e d
Radon-Nikodym
of a u n i f o r m weak amart. The main result is the following
162
u n i f o r m weak potential 3.6.7.
theorem:
Theorem.
Suppose If
convergence
~
~
has the R a d o n - N i k o d y m
property and a separable dual.
is a T - b o u n d e d u n i f o r m weak potential,
w - l i m Dn~ n Proof.
Since
(Lemma 3.5.10)
~
=
0
then
a.e.
is T-bounded,
it follows from the maximal
that there exists a null set
A CL
inequality
such that the value
II (Dn~ n) (~) [I
su~
is finite for all
~ C Q~A
.
Now consider a sequence For each
{ e~ E ~ ' [ J 6 ~ } which is dense in 3 , there exists a null set A~ 6 L such that J
j 6~
lim
(e3Dn~ n) (~)
=
lim
(Dnei~ n) (~)
=
0
~ E ~kA. • by T h e o r e m 3.1.11 and T h e o r e m 2.5.13. 3 and ~ 6 ~ A ~ ( U ~ Aj) , the right hand side of for each e' E~'
holds for all Thus,
the inequality I (e'Dn~ n) (~) l
< --
II e'-e'j II II (Dn~ n) (~) II + l(e~Dn~ n) (c0) l
can be made a r b i t r a r i l y
small by a suitable choice of
all
large. This yields
n £~
sufficiently
lim for all
(e'Dn~n) (~)
e' 6 ~ '
and
=
0
~ E ~A~(
j 6~
and for
,
U ~ A 5) , from which the assertion
follows. By the Riesz decomposition, is a c o n s e q u e n c e weak potential 3.6.8. Suppose If
~
the u n i f o r m weak amart conver@ence
of the m a r t i n g a l e
convergence
convergence
theorem
theorem and the u n i f o r m
theorem:
Corollary. ~
has the R a d o n - N i k o d y m
property
is a T - b o u n d e d u n i f o r m weak amart,
w-lim Dn~ n
=
D
w - l i m ~n
a.e.
and a separable dual. then
163
It can be seen from E x a m p l e t h e o r e m does not e x t e n d
since every
strong a m a r t theorem
theorem:
Corollary.
Suppose ~
~
has the R a d o n - N i k o d y m
is a T - b o u n d e d
=
D
extend
of a T - b o u n d e d
fails to c o n v e r g e
in every strong
due to W.J.
strong amarts.
infinite
Chacon
strong a m a r t
strongly.
dimensional
Davis
and q u o t e d by Chacon
Banach
this
theorem
and S u c h e s t o n
in a r e f l e x i v e
More generally,
amart w h i c h does not c o n v e r g e
We c o n c l u d e
dual.
a.e.
[33] that the strong a m a r t c o n v e r g e n c e
to ~ - b o u n d e d
an example
and a s e p a r a b l e
then
w - l i m ~n
It can be seen from an e x a m p l e and S u c h e s t o n
property
strong amart,
w-lim DnB n
which
However,
the u n i f o r m w e a k a m a r t c o n v e r g e n c e
the stron9 a m a r t c o n v e r g e n c e
3.6.9.
If
that the u n i f o r m w e a k amart c o n v e r g e n c e
to weak amarts.
is a u n i f o r m w e a k amart, conta i n s
3.6.2
space
Banach
Bellow
there exists
does not
[33] also gave space
[7] p r o v e d
that
a T-bounded
strongly.
s e c t i o n w i t h a brief
discussion
of weak
sequential
amarts.
A set f u n c t i o n { ~Tn (~) { ~n E T
I n6~
3.6.10.
Next,
}
~
is a w e a k
is w e a k l y
amart
for each
if the s e q u e n c e
increasing
sequence
} .
sequential
amart
is a u n i f o r m amart.
F i r s t note that every weak consider
n £~(k)
sequential
convergent
Theorem.
E very weak
Proof.
process
I n 6~
A £ F
and choose
, define a stopping
i T
n
(~)
time
sequential
k 6~ r
n
£ T
n
,
if
~ £A
k
,
if
~ 6 ~A
amart
such that
is a weak
A £ Fk
. For all
by letting
:=
Then we have
~n(A)
:
~T
(~) - ~k ( ~ A ) n
for all
n £~(k)
Since the sequence
{ rn6T
I n6~(k)
amart.
}
~is
164
increasing,
it follows that the sequence
{ ~n (A)
I n 6~
}
is w e a k l y
convergent,
m
This yields the weak sequential amart c o n v e r g e n c e theorem:
3.6.11. Suppose If
~
Corollary. ~
has the R a d o n - N i k o d y m p r o p e r t y and a separable dual.
is a T - b o u n d e d w e a k sequential amart, then
w - l i m Dn~ n
=
D
w - l i m ~n
a.e.
F r o m an example given by Brunel and S u c h e s t o n
[31] it can be seen that
the s e p a r a b i l i t y of the dual is a n e c e s s a r y c o n d i t i o n in the w e a k sequential amart c o n v e r g e n c e theorem.
In v i e w of these examples,
all a s s u m p t i o n s made in the u n i f o r m weak
amart c o n v e r g e n c e t h e o r e m are n e c e s s a r y c o n d i t i o n s and the c o n c l u s i o n c a n n o t be improved.
3.7.
R e m a r k s .
For v e c t o r m e a s u r e s of b o u n d e d variation,
it is also p o s s i b l e to prove a
Yosida-Hewitt decomposition. A vector measure be p u r e l y f i n i t e l y a d d i t i v e if its v a r i a t i o n additive.
~ £ bva(F,~) II ~ II
is said to
is purely f i n i t e l y
It is easy to see that a r e s u l t a n a l o g o u s to L e m m a 3.1.8 holds
for c o u n t a b l y a d d i t i v e v e c t o r m e a s u r e s and for purely finitely a d d i t i v e vector measures.
R e p l a c i n g ~ - c o n t i n u o u s and ~ - s i n g u l a r v e c t o r m e a s u r e s
by c o u n t a b l y a d d i t i v e and p u r e l y f i n i t e l y a d d i t i v e v e c t o r m e a s u r e s in T h e o r e m 3.1.9 yields the Y o s i d a - H e w i t t d e c o m p o s i t i o n ; slightly more general result,
see
for a proof of a
[49; T h e o r e m 1.5.8].
C h a t t e r j i ' s c o n v e r g e n c e t h e o r e m for v e c t o r - v a l u e d m a r t i n g a l e s e x t e n d e d to u n i f o r m amarts by B e l l o w processes;
see also
[10,12]. In
[34] was
[11] who also i n t r o d u c e d these
[37], C h a t t e r j i stated the c o n v e r g e n c e
theorems for the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s of m a r t i n g a l e s of measures
(Corollary 3.3.9)
and of u n i f o r m amarts of m e a s u r e s
(Corollary
3.5.13), thus a n t i c i p a t i n g the u n i f o r m a m a r t c o n v e r g e n c e theorem.
W e a k c o n v e r g e n c e theorems were proven by C h a c o n and S u c h e s t o n strong a m a r t s and by Brunel and S u c h e s t o n
[33] for
[29] for weak sequential amarts.
These results are slightly g e n e r a l i z e d by the u n i f o r m weak amart convergence theorem
(Corollary 3.6.8). Let us also remark that, by the
general optional sampling theorem,
every strong amart of r a n d o m v a r i a b l e s
is a weak sequential amart. For strong amarts of measures, however, we only have the optional
sampling theorem in the form of T h e o r e m 2.7.3 in
which the growth c o n d i t i o n be omitted.
r
E T(n) , for all n 6~ , apparently cannot n Therefore, we c a n n o t prove that every strong a m a r t of m e a s u r e s
is a w e a k sequential amart, and it was this o b s t a c l e w h i c h led us to i n t r o d u c e u n i f o r m w e a k amarts, w h i c h g e n e r a l i z e strong amarts as well as w e a k sequential amarts.
The R a d o n - N i k o d y m p r o p e r t y is e s s e n t i a l only for p r o v i n g the c o n v e r g e n c e t h e o r e m for u n i f o r m l y l - c o n t i n u o u s m a r t i n g a l e s
(Theorem 3.3.7);
see
[49]
for a d e t a i l e d d i s c u s s i o n of the R a d o n - N i k o d y m p r o p e r t y and m a r t i n g a l e convergence.
In the c o n v e r g e n c e theorems for m a r t i n g a l e s w i t h l - s i n g u l a r
limit m e a s u r e and for p o t e n t i a l s of any type, the R a d o n - N i k o d y m p r o p e r t y is only n e e d e d in the d e f i n i t i o n of the g e n e r a l i z e d R a d o n - N i k o d y m d e r i v a t i v e s and it can thus be omitted in the case where these set f u n c t i o n p r o c e s s e s are o b t a i n e d by i n t e g r a t i n g stochastic processes. For the role of the R a d o n - N i k o d y m p r o p e r t y of the dual and the
186
s e p a r a b i l i t y of the dual in the c o n t e x t of weak convergence, and J o h n s o n
[47], and Brunel and S u c h e s t o n
see Davis
[31].
A l t h o u g h strong amarts and u n i f o r m amarts are clearly d i s t i n g u i s h e d by their r e s p e c t i v e d i f f e r e n c e properties,
w h i c h do not involve the limit
m e a s u r e and w h i c h also show that the d i f f e r e n c e b e t w e e n strong amarts and u n i f o r m amarts p r e c i s e l y c o r r e s p o n d s to the d i f f e r e n c e b e t w e e n the s e m i v a r i a t i o n and the v a r i a t i o n of a v e c t o r measure,
the role of the
limit m e a s u r e and the Riesz d e c o m p o s i t i o n seems to be m o r e important, at least in the f r a m e w o r k of set f u n c t i o n processes. The d i f f e r e n t types of c o n v e r g e n c e to the limit m e a s u r e and the d i f f e r e n t types of p o t e n t i a l s r e s u l t i n g from the r e s p e c t i v e Riesz d e c o m p o s i t i o n s e x p l a i n in w h i c h way u n i f o r m amarts,
strong amarts, and u n i f o r m weak amarts a s y m p t o t i c a l l y
a p p r o a c h martingales.
For strong amarts,
the role of the Riesz
d e c o m p o s i t i o n was e m p h a s i z e d by E d g a r and S u c h e s t o n as p o i n t e d out by B e l l o w
[11,12],
it m o t i v a t e d
[58,59]. Moreover,
"to a large degree" the
n o t i o n of a u n i f o r m amart, and we can also say this for the n o t i o n of a u n i f o r m w e a k amart.
The d i s t i n c t i o n b e t w e e n u n i f o r m amarts,
strong amarts,
and u n i f o r m weak
amarts by d i f f e r e n t types of c o n v e r g e n c e to the limit m e a s u r e suggests g e n e r a l i z i n g the n o t i o n of a set f u n c t i o n process and to study also a d a p t e d sequences of b o u n d e d v e c t o r m e a s u r e s or even a d a p t e d sequences of v e c t o r m e a s u r e s having b o u n d e d scalar variations. of course,
These correspond,
to stochastic p r o c e s s e s of Pettis i n t e g r a b l e or s c a l a r l y
i n t e g r a b l e random variables. V e c t o r - v a l u e d amarts of Pettis integrable r a n d o m v a r i a b l e s w e r e studied by Bru and H e i n i c h Sucheston Uhl
[29,30,31], E d g a r and S u c h e s t o n
[25,27,28], B r u n e l and
[58,59,60], and G h o u s s o u b
[71];
[129] studied the Pettis m e a n c o n v e r g e n c e of v e c t o r - v a l u e d amarts.
In most c o n v e r g e n c e theorems, however,
Pettis i n t e g r a b l e r a n d o m v a r i a b l e s
are forced to be B o c h n e r integrable by the c o n d i t i o n of T - b o u n d e d n e s s .
The p r o p e r t i e s of v e c t o r - v a l u e d amarts are closely c o n n e c t e d w i t h the p r o p e r t i e s of B a n a c h spaces or, more generally,
those of b o u n d e d linear
o p e r a t o r s b e t w e e n B a n a c h spaces. C h a r a c t e r i z a t i o n s of finite dimensional, reflexive, Bellow Egghe
and w e a k l y s e q u e n t i a l l y c o m p l e t e B a n a c h spaces were g i v e n by
[7], Brunel and S u c h e s t o n [66]. Brunel and S u c h e s t o n
[29,30], Edgar and S u c h e s t o n
[60], and
[31] also gave a c h a r a c t e r i z a t i o n of
the s e p a r a b i l i t y of the dual. The c h a r a c t e r i z a t i o n of a b s o l u t e l y summing o p e r a t o r s in terms of strong amarts is due to G h o u s s o u b Edgar
[73]. Moreover,
[55] gave a c h a r a c t e r i z a t i o n of A s p l u n d o p e r a t o r s in terms of weak
sequential amarts.
4.
Amar
ts
in
a
Banach
lattice
.
The theory of amarts in a B a n a c h lattice is p a r t i c u l a r l y rich. Due to the e x i s t e n c e of a partial ordering,
the notions of s u b m a r t i n g a l e s and
s u p e r m a r t i n g a l e s make sense but, d e v i a t i n g from the real case, p o s i t i v e s u b m a r t i n g a l e s and p o s i t i v e s u p e r m a r t i n g a l e s possess quite d i f f e r e n t properties.
This d i f f e r e n c e is e s s e n t i a l l y the same as the d i f f e r e n c e
e x i s t i n g b e t w e e n u n i f o r m amarts and strong amarts.
The a p p r o p r i a t e g e n e r a l i z a t i o n of s u b m a r t i n g a l e s and s u p e r m a r t i n g a l e s is given by the notion of order amarts w h i c h are d e f i n e d in terms of order convergence. Although,
in an order c o n t i n u o u s B a n a c h lattice,
every o r d e r amart is a strong amart, order amarts have the a d v a n t a g e of forming a v e c t o r lattice, and the same is true for order potentials. By the Riesz d e c o m p o s i t i o n for order amarts and the lattice p r o p e r t y of order potentials,
the c o n v e r g e n c e p r o b l e m for T - b o u n d e d order amarts
reduces to the c o n v e r g e n c e p r o b l e m for T - b o u n d e d p o s i t i v e order potentials.
These are, in particular,
p o s i t i v e u n i f o r m weak potentials,
for which weak c o n v e r g e n c e obtains u n d e r less r e s t r i c t i v e c o n d i t i o n s on the dual than this is the case for general u n i f o r m w e a k potentials. Consequently,
the order amart c o n v e r g e n c e t h e o r e m is v a l i d in a larger
class of B a n a c h lattices than the strong amart c o n v e r g e n c e t h e o r e m is.
A n o t h e r interesting point is that, for p o s i t i v e set function processes, cone a b s o l u t e l y summing o p e r a t o r s and A L - s p a c e s play the same role as a b s o l u t e l y summing o p e r a t o r s and finite d i m e n s i o n a l B a n a c h spaces do for general set f u n c t i o n processes.
This will b e c o m e clear from various
c h a r a c t e r i z a t i o n s of cone a b s o l u t e l y summing o p e r a t o r s and A L - s p a c e s in terms of set function processes.
168
The i n v e s t i g a t i o n of amarts in a B a n a c h lattice r e q u i r e s some a d d i t i o n a l i n f o r m a t i o n on v e c t o r m e a s u r e s w h i c h e s s e n t i a l l y c o n c e r n s the e x i s t e n c e of a J o r d a n d e c o m p o s i t i o n for v e c t o r m e a s u r e s of b o u n d e d variation. We shall thus start again with vector m e a s u r e s
(Section 4.1). A f t e r a brief
d i s c u s s i o n of general set f u n c t i o n p r o c e s s e s and p o s i t i v e semiamarts (Section 4.2), we shall study s u b m a r t i n g a l e s amarts and p o s i t i v e strong p o t e n t i a l s p o s i t i v e weak p o t e n t i a l s
(Section 4.3), u n i f o r m
(Section 4.4), w e a k amarts and
(Section 4.5), and order amarts
(Section 4.6).
Again, we shall c o n c l u d e this chapter w i t h some remarks and c o m p l e m e n t s (Section 4.7).
T h r o u g h o u t this chapter, ordering
<
and n o r m
let
~
be a (real) Banach lattice w i t h partial
II. II . For some d e f i n i t i o n s and p r o p e r t i e s of
specific Banach lattices, we refer to the a p p e n d i x on Banach lattices at the end of these notes. F u r t h e r details may be found in the books by Schaefer
[109] and by L i n d e n s t r a u s s and Tzafriri
[91].
4.1.
This
V e c t o r
section
relation
is c o n c e r n e d
to vector
tool will
m e a s u r e s
with
measures
.
order bounded
of b o u n d e d
again be the representation
vector
variation. of v e c t o r
measures
and their
The principal
measures
technical
by linear
operators.
Let
F
be an algebra
A set function
~
~(A)
holds
in
denote Endowed
. Clearly, bounded
the class with
4.1.1.
.
is p o s i t i v e
if
it is o r d e r
bounded
if
l~(A) I
order
and order,
Suppose
~
> ~
A 6 F , and
supF
and every
: F
~
6 ]E +
for all
exists
on a set
every
positive
vector
of a l l o r d e r b o u n d e d
the pointwise
the class
vector
measure
defined
oba(F, ~)
measure
is b o u n d e d . vector
addition,
is o r d e r b o u n d e d ,
Let
measures
oba(F, ~) in
a(F, ~ )
multiplication
is a n o r d e r e d
vector
.
by scalars,
space.
Lemma. ~
is o r d e r
Then the class
complete.
oba(F, ~)
is a v e c t o r
lattice,
(~v~)(A)
=
s u p F (A)
(~(B) + ~ ( A ~ B ) )
(~^~)(A)
=
inf F (A)
(~(B) + ~ ( A ~ B ) )
and the
identities
and
hold
for all
i is a l a t t i c e
The proof
>
norm
of t h i s
lattice
order bounded
A £ F o Moreover,
the map
II l~i (n) II on
oba(F, ~)
lemma
It can also be proven Banach
and
~, ~ £ o b a ( F , ~ )
is t h e
by direct
for the norm vector
s a m e a s in t h e r e a l c a s e
measures
methods
II l.t(~) II
that the class if
are bounded,
~
is o r d e r
(Lemma
2.1.1).
oba(F, ~) complete.
they can be represented
is a Since by
170
b o u n d e d linear o p e r a t o r s on the s u p - n o r m c o m p l e t i o n of the class of all simple functions,
and we shall use this a p p r o a c h for proving further
results on order b o u n d e d vector measures.
In order to c h a r a c t e r i z e order b o u n d e d v e c t o r m e a s u r e s in terms of their r e p r e s e n t i n g linear operators,
let us recall the f o l l o w i n g definition:
If
is a B a n a c h lattice and
~
then a b o u n d e d linear o p e r a t o r
~
~
order b o u n d e d sets in operator
~
S 6 ~ ( ~ , ~)
is an order c o m p l e t e B a n a c h lattice, • R
is regular if it maps the
into the order b o u n d e d sets in
~
iSi
is regular if and only if its m o d u l u s
exists in the o r d e r e d v e c t o r space of all linear o p e r a t o r s For a regular o p e r a t o r bounded,
. A linear
S 6 ~ ( ~ , ~)
, the m o d u l u s
ISi
~
> ~
.
is a u t o m a t i c a l l y
and we define
II s I1
:=
r
11 Isl
11
Then the map
S
I
>
II s IIr
is a n o r m on the class
~ r ( ~ , ~)
of all regular o p e r a t o r s
~
• R
,
w h i c h is an order c o m p l e t e B a n a c h lattice for this norm. For details, see
[109; Section IV.l].
4.1.2.
Theorem.
Suppose
~
is order complete.
Then the class norm
oba(F, ~)
il i.l(~) II , and the map
i s o m o r p h i s m of
oba(F, ~)
III~ III(~) holds for all
Proof.
is an o r d e r c o m p l e t e B a n a c h lattice for the
<
onto
X
is an isometric v e c t o r lattice
~ r ( D , ~)
III I~I lll(s)
=
. Moreover,
II l~l(s)
the r e l a t i o n
II
~ 6 oba(F, ~)
In order to p r o v e the theorem,
a vector measure
~£ba(F,
~)
it is s u f f i c i e n t to c o n s i d e r
and its r e p r e s e n t i n g linear o p e r a t o r
T E ~ ( D , ~) Suppose first that
~
is order bounded.
for each simple f u n c t i o n
iT( Z ~ i X A i ) I
Consider
Z ~iXAi E [-XA,X A]
=
I Z ~ i ~ ( A i) i
~
A £ F . Then we have,
,
Z
l~il I~(A i) i
~
I~I (A)
.
171
S i n c e the o r d e r
[- I~I (A) , I~I (A) ]
interval
is c l o s e d a n d
T
is b o u n d e d ,
this y i e l d s <
ITxl
for all
1~1 (A)
x £ [-XA,X A]
,
. Since each order bounded
in some m u l t i p l e
of the o r d e r i n t e r v a l
above
that
inequality
T
is r e g u l a r .
set in
[-X~,XQ] Therefore,
]9
is c o n t a i n e d
, it f o l l o w s ITJ
exists
f r o m the a n d we
have
ITIx A for all
<
A£ F . Moreover,
II lu,l(n) II
II iTi II
=
suP[0,Xn]
if
for all
order bounded,
T
is r e g u l a r ,
=
ITXBI
A £ F
and
~
II iT1x II
=
tl ITIX~ II
then
ITIX A
B 6 F(A)
. This relation
implies
that
~
is
of
I~I
,
a n d it a l s o y i e l d s
i~i(A)
~
ITIx A
,
A6 F , hence
II I~I(~) II Thus,
,
of
~(B) I
for all
we have
<
Conversely,
holds
,
II
II ITI
because
11~I (A)
<
II ITI II
we h a v e
I~I which means
=
that
ITI o X
ITI
,
is the r e p r e s e n t i n g
a n d we a l s o h a v e
It l ~ l ( n )
II
=
II ITI
I~
=
~I T II r
linear operator
172
This proves the a s s e r t i o n c o n v e r n i n g
oba(F, ~)
and
X . Finally, we
have
III ~ III(n)
=
II T II
<
II ITI
II
=
III I~I III (n)
,
by T h e o r e m 3.1.1.
The n e x t result i l l u s t r a t e s that, m o d u l o a mild c o n d i t i o n on the B a n a c h lattice, the o r d e r b o u n d e d v e c t o r m e a s u r e s form an i n t e r m e d i a t e c o n c e p t b e t w e e n the v e c t o r m e a s u r e s of b o u n d e d v a r i a t i o n and the b o u n d e d v e c t o r measures. 4.1.3.
Theorem.
Suppose
~
has p r o p e r t y
Then the class norm
H.
bva(F, ~)
is an order c o m p l e t e Banach lattice for the
If(Q) , and the map
i s o m o r p h i s m of an ideal in
bva(F, ~)
oba(F, ~)
II I~I(Q) II holds for all
Proof.
(P).
<
~
~)
. Moreover,
bva(F, ~)
~)
(P), the class
~I(D,
~)
tl. 111 , and it also is an ideal in
by
[109; T h e o r e m IV.4.3]. NOW the a s s e r t i o n c o n c e r n i n g
X
follows from T h e o r e m 3.1.2 and T h e o r e m 4.1.2. Moreover, £ b v a ( F , ~)
is a B a n a c h ~ r ( D , ~)
bva(F, ~)
=
III I~I lll(s)
by T h e o r e m 4.1.2 and since
II. ll(~)
<
II I~I ll(n)
=
II ~ll(S)
,
is a lattice norm.
Corollary.
The f o l l o w i n g are equivalent: (a)
~
(b)
bva(F, ~)
Moreover,
if
and
is an AL-space.
oba(F, ~)
Proof. IV.4.5].
~
is an AL-space.
is an AL-space,
then the B a n a c h lattices
bva (F, ]E)
are identical.
Note that
D
is an A M - s p a c e and apply
[109; P r o p o s i t i o n
,
and
for all
, we have
II l~l(s) II
4.1.4.
is
II u, ll(n)
has p r o p e r t y
lattice for the n o r m
is an isometric v e c t o r lattice ~I(D,
, and the i n e q u a l i t y
~£bva(F,
Since
X
onto
173 We shall now c h a r a c t e r i z e cone a b s o l u t e l y summing o p e r a t o r s and,
in
particular, A L - s p a c e s in terms of p o s i t i v e vector m e a s u r e s and in terms of order b o u n d e d v e c t o r measures.
These results are c o u n t e r p a r t s of the
c h a r a c t e r i z a t i o n of a b s o l u t e l y summing operators and finite d i m e n s i o n a l Banach spaces given in Section 3.1.
4.1.5.
Theorem.
Suppose
~
is a B a n a c h lattice,
~
is a B a n a c h space, and
S :
>14
is a b o u n d e d linear operator. Then the f o l l o w i n g are equivalent: (a) S is cone a b s o l u t e l y summing. (b)
There exists a c o n s t a n t [I S~ {l (Q) < O I I ~(~) II
p £]R+
each p o s i t i v e v e c t o r m e a s u r e
(c)
vector m e a s u r e if
holds with
Proof.
S
~6a(F,
Suppose first that
S
T 6 ~ ( D , ~)
~ E is bounded).
(note that
cone a b s o l u t e l y summing, we have,
hence
S 0T
and its r e p r e s e n t i n g linear o p e r a t o r Since
II s II1
II z Tx i II
<_
II s II1
II T II II Z x i II
_<
II s II1
II T II
S
,
Since
,
S 0 T
is the r e p r e s e n t i n g linear
S~ , the above i n e q u a l i t y y i e l d s
<_
II s II1
III~ III(~)
by T h e o r e m 3.1.2, T h e o r e m 3.1.1, and since (a) implies
(b) implies
4.1.6 that
is p o s i t i v e and
is cone a b s o l u t e l y summing and satisfies the inequality
II S ~ l l ( n )
Clearly,
T
for each finite c o l l e c t i o n
_<
[109; P r o p o s i t i o n IV.3.3].
Therefore,
then the inequality in (b)
,
II S T x i II
II S 0 T II1
o p e r a t o r of
and for each p o s i t i v e
is cone a b s o l u t e l y summing. C o n s i d e r a
~ 6 a(F, ~)
z
F
.
positive v e c t o r measure
{Xl,X 2 .... ,Xk} c D +
and for
~)
is cone a b s o l u t e l y summing,
O = II s II l
F
~ £ a (F, ~)
S~ E bva(F, IF:.) holds for each algebra
Moreover,
by
such that
holds for each a l g e b r a
=
II s II1
~
II ~ ( n )
II
,
is positive.
(b).
(c), and it follows from the s u b s e q u e n t Example
(c) implies
(a).
is
174
4.1.6. Suppose
Example. ~
is a B a n a c h lattice,
~
is a B a n a c h space, and
S :~
>
is a b o u n d e d linear o p e r a t o r w h i c h is not cone a b s o l u t e l y summing. Then there exists a summable sequence sequence
{ Sx n
i n 6~
}
C o n s i d e r the algebra ,
I n 6~
}
such that the
is not a b s o l u t e l y summable.
F
Bn, 2 := [2-n,2 -n+1)
{ xn E~+
on
[0,1)
n£~
w h i c h is g e n e r a t e d by the sets
, and define a vector m e a s u r e
~ 6 a(F, ~)
by letting
~(Bn, 2) for all
n 6~
measure
SB
4.1.7.
:=
xn
. Then
~
, is positive,
hence bounded, but the v e c t o r
does not have b o u n d e d variation.
Corollary.
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
(as a topological v e c t o r lattice)
to an
AL-space.
II ~(~)
(b)
II = H ~ ll(~)
holds for each a l g e b r a
positive vector measure (c)
~Ebva(P,
~)
vector measure
Proof.
holds for each algebra
i~
P
and for each p o s i t i v e
[109; T h e o r e m IV.2.7],
(as a t o p o l o g i c a l v e c t o r lattice)
if the identity map
and for each
~ 6 a(F, ~)
By S c h l o t t e r b e c k ' s t h e o r e m
isomorphic
F
~ E a(F, ~)
~
is
to an A L - s p a c e if and only
is cone a b s o l u t e l y summing, and in this case
we have
1
:
lli%ll
--
Ill%If
I
N o w the a s s e r t i o n follows from T h e o r e m 4.1.5.
4.1.8. Suppose
Corollary. ~
is an AL-space.
Then the identity
III~ lll(n)
=
II~(Q) II
--
II ~ ll(n)
holds for each p o s i t i v e v e c t o r m e a s u r e Proof.
~ 6 a(P, ~)
The first identity follows from T h e o r e m 4.1.2 and the second
one is given in C o r o l l a r y 4.1.7.
m
175
The f o l l o w i n g results are v a r i a n t s of T h e o r e m 4.1.5 and C o r o l l a r y 4.1.7, with order b o u n d e d v e c t o r m e a s u r e s in the place of p o s i t i v e v e c t o r measures:
4.1.9.
Theorem.
Suppose and
~
is an o r d e r c o m p l e t e B a n a c h lattice,
S :~
> ~
~
is a B a n a c h space,
is a b o u n d e d linear operator. T h e n the f o l l o w i n g are
equivalent: (a)
S
is cone a b s o l u t e l y summing.
(b)
There exists a c o n s t a n t
p E~+
i[ S~ II(~) < p II [~I(~) [[
such that
holds for each a l g e b r a
F
and for
m
each vector measure (c)
S~ £ bva(F, ~) measure
Moreover,
if
holds with Proof.
S
F
and for each v e c t o r
is cone a b s o l u t e l y summing, then the inequality in (b)
p = II S II1 .
f' 6 ~
satisfying
II Sx II
holds for all have,
~)
~ E oba(F, ~)
Suppose first that
exists
~£oba(F,
holds for each a l g e b r a
<
x £~
is cone a b s o l u t e l y summing. Then there II S II1
and such that
f' (Ixl)
. Consider a vector measure
for e a c h p a r t i t i o n
z
S
Jl f' II ~
II S~(A i) II
{AI,A 2 .... , % } 6 P(~)
<_
f' ( Z I~(A i) I)
<
f'(l~l(~))
_<
II s II1 II lul(n) II
w
~ £ oba(F, ~)
. Then we
,
This yields
II s~ll(n) Therefore, Clearly, implies
Again,
<_
(a) implies
(b) implies
II s II1 II I~I(~) II (b).
(c), and it follows from E x a m p l e 4.1.6 that
(c)
(a).
the f o l l o w i n g c h a r a c t e r i z a t i o n of A L - s p a c e s
S c h l o t t e r b e c k ' s theorem:
is a c o n s e q u e n c e of
176
4.1.10. suppose (a)
Corollary. is order complete. is isomorphic
Then the f o l l o w i n g are equivalent:
(as a t o p o l o g i c a l vector lattice)
to an
AL-space. (b)
II I~I(Q) II = II ~ II(~) vector measure
(c)
bva(F, ~)
holds for each algebra
F
and for each
~ E oba(F, ~)
= oba(F, ~)
holds for each algebra
F .
4.2.
S e t
The main
purpose
cone absolutely function
Let
F
f u n c t i o n
of this summing
operators
and AL-spaces
in t e r m s
of positive
be a stochastic
~
basis
is a B a n a c h
is a b o u n d e d
linear
(a)
S
is c o n e
(b)
S
maps
on a set
in (c)
S
~
set
~
.
maps
Suppose
space,
following
are
and
S
:
equivalent:
summing. in
~
into the T-bounded
II S ~ T ll(~)
in
~
first
(on a n a r b i t r a r y set function
processes
that
~
Therefore,
(a)
implies
It f o l l o w s
from the
S
in
in
basis)
semiamarts
into
in
~
the T-bounded
(on a n set function
.
~
is f i n i t e ,
process
positive
stochastic
semiamart
set function
is a B a n a c h the
semiamarts
the N-bounded
processes
Proof.
absolutely
basis)
•
Then
.
arbitrary
a positive
lattice,
operator.
the positive
stochastic
~
is c o n e a b s o l u t e l y . Then
sup T
by Theorem
summing.
II ~ T ( ~ )
4.1.5,
hence
Consider
is f i n i t e ,
II
S~
hence
is a T - b o u n d e d
.
(b).
subsequent
Example
4.2.2
that
(c)
implies
(a).
o
Example.
4.2.2.
Suppose
~
is a B a n a c h
is a b o u n d e d there
sequence
linear
exists { Sx n
On the standard process
~
lattice,
operator
a summable
I n £~
}
stochastic
~
which
is a B a n a c h is n o t c o n e
sequence
basis
on
[0,1)
space,
and
absolutely
{ x n 6~+
is n o t a b s o l u t e l y
I n 6~
summable. , define
} Let
S
:F
n Z
xj
,
if
k = 1
such that the x
:= Z x n
a set function
Xn
,
if
k
0
,
otherwise
9=I ~ n (Bn ,k )
~
:=
is a p o s i t i v e
set function
process.
=
2
Moreover,
)
summing.
by letting
X--
Then
of
Theorem.
Suppose
Then
is t o g i v e a f i r s t c h a r a c t e r i z a t i o n
processes.
4.2.1.
sup T
section
p r o c e s s e s .
it c a n b e
seen
178 from Example 3.2.5 that function process constant
p 6~+
S~
~
is an P - b o u n d e d
is not T - b o u n d e d
s e m i a m a r t and that the set
(for the p r e s e n t example,
may be chosen to be equal to
the
II x II ).
For later reference, we also remark that the set function process is a strong potential, { ~T (~)
I T £ T }
by Example 3.2.5, and that the net
d e c r e a s e s to
0 , which means that
potential
(see Section 4.3 for the definition).
4.2.3.
Corollary.
~
is a Doob
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
(as a t o p o l o g i c a l vector lattice)
to an
AL-space. (b)
Every p o s i t i v e semiamart is T-bounded.
(c)
Every P - b o u n d e d p o s i t i v e semiamart is T-bounded.
The f o l l o w i n g e l e m e n t a r y lemma can be proven as in the real case (Lemma 2.2.1):
4.2.4. Suppose If
~
Lemma. ~
has p r o p e r t y
(P).
is a set f u n c t i o n process and
r
is a b o u n d e d stopping time,
then the identity
i ~ T I (A) holds for all
=
Z I~pl (An{~=p}) p=l
A E F
The f o l l o w i n g result is obvious:
4.2.5. Suppose
Theorem. ~
has p r o p e r t y
(P).
Then the class of all T - b o u n d e d set f u n c t i o n p r o c e s s e s is a B a n a c h lattice for the n o r m
lJ. II T "
For the r e m a i n d e r of this chapter,
A. :
F=,
'>... [ 0 , 1 ]
be a fixed p r o b a b i l i t y measure.
let
4.3.
S U b
m a r t i n ~ a 1 e s .
S u b m a r t i n g a l e s c e r t a i n l y c o n s t i t u t e the m o s t important class of set function p r o c e s s e s w h i c h are defined in terms of the partial o r d e r i n g of the B a n a c h lattice.
P a r t i c u l a r l y i n t e r e s t i n g are, of course, p o s i t i v e
s u b m a r t i n g a l e s since they may arise as the modulus of a martingale. We shall see that p o s i t i v e s u b m a r t i n g a l e s have many p r o p e r t i e s w i t h u n i f o r m amarts. For n e g a t i v e submartingales, is quite different:
Usually,
however,
in common the situation
these p r o c e s s e s only share the p r o p e r t i e s
of strong amarts but not those of u n i f o r m amarts, e x c e p t for the case w h e r e the B a n a c h lattice is an AL-space.
A set f u n c t i o n p r o c e s s if the net
{ ~z(Q)
~
is a s u b m a r t i n ~ a l e
I ~ E T }
is increasing
is a Doob p o t e n t i a l if the net
{ ~T(~)
(resp. supermartin~ale)
(resp. decreasing),
I T £T }
The following c h a r a c t e r i z a t i o n of s u b m a r t i n g a l e s real case
(Theorem 2.4.1):
4.3.1.
Theorem.
For a set f u n c t i o n process (a)
~
d e c r e a s e s to
and it 0 .
is the same as in the
, the f o l l o w i n g are equivalent:
~
is a submartingale.
(b)
~
~ R ~O
holds for all
T £ T
and
o6T(T)
(c)
~n ~ RnBm
holds for all
n 6~
and
m £ ~ (n)
(d)
~n ~ Rn~n+1
holds for all
n E~
.
Let us first study p o s i t i v e submartingales.
4.3.2. If
~
II ~ II Proof.
Theorem. is a p o s i t i v e submartingale,
then the set function process
is a real submartingale.
For all
II ~T li(A)
as was to be shown.
T 6T
,
o C T(~)
and
A C F T , we have
=
suppT(A)
Z
II ~T(Ai)
II
<
SUppo(A)
Z
II ~o(Ai) II
=
II ~o il(A) D
180
4.3.3.
Corollary.
For a p o s i t i v e s u b m a r t i n g a l e (a)
~
is ~ - b o u n d e d .
(b)
~
is T-bounded.
Moreover,
if
~
~
is ~ - b o u n d e d ,
, the following are equivalent:
then
II ~ I~ = II ~ IIT
The next result is the Riesz d e c o m p o s i t i o n for ~ - b o u n d e d p o s i t i v e submartingales:
4.3.4. Suppose
Theorem. ~
is a KB-space.
Then every ~ - b o u n d e d p o s i t i v e s u b m a r t i n g a l e is the d i f f e r e n c e of an ~ - b o u n d e d
p o s i t i v e m a r t i n g a l e and a T - b o u n d e d Doob potential.
The d e c o m p o s i t i o n
Proof. all
is unique.
C o n s i d e r an ~ - b o u n d e d
A E F
, the sequence
increasing,
positive submartingale
{ ~n(A)
[ n 6~
hence c o n v e r g e n t since
~(A)
:=
exists for a l l
lim ~n(A)
A E F
~
:
su b
Therefore,
_~
:=
{ Rn~
~
for
Thus,
~n(A) 5 £ a(F,
it can be shown that
g e n e r a t e s an ~ - b o u n d e d
I n ql~ }
. Then,
is n o r m b o u n d e d and
is a KB-space.
and defines a v e c t o r m e a s u r e
As in the proof of T h e o r e m 3.4.4, variation.
}
~
~
~)
has b o u n d e d
positive martingale
,
and it is then clear that the set function process
_~
::
_~-_~
is a Doob p o t e n t i a l w h i c h is T - b o u n d e d since
~
and
~
are T-bounded,
by C o r o l l a r y 4.3.3.
[]
The Riesz d e c o m p o s i t i o n will be used in the proof of the strong c o n v e r g e n c e t h e o r e m for ~ - b o u n d e d
p o s i t i v e submartingales,
we shall
also need the following result on B a n a c h lattices:
4.3.5.
Lemma.
Suppose
~
If, for
u 6~+
to
has the R a d o n - N i k o d y m property. , the sequence
{ u n £ [0,u]
u , then it c o n v e r g e s strongly to
u
I n6~
}
converges weakly
181
Proof.
Consider
0
<
z £ [0,u]
u - u vz
--
for
all
n 6~
and
we also
all
z
=
<
u
z - Un^Z
n E~
, which
,
I n 6~
u ^z n
} c
=
(UnVZ-U)
+
( u - u n)
point
[0,u]
,
yields
=
is a s t r o n g l y
{ UnAZ
n
yields
u vz n
w-lim
If
u - u
--
, which
we have
have
0
for
<
n
w-lim
. Then
z
exposed
[0,u]
converges
of
strongly
, then to
z
the
sequence
, which
yields
+ lim
If
z
Zl,
z2,
( z - u n)
is a c o n v e x ...,
=
(Z-UnAZ)
combination
z k 6 [0,u]
~i(zi-
lim
, then
u n)
=
Z ~izi we
of
0
strongly
exposed
points
have
u n)
~
X ~i(zi-
<
Z ~ i ( z i - Un)
and + 0
,
hence
_
0
for
all
<
n £~
=
(Z-Un)+
, which
( Z~i(zi-u
n)
)+
<
Z ~i(zi-Un)
+
,
yields +
lim
By Phelps' the from
theorem
closed the
(z - Un)
convex
[49; hull
=
0
Theorem of
its
VII.3.3], strongly
the
order
exposed
points.
inequality + 0
<__
u-
un
<_
(u-z)
+
(z-u
n)
interval
,
Now
[0,u] it follows
is
182
for all to
u
n E]q , t h a t
sequence
{ un
I n E]q }
converges
strongly
.
o
We can now prove
4.3.6.
the p o s i t i v e
submartin~ale
convergence
theorem:
Theorem.
Suppose If
the
_~
IE
has
the
Radon-Nikodym
is an ]q-bounded
lim Dn~ n Proof.
positive
=
D
By the R i e s z
measure
~
and
l i m ~n
:=
{ Rn~
Y
:=
D~
then
a.e.
decomposition,
increases
_~
property. submartingale,
the
to the ] q - b o u n d e d
submartingale positive
_~
has
a limit
martingale
I n E]q }
Define
and,
for all
n q]q ,
Xn
:=
DnB n
Yn
:=
Dn~n
and
By the martingale
lim Y
Since
convergence
=
n
Y
the m a r t i n g a l e
theorem
(Corollary
3.3.9),
we have
a.e.
~
majorizes
the p o s i t i v e
submartingale
have
0
<
X
--
< n
Y
--
a.e., n
hence
0
<
X
--
for all
vY
- Y
n
n E]q , w h i c h
lim X
n
vY
<
Y
--
yields
=
Y
a.e.
VY n
- YvY
< --
IY
-YJ n
a.e.,
~
, we
183
In o r d e r
to prove
lim X
by the vector n £~
AY
n
X
by Lemma
4.3.5.
In order
to prove
Y
=
lattice
, it is e v e n
w-lim
to
lim X
= Y
n Y
identity.
AY
--
that
the
is a b o u n d e d
real
e' E ~ '
set function
limit measure Theorem
Since
n
^ Y £ [0,Y]
{ Xn ^ Y
for
e' £ ~
submartingale
e'5
X
sequence that,
for
with
to prove
a.e.
holds
for all
a.e.,
e'~
with
sufficient
to prove
Y
, let us first note
, the
it is t h u s
a.e.,
sufficient
n
a.e.,
with
process
I n 6~ , the
converges
limit measure
e'~
, and the amart
}
set function e'~
is a b o u n d e d
convergence
. Hence,
real
theorem
weakly
process
amart
yields,
3.1.11,
lim e'Dn~ n
=
lim Dne'~ n
=
D e'~
=
e'D ~
a.e.,
hence
lim e'X
and
n
=
e'Y
therefore
lim e'(X n AY)
Next, has
a.e.,
for almost
all
hence
convergent
lim e'(X n +Y-X
~ £ fl , w e h a v e
the Radon-Nikodym
compact,
=
weakly
subsequence
property,
=
e'Y
(X n ^ Y) (to) £ [0,Y(to)]
the order
sequentially
n v Y)
interval
compact,
. Since
[0,Y(to)]
and we may
{ (Xnk ^ Y) (to) I k 6 ~
a.e.
select
} , depending
on
is w e a k l y a weakly to .
Define
X(to)
:=
w-lim
Then we have,
for all
depending
e'
on
(Xnk ^ Y) (to)
e' E ~ '
and
toEfl
for all
outside
,
e' (X(t0))
=
lira e' ((Xnk ^ Y ) (to))
=
lime'
((X n ^ Y ) (~))
=
e' (Y(to))
a null
set
184
Therefore, n E~
X
is w e a k l y m e a s u r a b l e .
, are a l m o s t
separable.
Then
X
separably X
=
valued,
Y
every weak c l u s t e r a.e.,
sequence.
result
the o r d e r
interval
seque n c e
{ Xn ^ Y
cluster
hence
point
I n 6~
Y
Doob p o t e n t i a l
submartingale
Doob potentials,
4.3.7. Suppose
property,
is w e a k l y }
is
{ Xn ^ Y
LI(L,JI,
and,
~)
and it f o l l o w s
convergent
I n 6~
p o i n t of this
is o r d e r c o n t i n u o u s
. Thus,
t h a t the
to its u n i q u e weak a
submartingales.
We have p o i n t e d
in the Riesz d e c o m p o s i t i o n is T-bounded.
This
out that the
of an ~ - b o u n d e d
is not the case
as c a n be seen f r o m the f o l l o w i n g summing
operators
and AL-spaces:
lattice,
~
for a r b i t r a r y
characterization
of
Theorem. ~
is a B a n a c h
linear operator.
S
is cone a b s o l u t e l y
(b)
S
maps
S
maps
(d)
S
maps
Proof.
Suppose
the p o s i t i v e
r £ T
that
, hence
(c) implies
By E x a m p l e
4.2.2,
subsequent
Example
~
<
martingales in
S in
in
~
~
in
S :F
into the
in
in
4.3.8 that
into
~
into the ~ - b o u n d e d
.
. Then we have,
<
summing.
Consider
by T h e o r e m
4.1.5,
II s II 1 I l U l (Q) II
is T-bounded.
(b) and
•
in
(c).
(b) implies
>
.
is cone a b s o l u t e l y ~
~
supermartingales
processes
II s 111 II ~T (n) II S~
(a) implies
positive
processes
first
II s ~ x I I ( n )
Therefore,
processes
set f u n c t i o n
supermartingale
and
are equivalent:
Doob p o t e n t i a l s
set f u n c t i o n
set f u n c t i o n
space,
summing.
the ( ~ - b o u n d e d )
the T - b o u n d e d
is a B a n a c h
Then the f o l l o w i n g
the ( ~ - b o u n d e d )
T-bounded (c)
Clearly,
~
compact,
is w e a k l y
(a)
for all
p o i n t of the s e q u e n c e
is the u n i q u e weak c l u s t e r
is a b o u n d e d
positive
~
, as was to be shown,
occuring
cone a b s o l u t e l y
that
,
and we h a v e
[136], the same is true for
[0,Y]
Let us now study g e n e r a l
positive
Y
By the R a d o n - N i k o d y m
by C a r t w r i g h t ' s
we may and do a s s u m e
Xn A Y
a.e.
is equal
Y
the r a n d o m v a r i a b l e s
is s t r o n g l y m e a s u r a b l e ,
More generally, to
Since
(d). (a), and it can be seen from the (d) implies
(a).
,
a
}
185
4.3.8.
Example.
Suppose
~
is a B a n a c h
is a b o u n d e d Then
there
exists
sequence On the
linear
{ Sx n
_~
}
sequence
stochastic
basis
n Z
:=
is a p o s i t i v e n Z j=1
]I Sx. l] 3
The
the
and
I n 6~
summable. , define
S
:~
}
such
Let
x
t h a t the
:= Z x n
a set f u n c t i o n
n Z
<
,
if
k = 1
,
if
k 6 {2 j-1 + I ..... 29 }
3
Xn+l_ j
For
all
n 61~ , we h a v e n
--j=l
2J-I11 21-J
Xn+l_j
H
+
II x - Z
xj II 9=1
S ~ n II (0)
II
set f u n c t i o n
process
S~
~
is n o t ~ - b o u n d e d .
are e q u i v a l e n t : is i s o m o r p h i c
(as a t o p o l o g i c a l
vector
to an
lattice)
AL-space. (b)
Every (~-bounded)
Doob
(c)
Every (~-bounded)
positive
(d)
Every
It s h o u l d
be n o t e d
positive
that
potential
martingale
even
is T - b o u n d e d .
supermartingale
is T - b o u n d e d .
is ~ - b o u n d e d .
in an A L - s p a c e
a positive
submartingale
n e e d n o t be H - b o u n d e d .
For
a wide
class
of B a n a c h
lattices,
every
Doob
potential
is a s t r o n g
potential:
4.3.10. The
Theorem.
following
are equivalent:
(a)
~
is o r d e r
(b)
~
is c o u n t a b l y
is a s t r o n g
Proof.
Suppose
countably
order
continuous. order
complete
and every
Doob
potential
potential.
first
that
complete.
If
~
is o r d e r
~
is a D o o b
continuous. potential,
>
summing.
Corollary.
following
(a)
x.
martingale.
=
4.3.9.
{ x n £F+
[0,1)
9=i 21-j
Therefore,
on
space,
absolutely
by l e t t i n g
%in (Bn, k)
_~
is a B a n a c h is n o t c o n e
is n o t a b s o l u t e l y
x -
Then
~
which
a summable I n 6~
standard
process
lattice,
operator
Then
~
is
then we have
186
lim
III UT III(~)
=
by T h e o r e m 4.1.2 and since Conversely,
lira II ~T(n) II
~
=
0
,
is order continuous.
if every Doob p o t e n t i a l is a strong potential,
then this is
in p a r t i c u l a r true for Doob p o t e n t i a l s on the trivial stochastic basis, hence
~
is c o u n t a b l y order continuous.
order complete,
it follows that
~
~
is a l s o c o u n t a b l y
is order continuous.
We shall see that even an ~ - b o u n d e d potential.
Since
Q
Doob potential need not be a u n i f o r m
This will be made precise by another c h a r a c t e r i z a t i o n of cone
a b s o l u t e l y summing o p e r a t o r s and AL-spaces.
Let us first prove the Riesz
d e c o m p o s i t i o n for H - b o u n d e d n e g a t i v e submartingales:
4.3.11. Suppose
Theorem. ~
is order continuous.
Then every ~ - b o u n d e d n e g a t i v e s u b m a r t i n g a l e is the d i f f e r e n c e of an H - b o u n d e d m a r t i n g a l e and an ~ - b o u n d e d
Doob potential.
The d e c o m p o s i t i o n is unique. If the n e g a t i v e s u b m a r t i n g a l e is T-bounded,
Proof. all
then so is the Doob potential.
C o n s i d e r an Y - b o u n d e d n e g a t i v e s u b m a r t i n g a l e
A 6 F , the sequence
~(A) exists since
:= ~
su~
{ ~n (A)
}In (A)
_<
I n E~
}
~ . Then,
is m a j o r i z e d by
for
0 , hence
0
is order complete, and we have
~(A)
=
lim ~n(A)
,
by the s u b m a r t i n g a l e p r o p e r t y and since is a v e c t o r m e a s u r e in
a(F
, ~)
~
is order continuous. Hence
. The r e m a i n d e r of the proof is the
same as in the proof of T h e o r e m 4.3.4.
o
Let us remark that, d i f f e r e n t l y from the case of an ~ - b o u n d e d p o s i t i v e submartingale,
the Doob p o t e n t i a l o c c u r i n g in the Riesz d e c o m p o s i t i o n
of an ~ - b o u n d e d n e g a t i v e s u b m a r t i n g a l e n e e d not be T-bounded, by C o r o l l a r y 4.3.9.
4.3.12. Suppose
Corollary. ~
is order continuous.
Then every ~ - b o u n d e d n e g a t i v e s u b m a r t i n g a l e p o s i t i v e supermartingale)
is a strong amart.
(and thus: every ~ - b o u n d e d
187 The f o l l o w i n g c h a r a c t e r i z a t i o n of cone a b s o l u t e l y summing o p e r a t o r s and A L - s p a c e s is similar to T h e o r e m 4.3.7 and C o r o l l a r y 4.3.9:
4.3.13. Suppose and
Theorem. ~
is an order c o n t i n u o u s B a n a c h lattice,
S : ~ ---~
~
is a B a n a c h space,
is a b o u n d e d linear operator. T h e n the f o l l o w i n g are
equivalent: (a)
S
is cone a b s o l u t e l y summing.
(b)
S
maps the (~-bounded)
p o t e n t i a l s in (c)
S
~
in
~
(a) implies
an ~ - b o u n d e d
S
in
~
is cone a b s o l u t e l y summing.
If
into the
~
is
II S H 1 lim II ~T(~) II ~
=
0
,
is order continuous.
(b).
By the Riesz d e c o m p o s i t i o n , if
positive supermartingales .
S
<
by T h e o r e m 4.1.9 and since Therefore,
into the u n i f o r m
, then we have
II S~ T If(Q)
lim
Finally,
~
Suppose first that
a Doob p o t e n t i a l
~
.
maps the ~ - b o u n d e d
u n i f o r m amarts in
Proof.
Doob p o t e n t i a l s in
(b) implies
(c).
is not cone a b s o l u t e l y summing,
positive supermartingale
T-bounded, by T h e o r e m 4.3.7, h e n c e
S~
~
in
then there exists
~
such that
S~
is not
fails to be a u n i f o r m amart, by
C o r o l l a r y 3.5.4.
4.3.14.
o
Corollary.
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
(as a t o p o l o g i c a l v e c t o r lattice)
to an
AL-space. (b)
E v e r y (~-bounded)
(c)
Every ~ - b o u n d e d
Proof.
Since every A L - s p a c e is order continuous,
(c), by T h e o r e m 4.3.13.
Doob p o t e n t i a l
is a u n i f o r m potential.
positive supermartingale
is a u n i f o r m amart.
(a) implies
(b) and
The c o n v e r s e i m p l i c a t i o n s f o l l o w from C o r o l l a r y
4.3.9, as in the proof of T h e o r e m 4.3.13.
The Riesz d e c o m p o s i t i o n for ~ - b o u n d e d g e n e r a l i z e d to those ~ - b o u n d e d
o
n e g a t i v e s u b m a r t i n g a l e s can be
s u b m a r t i n g a l e s w h i c h are not n e c e s s a r i l y
n e g a t i v e but m a j o r i z e d by an ~ - b o u n d e d m a r t i n g a l e ;
in a similar way,
C o r o l l a r y 4.3.12, T h e o r e m 4.3.13, and C o r o l l a r y 4.3.14 can be modified. In a w e a k l y s e q u e n t i a l l y c o m p l e t e B a n a c h lattice,
even this latter
188
condition
can be dropped,
general ~-bounded
4.3.15. ~
is a KB-space.
martingale
submartingale
and an ~ - b o u n d e d
The d e c o m p o s i t i o n
is the d i f f e r e n c e
of an ~ - b o u n d e d
Doob potential.
is unique.
If the s u b m a r t i n g a l e
4.3.4,
for
submartingales:
Then e v e r y ~ - b o u n d e d
process
the Riesz d e c o m p o s i t i o n
Theorem.
Suppose
Proof.
and we thus o b t a i n
is T-bounded,
then
so is the Doob potential.
C o n s i d e r an ~ - b o u n d e d s u b m a r t i n g a l e ~ . Then the set f u n c t i o n + ~ is an ~ - b o u n d e d p o s i t i v e submartingale, hence, by T h e o r e m + ~ is the d i f f e r e n c e of an ~ - b o u n d e d p o s i t i v e m a r t i n g a l e
and a T - b o u n d e d
<
Doob potential.
~
+
<
~
Then we have
,
hence
~_
:=
is an ~ - b o u n d e d difference
_~-~_ negative
submartingale.
of an ~ - b o u n d e d
martingale
By T h e o r e m ~
4.3.11,
and an ~ - b o u n d e d
~
is the
Doob p o t e n t i a l
. Thus we have
=
from w h i c h
~_ + ~_
=
_~ + ~_ - ~_
the a s s e r t i o n
follows.
F r o m the Riesz d e c o m p o s i t i o n following
characterization
4.3.16.
Theorem.
The f o l l o w i n g
,
for ~ - b o u n d e d
the
are e q u i v a l e n t :
~
(b)
Every ~-bounded
submartingale
(c)
Every ~ - b o u n d e d
positive
Proof.
is a K B - s p a c e .
Suppose
submartingale
then this
we o b t a i n
of KB-spaces:
(a)
Conversely,
submartingales,
first
that
is a strong
~
is in p a r t i c u l a r
is a KB-space.
amart,
if every ~ - b o u n d e d true
is a strong amart.
submartingale
by T h e o r e m
positive
is a strong amart.
Then e v e r y ~ - b o u n d e d
4.3.15
and T h e o r e m
submartingale
for e v e r y ~ - b o u n d e d
4.3.10.
is a strong amart,
positive
submartingale
189
on the trivial submartingale
stochastic
basis,
on the trivial
and thus for e v e r y ~ - b o u n d e d
stochastic
basis,
hence
~
is a
KB-space.
Since every B a n a c h the strong convergence
theorem
that a s l i g h t l y the p o s i t i v e
Also
lattice w i t h the R a d o n - N i k o d y m
amart convergence
for ~ - b o u n d e d
more g e n e r a l
weak potential
from the Riesz
characterization
cone a b s o l u t e l y
submartingales.
convergence
decomposition,
we o b t a i n
submartingales,
summing
theorem;
see, however,
as a c o n s e q u e n c e
see S e c t i o n
of
4.5.
the f o l l o w i n g
summing o p e r a t o r s using earlier
operators
is a KB-space,
to give a weak
We shall
r e s u l t can be o b t a i n e d
of cone a b s o l u t e l y
terms of ~ - b o u n d e d
property
t h e o r e m can n o w be used
and A L - s p a c e s
and A L - s p a c e s
characterizations in terms
in of
of Doob
potentials:
4.3.17.
Theorem.
Suppose
~
is a KB-space,
~{
is a B a n a c h
linear operator.
(a)
S
is cone a b s o l u t e l y
summing.
(b)
S
maps
submartingales
S
maps
amarts
4.3.18.
the f o l l o w i n g
the~-bounded
set f u n c t i o n
(c)
Then
space,
a bounded
processes
the ~ - b o u n d e d in
~
in
~
and
S :~
is
>
are equivalent:
in
into the T - b o u n d e d
in
into the u n i f o r m
.
submartingales
.
Corollar Y .
The f o l l o w i n g (a)
~
are equivalent: is isomorphic
(as a t o p o l o g i c a l
vector
lattice)
to an
AL-space. (b)
Every P-bounded
submartingale
is T-bounded.
(c)
Every ~-bounded
submartingale
is a u n i f o r m amart.
F r o m the u n i f o r m a m a r t c o n v e r g e n c e
4.3.19. Suppose If
~
t h e o r e m we thus obtain:
Cor01!ary. ~
is i s o m o r p h i c
is an ~ - b o u n d e d
lim D n ~ n
=
(as a t o p o l o c i a l
submartingale,
D
lim ~n
It can be seen from an e x a m p l e that the c o n d i t i o n
vector
to
It(F)
a.e.
g i v e n by B e n y a m i n i
on the B a n a c h
lattice)
then
lattice
and Ghoussoub
c a n n o t be relaxed.
[20]
190
We shall now see that in the p r e v i o u s results ~ - b o u n d e d
submartingales
can be r e p l a c e d by s u b m a r t i n g a l e s for w h i c h the p o s i t i v e part exists and is ~ - b o u n d e d .
A set f u n c t i o n process ~ satisfies the Doob c o n d i t i o n if its p o s i t i v e + part ~ exists and is ~ - b o u n d e d , and it has a K r i c k e b e r ~ d e c o m p o s i t i o n if it is the d i f f e r e n c e of a n ~ - b o u n d e d p o s i t i v e supermartingale.
For example,
in a Banach lattice w i t h p r o p e r t y
p o s i t i v e m a r t i n g a l e and a every n e g a t i v e s u b m a r t i n g a l e and,
(P), every ~ - b o u n d e d
submartingale
satisfies the Doob c o n d i t i o n and has a K r i c k e b e r g d e c o m p o s i t i o n . For general submartingales,
the Doob c o n d i t i o n and the e x i s t e n c e of a
K r i c k e b e r g d e c o m p o s i t i o n are r e l a t e d as follows:
4.3.20.
Theorem.
Suppose If
~
~
is a KB-space.
is a submartingale,
(a)
then the f o l l o w i n g are equivalent:
~
satisfies the Doob condition.
(b)
~
has a K r i c k e b e r g d e c o m p o s i t i o n .
(c)
~
is m a j o r i z e d by an ~ - b o u n d e d
Moreover,
if
~
p o s i t i v e martingale.
satisfies the Doob condition,
positive martingale majorizing +
~
then the smallest ~ - b o ~ n d e d
is given by the m a r t i n g a l e of the Riesz
d e c o m p o s i t i o n of
+ Proof.
Suppose first that
is an ~ - b o u n d e d
satisfies the Doob condition. + p o s i t i v e submartingale. By T h e o r e m 4.3.4, ~
d i f f e r e n c e of an ~ - b o u n d e d potential
~ ~
hence
~
~
positive martingale
<
~
+
=
~-~
<
~
and a T - b o u n d e d Doob
,
is the d i f f e r e n c e of the ~ - b o u n d e d
Therefore,
(a) implies
(b).
Obviously,
(b) implies
(c).
if
is the
. Thus we have
the p o s i t i v e s u p e r m a r t i n g a l e
Finally,
~
Then
positive martingale
~
and
positive martingale
~ ,
~ - ~ .
~
is m a j o r i z e d by an ~ - b o u n d e d
<
~
then we have + 0
hence
~
=
~v0
<
~
,
satisfies the Doob condition.
The final a s s e r t i o n is then obvious.
a
191 The K r i c k e b e r g d e c o m p o s i t i o n leads to the f o l l o w i n g c h a r a c t e r i z a t i o n of cone a b s o l u t e l y summing o p e r a t o r s and AL-spaces:
4.3.21. Suppose
Theorem. ~
is a KB-space,
a b o u n d e d linear operator.
~
is a B a n a c h space, and
S :~
> ~
is
Then the f o l l o w i n g are equivalent:
(a)
S
is cone a b s o l u t e l y summing.
(b)
S
maps the s u b m a r t i n g a l e s s a t i s f y i n g the Doob c o n d i t i o n in
into the T - b o u n d e d set f u n c t i o n p r o c e s s e s in
Proof.
A p p l y T h e o r e m 4.3.20 and T h e o r e m 4.3.7.
4.3.22.
Corollary.
~
.
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
to an
(as a t o p o l o g i c a l vector lattice)
AL-spaCe.
(b) (c)
E v e r y s u b m a r t i n g a l e s a t i s f y i n g the Doob c o n d i t i o n is T-bounded. E v e r y s u b m a r t i n g a l e s a t i s f y i n g the Doob c o n d i t i o n is a u n i f o r m amart.
Proof.
By T h e o r e m 4.3.21 and C o r o l l a r y 4.3.9,
equivalent.
If
~
is a n A L - s p a c e ,
(a) and
(b) are
then e v e r y s u b m a r t i n g a l e s a t i s f y i n g
the Doob c o n d i t i o n i s ~ - b o u n d e d ,
by
(b), h e n c e it is a u n i f o r m amart,
by C o r o l l a r y 4.3.18. Conversely,
if
~
is not an AL-space,
then there
exists a n e g a t i v e s u b m a r t i n g a l e w h i c h is not a u n i f o r m amart, by C o r o l l a r y 4.3.14.
4.3.23. Suppose If
~
a
Corollary~ ~
is i s o m o r p h i c
(as a t o p o l o g i c a l v e c t o r lattice)
is a s u b m a r t i n g a l e s a t i s f y i n g the Doob condition,
lim DnB n
=
D
lim ~n
to
11 (r)
then
a.e.
We c o n c l u d e this s e c t i o n on s u b m ~ r t i n g a l e s w i t h a brief d i s c u s s i o n of quasimartingales.
4.3.24.
Theorem.
The f o l l o w i n g are equivalent: (a)
~
is i s o m o r p h i c
(as a t o p o l o g i c a l v e c t o r lattice)
to an
AL-space. (b)
Every ~ - b o u n d e d
s u b m a r t i n g a l e is a q u a s i m a r t i n g a l e .
(c)
Every ~ - b o u n d e d
p o s i t i v e s u b m a r t i n g a l e is a q u a s i m a r t i n g a l e .
192
Proof.
Suppose
submartingale
~
0 hence,
~
first
that
~
R n B n + 1 - Bn
for all
is an A L - s p a c e .
. Then we have,
m E~
for all
n 6~
Consider
an ~ - b o u n d e d
,
'
,
m
m
I
II ~n - Rn~n+1
II (n)
=
II ~n+1(n) - ~ n ( ~ )
Z
n=l
II
n=l
II ~m+l (~) - ~'I (s) II 2 li~l~ which
means
that
The converse
4.3.25. ~
exists
absolutely
a summable
summable.
standard
process
~
f o r all
Let
n E~
means
Example
4.3.25.
and
~
~n (Bn,k) n E~
and
decomposition
potential
n on
basis
[0,1)
,
k 6 K(n)
n E~
}
which
is n o t
, define
a set f u n c t i o n
. Then
_~
II (~)
fails
~- ~
amart.
=
X k E K(n)
=
II X n + 1 II
that ~
~
II ~ n + 1 ( B n , k )
- ~ n ( B n , k ) II
,
smallest
martingale
majorizing
by
, Clearly,
, and
is a u n i f o r m
the
given
2 -n x
k E K(n), of
positive
to b e a q u a s i m a r t i n g a l e .
process
:=
is a n H - b o u n d e d
n El~ , w e h a v e
let us r e m a r k
set f u n c t i o n
for all
{ x n£~+
sequence := X x
n Z x. j:l 3
For all
that
In a d d i t i o n ,
uniform
subsequent
2-n
:=
II ~ n - Rn~n+l
Riesz
the
by l e t t i n g
submartingale.
is the
x
stochastic
~ n (Bn ,k )
which
from
is n o t an A L - s p a c e .
there
On the
is a q u a s i m a r t i n g a l e . seen
Example.
Suppose Then
~
c a n be
,
~
is the m a r t i n g a l e
it is e a s y
potential,
to see t h a t
which
means
of the
the D o o b
that
~
is a
4.4.
U n i f o r m and
In this
section,
of p o s i t i v e
summing
p o t e n t i a l s .
9
of the c l a s s e s
u n i f o r m amarts.
operators
of all
We also
and A L - s p a c e s
in terms
strong p o t e n t i a l s .
u n i f o r m potentials.
Theorem.
Suppose
~
has p r o p e r t y
Then the c l a s s norm
stron
and of all H - b o u n d e d
cone a b s o l u t e l y
Let us first c o n s i d e r
4.4.1.
t s
we study the order p r o p e r t i e s
uniform potentials characterize
amar
p o s i t i v e
H.
func t i o n
of all u n i f o r m p o t e n t i a l s
IIT , and it is an ideal
is a B a n a c h
in the v e c t o r
lattice
lattice
for the
of all
set
processes.
Proof. Banach
(P).
By T h e o r e m
3.5.2,
space for the n o r m
the class II. 11T . If
of all u n i f o r m p o t e n t i a l s ~
is a
is a u n i f o r m potential,
then
we h a v e
lim
since
10 I ~
~i. tI(~)
is a u n i f o r m
H (~)
=
lim
is a lattice
potential.
II ~r H (n)
norm,
by T h e o r e m
The r e m a i n i n g
For the class of all ~ - b o u n d e d
=
uniform
assertion
amarts,
0
4.1.3.
Therefore,
~I
is obvious.
D
we have a less g e n e r a l
result:
4.4.2.
Theorem.
Suppose
~
is an AL-space.
Then the class of all H - b o u n d e d the n o r m
It. II T
Proof.
By T h e o r e m
is a B a n a c h amart
~
uniform amarts
is a B a n a c h
lattice
for
" 3.5.7,
the class of all H - b o u n d e d
space for the n o r m
. By the Riesz
II. itT
decomposition,
• Consider ~
u n i f o r m amarts
an H - b o u n d e d
uniform
is the sum of an H - b o u n d e d
martingale ~ and a u n i f o r m p o t e n t i a l ~ . Then the set f u n c t i o n .+ process ~ is an ~ - b o u n d e d p o s i t i v e submartingale. By C o r o l l a r y 4.3.18, .+ is a u n i f o r m amart, h e n c e it is the sum of an ~ - b o u n d e d m a r t i n g a l e and a u n i f o r m p o t e n t i a l
~
. Thus we have
194
t~ ÷ - ~ i
~
By T h e o r e m 4.4.1,
t~+ - ~ + i
{~i
u n i f o r m potential, hence + set f u n c t i o n process ~
+ ~
<
t~-~t
+ ~
=
t~i +
is a u n i f o r m p o t e n t i a l and thus + ~ -~ is a u n i f o r m potential. is
an ~-bounded
uniform
amart,
{~+-~l
is a
Therefore,
the
from
the
which
a s s e r t i o n follows.
D
The following c h a r a c t e r i z a t i o n of cone a b s o l u t e l y summing o p e r a t o r s and A L - s p a c e s in terms of p o s i t i v e strong p o t e n t i a l s is similar to T h e o r e m 3.4.5 and C o r o l l a r y 3.4.6, and to T h e o r e m 3.5.8 and C o r o l l a r y 3.5.9:
4.4.3. Suppose
Theorem. ~
is a Banach lattice,
~
is a Banach space, and
S :
is a b o u n d e d linear operator. Then the following are equivalent: (a) S is cone a b s o l u t e l y summing. (b)
S
maps the (~-bounded)
p o s i t i v e strong p o t e n t i a l s in
~
into
~
into
~
is a
the T - b o u n d e d set function p r o c e s s e s in
(c)
S
maps the (~-bounded)
p o s i t i v e strong p o t e n t i a l s in
the u n i f o r m p o t e n t i a l s in Proof.
Suppose first that
S
p o s i t i v e strong p o t e n t i a l in
0
<
lim
II S~T II(n)
by T h e o r e m 4.1.5, hence Conversely, an ~ - b o u n d e d
if
S
~
S~
~
.
is cone a b s o l u t e l y summing.
<
II S II1 lim
II ~T(~)
<
II S II1 lim
III ~T III(n)
II =
0
,
is a u n i f o r m p o t e n t i a l and thus T-bounded.
is not cone a b s o l u t e l y s u m m i n g , then there exists
p o s i t i v e strong p o t e n t i a l
~
in
~
T - b o u n d e d and thus cannot be a u n i f o r m potential, 4.4.4.
If
, then we have
such that
S~
is not
by Example 4.2.2.
Corollary.
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
(as a t o p o l o g i c a l v e c t o r lattice)
to an
AL-space. (b)
Every (~-bounded)
p o s i t i v e strong p o t e n t i a l
(c)
Every (~-bounded)
p o s i t i v e strong p o t e n t i a l is a u n i f o r m
is T-bounded.
potential.
F r o m the u n i f o r m p o t e n t i a l c o n v e r g e n c e t h e o r e m we thus obtain:
[]
195
4.4.5. Suppose If
~
Corollary. ~
is isomorphic
is a p o s i t i v e strong potential,
lim DnB n
=
For Doob potentials,
4.4.6. Suppose If
~
(as a t o p o l o g i c a l v e c t o r lattice)
0
to
It(F)
to
II(F)
then
a.e.
this yields:
Corollary. ~
is isomorphic
(as a t o p o l o g i c a l v e c t o r lattice)
is a Doob potential,
lim DnB n
=
0
then
a.e.
The p r e v i o u s result is also c o n t a i n e d in C o r o l l a r y 4.3.19. Again, can be seen from an example due to B e n y a m i n i and G h o u s s o u b the c o n d i t i o n on the B a n a c h lattice c a n n o t be relaxed.
it
[20] that
4.5.
Weak and
amarts p o s i t i v e
weak
p o t e n t i a l s .
Similar to the e x t e n s i o n of the strong c o n v e r g e n c e t h e o r e m for strong p o t e n t i a l s from finite d i m e n s i o n a l B a n a c h spaces to of p o s i t i v e strong potentials,
If(F)
in the case
there exists an e x t e n s i o n of the weak
c o n v e r g e n c e t h e o r e m for u n i f o r m weak p o t e n t i a l s in the case of p o s i t i v e u n i f o r m weak potentials.
The w e a k c o n v e r g e n c e t h e o r e m for p o s i t i v e
u n i f o r m w e a k p o t e n t i a l s is the main result of this section, and it yields weak c o n v e r g e n c e theorems for Doob p o t e n t i a l s and submartingales.
The f o l l o w i n g result is the B a n a c h lattice v e r s i o n of T h e o r e m 3.6.1:
4.5.1.
Theorem.
The f o l l o w i n g are equivalent: (a)
~
(b)
Every weak amart is a u n i f o r m weak amart.
is a KB-space.
Proof.
A B a n a c h lattice is a K B - s p a c e if and only if it is w e a k l y
s e q u e n t i a l l y complete. N o w the a s s e r t i o n follows from T h e o r e m 3.6.1.
a
We thus obtain the f o l l o w i n g w e a k amart c o n v e r g e n c e theorem:
4.5.2. Suppose If
~
Corollary. ~
has the R a d o n - N i k o d y m p r o p e r t y and a separable dual.
i s a T - b o u n d e d weak amart, then
w - l i m Dn~ n Proof.
=
D
w - l i m ~n
a.e.
A B a n a c h lattice h a v i n g the R a d o n - N i k o d y m p r o p e r t y does not
c o n t a i n a Banach sublattice isomorphic to
co
and h e n c e is a KB-space.
Now the a s s e r t i o n follows from T h e o r e m 4.5.1 and C o r o l l a r y 3.6.8. The p r e v i o u s results suggest the i n t r o d u c t i o n of a type of p o t e n t i a l c o r r e s p o n d i n g to weak amarts:
A set f u n c t i o n process { U~(n)
I T 6 T }
~
is a w e a k p o t e n t i a l if the net
w e a k l y c o n v e r g e s to
For weak p o t e n t i a l s w h i c h are positive, T h e o r e m 4.5.1:
0 .
there is a partial a n a l o g u e to
a
197
4.5.3.
Theorem.
Suppose
~
is a KB-space.
Then every p o s i t i v e weak p o t e n t i a l
Proof.
is a u n i f o r m weak potential.
Consider a positive weak potential
is a u n i f o r m weak amart. equal to
Since
0 , w h i c h means that
~
~
~
. By T h e o r e m 4.5.1,
is positive,
its limit m e a s u r e is
is a u n i f o r m w e a k potential,
o
Thus we have, by the u n i f o r m w e a k p o t e n t i a l c o n v e r g e n c e theorem:
4.5.4.
Corollary.
Suppose If
~
~
has the R a d o n - N i k o d y m p r o p e r t y and a separable dual.
is a T - b o u n d e d p o s i t i v e w e a k potential,
w - l i m Dn~ n
=
0
then
a.e.
The p r e v i o u s result can be improved. To this end, let us recall the f o l l o w i n g definition:
If
~
is a Banach lattice,
o r t h o g o n a l system of By
~
then a set
S c ~+
is a t o p o l Q @ i c a l
if the ideal g e n e r a t e d by
S
is dense in
~
.
[109; P r o p o s i t i o n II.6.2], e v e r y s e p a r a b l e B a n a c h lattice has a
q u a s i - i n t e r i o r point and thus p o s s e s s e s a t o p o l o g i c a l o r t h o g o n a l system c o n s i s t i n g of one point only. Therefore, theorems,
in the s u b s e q u e n t c o n v e r g e n c e
the c o n d i t i o n s imposed on the B a n a c h lattice are less
r e s t r i c t i v e than those in C o r o l l a r y 4.5.4. We can now state the p o s i t i v e w e a k p o t e n t i a l c o n v e r g e n c e theorem:
4.5.5. Suppose
Theorem. ~
has the R a d o n - N i k o d y m p r o p e r t y and a c o u n t a b l e t o p o l o g i c a l
o r t h o g o n a l system in its dual. If
~
is a T - b o u n d e d p o s i t i v e weak potential,
w - l i m Dn~ n Proof.
Since
(Lemma 3.5.10)
su~
~
=
0
a.e.
is T-bounded,
it follows from the maximal i n e q u a l i t y
that there exists a null set
II (Dn~ n) (~) II
is finite for all
~ £ ~A
then
.
A EL
such that the value
198
Now consider of
~'
a countable
. For each
lim
~ 6 ~A. for each
{ Z H e' ^ p e ~ converges Thus,
to set
e' E ~ H c~
0
becomes
{ e~ 6 ~ I J E~ 3 Aj £ L such that
a null set
(Dnei~ n) (~) 3.1.11
e' E ~ :
, the net
I P E~
,
H c~
=
0
and Theorem
finite
2.5.13.
}
and
~ £~A~(
~
Aj)
, we may choose
such that the last expression
(e'Dn~ n) (~)
<_
l e ' - r H e' Ape31
II (Dn~n) (~) [l +
<
[e' - Z H e' ^Pe3]
lJ (Dn~ n) (co) [J + p Z
lim
small for all
(e'Dn~ n) (~)
e' 6 ~
and
=
n q~
p 6~
and
of the inequality
<
arbitrarily
for all
lim
system
e'
for each
a finite
=
, by Theorem
3
orthogonal
, there exists
(eiDn~n) (co)
holds for all Furthermore,
j £~
topological
(Z H e' Ape3) (Dn~n) (~)
sufficiently
(e3Dn~ n) (co) large.
This yields
0
~ E ~A~(
~
AS)
, from which the assertion
follows. From an example given by Ghoussoub that the existence
of a countable
dual is a necessary
condition
and Talagrand topological
in the positive
[77] it can be seen
orthogonal
system
weak potential
in the
convergence
theorem. 4.5.6. Suppose
Corollary. ~
orthogonal If
~
has the Radon-Nikodym
is a T-bounded w-lim Dn~ n
Proof. Theorem
property
and a countable
topological
system in its dual. Doob potential, =
0
a.e.
Every Doob potential 4.3.10,
hence
then
is a positive
it is a positive
strong potential,
weak potential.
by O
}
199
4.5.7. Suppose
Corollary. ~
orthogonal If
~
has the Radon-Nikodym property and a countable topological system in its dual.
is a T-bounded submartingale,
w-lim Dn~ n Proof.
=
D
w-lim ~n
then
a.e.
Apply Theorem 4.3.15.
By the above-mentioned example, the existence of a countable topological orthogonal
system in the dual is also a necessary condition in Corollary
4.5.6 and Corollary 4.5.7. Further applications of the positive weak potential convergence theorem will be given in Section 4.6.
4.6.
O r d e r
Order
amarts
f o r m the n a t u r a l
of the p a r t i a l Banach does
in o r d e r
class
of the B a n a c h amart
amarts
order about
is m a i n l y
lattice
of B a n a c h
In w h a t
generalization
every
say v e r y m u c h
form a vector
.
ordering
lattice,
not
a m a r t s
follows,
we
due
than
shall
lattice.
is a s t r o n g
the p r o p e r t i e s to the f a c t
and admit
lattices
of s u b m a r t i n g a l e s
a weak
strong
In a n o r d e r amart,
but
of o r d e r
that
amarts
continuous
this
amarts.
theorem
observation The
the T - b o u n d e d
convergence
in t e r m s
interest
order
amarts
in a l a r g e r
do.
write
o-lim
for the o r d e r
limit
A set f u n c t i o n is o r d e r
Similar their
process
to real
amarts,
is o r d e r
(a)
order
if the n e t
{ ~T(0)
I ~ 6 T }
amarts
m a y be c h a r a c t e r i z e d
in t e r m s
of
complete. process,
is a n o r d e r
(b) (c)
o-lim
I~T-R ~ I ( 0 ) = 0
.
exists
measure
= o-lim
There ~(n)
Proof.
exists = o-lim
Suppose
modification the e x i s t e n c e
measures
by L e m m a
4.1.1.
remaining
amarts
a vector
a vector ~r(A)
following
are equivalent:
~ £ a(F
~ 6 a(F
for all
measure
A6
~ E a(F
, ~)
such
that
, ~)
such
that
such
that
F , ~)
~T(n)
first
that
limit
~T- R ~ As
measure
holds
a vector
of the p r o o f of the
the
amart.
exists
~(A) (d)
then
There
There
Order
amart
measure:
is a set f u n c t i o n
vector
is an o r d e r
n e t or s e q u e n c e .
Theorem.
Suppose
The
~
convergent
convergent.
limit
4.6.1 .
If
of an o r d e r
in the
implications
~
given
is
an order
in the
measure is o r d e r real are
~
and also
bounded
case,
amart.
real c a s e
shows
a n d thus
we o b t a i n
A straightforward
(Theorem that
2.5.4)
possesses
o-lim
a modulus,
I~T-Rr~I(Q)
obvious,
m a y a l s o be c h a r a c t e r i z e d
yields
e a c h of the
= 0 o
by a difference
property:
201
4.6.2.
Corollary.
Suppose If
~
~
is o r d e r complete.
is a set function process,
then the f o l l o w i n g are equivalent:
(a)
~
is an order amart.
(b)
o-lim suPT(T ) I~r-R ~al(~)
Proof.
Suppose first that
~
= 0
is an order amart w i t h limit m e a s u r e
. By T h e o r e m 4.6.1, there exists a net d i r e c t e d d o w n w a r d and d e c r e a s e s to
I~ z - RZ~I(~) holds for all downward,
~
~ qT
Consider
I ~ £T
}
w h i c h is
e
• E T . Since the net
and
~ q T ,
5
{ e T E~+
I ~ £ T }
is d i r e c t e d
e
v E T(K) z ET(K)
and
I(~ -Rx~a)(C)I hence,
[ eT 6~+
such that
we even have
I~v - Rv~l(n) for all
0
for all
A E Fz
~ and
aET(T)
lU - R x ~ I ( c ) B E FT(A)
. Then we have, for all
+ IRa~-~aI(C)
C E FT
,
,
I ~ - R ~l(n) + IRa~-~aI(Q) Therefore,
I~T-Rr~ol
exists,
is order complete,
since
and the
previous inequality yields
I~ z - RT~aI(~)
~
2 e
Thus we have
suPT(T) for all
T E T(M)
[~T- RT~c[(Q)
~
2 e
. The c o n v e r s e i m p l i c a t i o n is obvious.
In an order c o n t i n u o u s B a n a c h lattice, every order amart c l e a r l y is a strong amart. The r e l a t i o n b e t w e e n order amarts and u n i f o r m amarts is c l a r i f i e d by the f o l l o w i n g c h a r a c t e r i z a t i o n of cone a b s o l u t e l y summing o p e r a t o r s and AL-spaces:
202 4.6.3.
Theorem.
Suppose and
~
is an order c o n t i n u o u s B a n a c h lattice,
S :~
) ~
~
is a B a n a c h space
is a b o u n d e d linear operator. Then the f o l l o w i n g are
equivalent:
(a)
S
is cone a b s o l u t e l y summing.
(b)
S
maps the ~ - b o u n d e d
order a m a r t s in
set f u n c t i o n p r o c e s s e s in
(c)
S
maps the (~-bounded)
a m a r t s in
Proof.
~
f' 6 ~
.
o r d e r amarts in
~
into the u n i f o r m
S
is cone a b s o l u t e l y summing. Then there
such that
II Sx II holds for all
into the T - b o u n d e d
.
Suppose first that
exists
~
~
<
x E~
f' (Ixl)
. C o n s i d e r an order a m a r t
proof of T h e o r e m 4.1.9,
~
in
~
. As in the
it can be seen that
II S~ T - R T S ~ o ll(n)
=
II S ( ~ T - R T ~ o) If(Q)
f'(I~T-RT~oI(n))
f'(SUPT(T) holds for all properties, Therefore, Clearly,
T 6 T
and
this means that (a) implies
(c) implies
(b) implies 4.6.4.
o E T(T) S~
IBT-RTBoI(Q))
. By the r e s p e c t i v e d i f f e r e n c e is a u n i f o r m amart.
(c).
(b), and it can be seen from E x a m p l e 4.2.2 that Q
(a), since every Doob p o t e n t i a l is an o r d e r amart.
Corollary.
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
(as a t o p o l o g i c a l v e c t o r lattice)
to an
AL-space. (b)
Every ~ - b o u n d e d
(c)
Every (~-bounded)
4.6.5. Suppose If
~
order a m a r t is T-bounded. order a m a r t is a u n i f o r m amart.
Corollary. ~
is isomorphic
is an ~ - b o u n d e d
lim Dn~ n
=
(as a t o p o l o g i c a l vector lattice)
order amart,
D
l~m ~n
then
a.e.
to
II(F)
203 A set f u n c t i o n p r o c e s s all
T 6 T
Clearly,
~
is an o r d e r p o t e n t i a I if
and if the net
{ I~TI(~)
I T £ T }
I~ I
exists for
order c o n v e r g e s to
every Doob p o t e n t i a l is an order potential,
0
and we also have
the f o l l o w i n g result:
4.6.6.
Theorem.
Suppose
~
has p r o p e r t y
(P).
Then the class of all T - b o u n d e d order p o t e n t i a l s is a vector lattice, and it is an ideal in the v e c t o r lattice of all set f u n c t i o n processes.
Proof.
Since
~
has p r o p e r t y
(P), it follows from T h e o r e m 4.1.3
that the sum of two order p o t e n t i a l s a set f u n c t i o n process function process
I~I
~
is an order potential. Moreover,
is an order p o t e n t i a l
if and only if the set
is an order potential,
a
It is c l e a r that a n a l o g o u s results h o l d for the c l a s s of all o r d e r p o t e n t i a l s and for the class of all P - b o u n d e d
order potentials,
but the
case of T - b o u n d e d order p o t e n t i a l s is the i m p o r t a n t one. In o r d e r to prove an a n a l o g o u s result for the class of all T - b o u n d e d o r d e r amarts, let us first prove the Riesz d e c o m p o s i t i o n f o r ~ - b o u n d e d
4.6.7.
order amarts:
Theorem.
Suppose
~
is o r d e r continuous.
T h e n every P - b o u n d e d and an P - b o u n d e d
order amart is the sum of an P - b o u n d e d m a r t i n g a l e
order potential.
The d e c o m p o s i t i o n is unique. If the order amart is T-bounded,
Proof. Since
~
then so is the order potential.
C o n s i d e r an P - b o u n d e d
o r d e r amart
is order continuous,
~
d e c o m p o s i t i o n for P - b o u n d e d
~
w i t h limit m e a s u r e
is a strong amart. By the Riesz
strong amarts,
the limit m e a s u r e
b o u n d e d v a r i a t i o n and g e n e r a t e s an P - b o u n d e d m a r t i n g a l e
:=
{ Rn~
I n 6P
}
.
Then the set f u n c t i o n p r o c e s s
is an order potential, by T h e o r e m 4.6.1. The r e m a i n i n g a s s e r t i o n s are obvious.
~
has
~ .
204
We can now d e s c r i b e the s t r u c t u r e of the class of all T - b o u n d e d order amarts:
4.6.8.
Theorem.
Suppose
~
is a KB-space.
Then the class of all T - b o u n d e d order amarts is a v e c t o r lattice.
Proof.
C o n s i d e r a T - b o u n d e d order amart
decomposition,
~
T - b o u n d e d order p o t e n t i a l an Y - b o u n d e d
~
~
~
~
and a
By T h e o r e m 4.3.4,
positive martingale
~
~+
~+
is
is the
and a T - b o u n d e d Doob
. Thus we have
I~ + - ~ I Since + -~
. By the Riesz
. Then the set function process
p o s i t i v e submartingale.
d i f f e r e n c e of an ~ - b o u n d e d potential
~
is the sum of an Y - b o u n d e d m a r t i n g a l e
and
~
I~ + - ~ + I
+ ~
<
t~-~l
+ ~
=
I~I + ~
.
~
are T - b o u n d e d order potentials, the same is true for + . Therefore, the set f u n c t i o n process ~ is a T - b o u n d e d order
amart, f r o m w h i c h the a s s e r t i o n follows,
u
It is clear that an a n a l o g o u s result holds for the class of all Y - b o u n d e d o r d e r amarts.
The n e x t result is the o r d e r a m a r t c o n v e r g e n c e theorem:
4.6.9. Suppose
Corollary. ~
has the R a d o n - N i k o d y m p r o p e r t y and a c o u n t a b l e t o p o l o g i c a l
o r t h o g o n a l s y s t e m in its dual. If
~
is a T - b o u n d e d order amart,
w - l i m Dn~ n Proof.
=
D
lim ~n
then
a.e.
By the Riesz d e c o m p o s i t i o n ,
martingale
~
~
is the sum of a n Y - b o u n d e d
and a T - b o u n d e d o r d e r p o t e n t i a l
~ . Then we have, by
the v e c t o r lattice p r o p e r t y of the T - b o u n d e d order potentials, _~
=
_~ + _~+ - _~-
The m a r t i n g a l e c o n v e r g e n c e t h e o r e m y i e l d s
lim D n 5 n
=
D
lim ~n
a.e.,
205
since
~
and
~
have the same limit measure. Moreover,
order continuous,
the T - b o u n d e d p o s i t i v e order p o t e n t i a l s
since
~
~+
and
is ~-
are strong potentials, hence weak potentials. N o w the p o s i t i v e weak potentialconvergence
t h e o r e m yields
w - l i m Dn ~+n
=
0
a.e.
w - l i m Dn~ ~
=
0
a.e.
=
D~ lim ~n
and
Thus we have
w - l i m Dn~ n
a.e.,
as was to be shown.
We c o n c l u d e this section on order amarts with a brief d i s c u s s i o n of hypomartingales.
A set function process
is a h y p o m a r t i n g a l e if it is the d i f f e r e n c e of a
m a r t i n g a l e and a p o s i t i v e order potential. is an order amart. Also, a hypomartingale,
in a KB-space,
and every ~ - b o u n d e d
Clearly,
every h y p o m a r t i n g a l e
every ~ - b o u n d e d
s u b m a r t i n g a l e is
order amart is the d i f f e r e n c e of
two h y p o m a r t i n g a l e s .
we shall now see that the p r o p e r t i e s of p o s i t i v e h y p o m a r t i n g a l e s are similar to those of p o s i t i v e submartingales.
The first result is the Riesz d e c o m p o s i t i o n for ~ - b o u n d e d
positive
hypomartingales:
4.6.10. Suppose
Theorem. ~
is order continuous.
Then every H - b o u n d e d an ~ - b o u n d e d
p o s i t i v e h y p o m a r t i n g a l e is the d i f f e r e n c e of
p o s i t i v e m a r t i n g a l e and a T - b o u n d e d p o s i t i v e order potential.
The d e c o m p o s i t i o n is unique.
Proof.
C o n s i d e r an H - b o u n d e d
exists a m a r t i n g a l e
0
F i r s t of all,
<
~
~
~
=
positive hypomartingale
and a p o s i t i v e order p o t e n t i a l
_~-~
<
~ . Then there ~
satisfying
~
is c l e a r l y positive. Moreover,
if
~
is c o n s i d e r e d
206
as an order amart, then it is clear from the Riesz decomposition for order amarts that inequality, 4.6.11. Suppose If
~
~
~
is ~-bounded,
is T-bounded,
hence T-bounded.
hence
~ = ~- ~
By the above
is T-bounded.
Cgrollary. ~
is order continuous.
is a positive hypomartingale,
(a)
~
is ~-bounded.
(b)
~
is T-bounded.
The main result on hypomartingales
then the following are equivalent:
is the positive hypomartingale
convergence theorem: Theorem.
4.6.12.
Suppose If
~
~
has the Radon-Nikodym property.
is an ~ - h o u n d e d positive hypomartingale,
lim Dn~ n
=
D
lim ~n
then
a.e.
The proof is the same as the proof of the positive submartingale convergence theorem.
4.7.
R e m a r k s .
A vector measure two o r t h o g o n a l lattice,
has a Jordan
positive
a vector measure
it is o r d e r bounded, variation
measures.
has a J o r d a n
by L e m m a
4.1.1.
in an o r d e r c o m p l e t e
decomposition AL-space, Schmidt
decomposition
vector
by Faires [119]
and M o r r i s o n
submartingales
[69]
were
in his p a p e r on the c o n v e r g e n c e strong
convergence
theorem
by Szulga and W o y c z y n s k i
r e s u l t can be o b t a i n e d theorem which
of the K a d e c - K l e e Ghoussoub,
convergence
[26,28],
Bru and H e i n i c h
positive
hypomartingale
submartingale
Since the p r o p e r t i e s
theorem;
amart
4.3.24;
is a strong amart,
Theorem
4.3.16.
submartingales, Doob p o t e n t i a l
Thus,
submartingales
which
convergence
theorem,
Radon-Nikodym
amart,
question
uniform
of T h e o r e m
these B a n a c h
these Banach
l a tt i c e s
property.
in w h i c h B a n a c h
form a B a n a c h
4.4.2.
Since every generalize
since every u n i f o r m
have
by an H - b o u n d e d
Apparently,
to
lattices
to be K B - s p a c e s ,
for H - b o u n d e d
lattices
these q u e s t i o n s
every
is a
submartingale
to k n o w w h e t h e r
the B a n a c h
by
positive
lattices
martingale
in v i e w of the p o s i t i v e
includes
similar
to ask
ask in w h i c h B a n a c h
it w o u l d be i n t e r e s t i n g lattices
the p o s i t i v e
[74].
amarts
by the Riesz d e c o m p o s i t i o n
is m a j o r i z e d
extends
in
is a u n i f o r m amart.
on the o t h e r hand,
Furthermore,
this class of B a n a c h
theorem
Davis,
of the p o s i t i v e
are very
submartingale
one may e q u i v a l e n t l y
u n i f o r m potential.
proof
see also G h o u s s o u b
the H - b o u n d e d
is a u n i f o r m
by T h e o r e m
lattices,
and stated the
theorem which
as can be seen from the proof
AL-spaces,
Banach
convergence renorming
hypomartingales
it is a natural
lattice,
a lattice
introduced
positive
quasimartingale
submartingale
their
paper q u o t e d
latti c e s
lattice,
submartingale,
In an u n p u b l i s h e d
those of u n i f o r m amarts,
In such a Banach
The first
submartingales
the s u b m a r t i n g a l e s
[46] gave a d i f f e r e n t
of p o s i t i v e
every ~-bounded
[108]
theorem.
convergence
convergence
by Scalora
seen to be the sum of
positive
[82]. U s i n g
and by
(P).
martingales.
Since
for o r d e r c o n t i n u o u s
and L i n d e n s t r a u s s
submartingale
[127].
are easily
from the p o s i t i v e
type
property
vector-valued
and an ~ - b o u n d e d
is due to H e i n i c h
of a Jordan
in the case of an
lattice h a v i n g
of v e c t o r - v a l u e d
by Szulga and W o y c z y n s k i
if
of b o u n d e d
in the case of a KB-space,
for c e r t a i n
considered
martingale
[48]
of
vector
if and only
the e x i s t e n c e
already mentioned
was p r o v e n
an ~ - b o u n d e d
lattice,
and Faires
in the case of a B a n a c h
Vector-valued
decomposition
For v e c t o r m e a s u r e s
Banach
was p r o v e n by Diestel
if it is the d i f f e r e n c e
In an order c o m p l e t e
or not
having
have not been
the
2O8
a n s w e r e d in the literature. theM-bounded
C o r r e s p o n d i n g p r o b l e m s arise, of course,
p o s i t i v e h y p o m a r t i n g a l e s or, e q u i v a l e n t l y ,
for
for the p o s i t i v e
order p o t e n t i a l s w h i c h are m a j o r i z e d by an M - b o u n d e d martingale.
Order amarts were i n t r o d u c e d by H e i n i c h
[81] who studied these p r o c e s s e s
in an order c o m p l e t e v e c t o r lattice. G h o u s s o u b
[72] studied o r d e r am~rts
in a B a n a c h lattice and p r o v e d the order a m a r t c o n v e r g e n c e theorem. By the Riesz decomposition,
the proof of the order amart c o n v e r g e n c e
t h e o r e m can be reduced to the order potential c o n v e r g e n c e theorem:
4.7.1.
CQrollary.
Suppose
~
has the R a d o n - N i k o d y m p r o p e r t y and a c o u n t a b l e t o p o l o g i c a l
o r t h o g o n a l system in its dual. If
~
is a T - b o u n d e d order potential,
w - l i m Dn~ n
=
0
then
a.e.
By the v e c t o r lattice p r o p e r t y of the class of all T - b o u n d e d o r d e r potentials,
the o r d e r potential c o n v e r g e n c e theorem is an immediate
c o n s e q u e n c e of the p o s i t i v e w e a k p o t e n t i a l c o n v e r g e n c e theorem. We have p r e f e r r e d to prove this latter r e s u l t since it yields the weak c o n v e r g e n c e theorems for Doob p o t e n t i a l s and s u b m a r t i n g a l e s w i t h o u t appeal to order p o t e n t i a l s and order amarts.
For p o t e n t i a l s in a B a n a c h lattice,
the role of the R a d o n - N i k o d y m
p r o p e r t y is e s s e n t i a l l y the same as for p o t e n t i a l s Thus,
in a B a n a c h space.
in the case w h e r e the set f u n c t i o n p r o c e s s e s are o b t a i n e d by
integrating stochastic processes, weak c o n v e r g e n c e obtains for T - b o u n d e d positive uniform weak potentials
in an a r b i t r a r y B a n a c h lattice, hence
for T - b o u n d e d Doob p o t e n t i a l s and T - b o u n d e d p o s i t i v e order p o t e n t i a l s in an order c o n t i n u o u s Banach lattice, and for T - b o u n d e d p o s i t i v e weak p o t e n t i a l s and a r b i t r a r y T - b o u n d e d order p o t e n t i a l s in a KB-space; also,
strong c o n v e r g e n c e obtains for p o s i t i v e strong potential~,
potentials,
and order p o t e n t i a l s in an AL-space.
Furthermore,
Doob
it is
p o s s i b l e to prove c o n v e r g e n c e theorems for order b o u n d e d stochastic p r o c e s s e s in an order c o n t i n u o u s B a n a c h lattice; and L i n d e n s t r a u s s
[46], and G h o u s s o u b
G h o u s s o u b and T a l a g r a n d
see Davis, Ghoussoub,
[71,74]. Let us a l s o remark that
[76] p r o v e d the order c o n v e r g e n c e of order
b o u n d e d order amarts in an order c o n t i n u o u s B a n a c h lattice; Ghoussoub
see also
[74].
The first c h a r a c t e r i z a t i o n s of cone a b s o l u t e l y summing o p e r a t o r s and
209
AL-spaces Szulga
in terms of s u b m a r t i n g a l e s
[124,125,126]
were o b t a i n e d to the v a r i o u s and A L - s p a c e s following
and G h o u s s o u b
by Bru and H e i n i c h
given
[72];
[25,28]
characterizations already
and o r d e r a m a r t s w e r e given by further
results
and by E g g h e
of cone a b s o l u t e l y
in this chapter,
of this type
[67].
summing
let us also
In a d d i t i o n operators
include
the
results:
4.7.2.
Theorem.
Suppose
~
is a KB-space,
~
is a B a n a c h
a bounded
linear operator.
(a)
S
is cone
Then
(b)
S
maps the s u b m a r t i n g a l e s
(c)
S
Proof. implies
maps
set f u n c t i o n
By T h e o r e m
(a) implies
the n e g a t i v e
martingale
is
>
the Doob c o n d i t i o n
processes
satisfying
into the ~ - b o u n d e d
(a), c o n s i d e r
S :~
are equivalent:
satisfying
set f u n c t i o n
4.3.21,
and
summing.
the m a r t i n g a l e s
is the p o s i t i v e
4.7.3.
the f o l l o w i n g
absolutely
into the ~ - b o u n d e d
space,
in
~
the Doob c o n d i t i o n
processes
in
~
in
.
(b). In o r d e r to see that
martingale
constructed
~
:= -~
in Example
in
.
in
~
(c)
, where
4.3.8.
Corollary.
The f o l l o w i n g (a)
~
are equivalent: is isomorphic
(as a t o p o l o g i c a l
vector
lattice)
to an
AL-space. (b)
Every
(c)
Every martingale
The f o l l o w i n g
4.7.4.
results
satisfying
satisfying
concern
the Doob c o n d i t i o n
the Doob c o n d i t i o n
is ~ - b o u n d e d .
is ~ - b o u n d e d .
order potentials:
Theorem.
Suppose and
submartingale
is an o r d e r c o n t i n u o u s
S :~
> ~
is a b o u n d e d
Banach
lattice,
linear operator.
~
Then
is a B a n a c h the f o l l o w i n g
space, are
equivalent: (a)
S
is cone a b s o l u t e l y
(b)
S
maps
T-bounded
(c)
S
maps
set f u n c t i o n
proof
Apply of T h e o r e m
order p o t e n t i a l s processes
the ( ~ - b o u n d e d )
uniform potentials
Proof.
summing.
the (~-bounded)
Theorem 4.6.3.
4.4.3
in
in
~
order p o t e n t i a l s ~
in
into the
in
into the
.
.
and E x a m p l e
4.2.2,
or p r o c e e d
as in the []
210
4.7.5.
Cprollary.
The f o l l o w i n g are equivalent: (a)
~
is isomorphic
to an
(as a t o p o l o g i c a l vector lattice)
AL-space. (b)
Every (~-bounded)
order p o t e n t i a l is T-bounded.
(c)
Every (~-bounded)
order potential
4.7.6. Suppose If
~
is a u n i f o r m potential.
Corollary. ~
is isomorphic
(as a t o p o l o g i c a l vector lattice)
is an order potential,
lim Dn~ n
=
0
ll(r)
to
then
a.e.
We remark that in these c h a r a c t e r i z a t i o n s of cone a b s o l u t e l y summing o p e r a t o r s and A L - s p a c e s only a very limited number of set f u n c t i o n p r o c e s s e s is r e q u i r e d to p r o v i d e the n e c e s s a r y c o u n t e r e x a m p l e s .
It is w e l l - k n o w n that an A L - s p a c e has the R a d o n - N i k o d y m p r o p e r t y if and only if it is isomorphic
(as a t o p o l o g i c a l vector lattice)
to
II(F)
There exist also c h a r a c t e r i z a t i o n s of these Banach lattices in terms of s u b m a r t i n g a l e s and order amarts. theorems in
II(F)
These results are r e l a t e d to c o n v e r g e n c e
, and the c o u n t e r e x a m p l e s w h i c h are i n v o l v e d are
c l e a r l y p r o b a b i l i s t i c and not m e a s u r e t h e o r e t i c a l ones; and G h o u s s o u b
[20], Egghe
[67], G h o u s s o u b
see B e n y a m i n i
[72,74], and Szulga
[124,125].
It w o u l d c e r t a i n l y be i n t e r e s t i n g to d e v e l o p a p u r e l y v e c t o r lattice t h e o r e t i c a l theory of order amarts in an order c o m p l e t e v e c t o r lattice w i t h o u t topology.
For order amarts of order b o u n d e d v e c t o r measures,
this has b e e n done by H e i n i c h
[81]. The critical point, however,
is the
a p p l i c a t i o n of these results to order amarts of r a n d o m variables. In order to a v o i d the i n t r o d u c t i o n of a topology, Bru
[24] c o n s t r u c t e d
an order integral for r a n d o m v a r i a b l e s in a wide class of o r d e r c o m p l e t e v e c t o r lattices.
In an order c o n t i n u o u s B a n a c h lattice,
the v e c t o r
lattice of all order integrable r a n d o m v a r i a b l e s in the sense of Bru c o i n c i d e s with the class of all random v a r i a b l e s p o s s e s s i n g a Pettis integrable modulus, w h i c h were also c o n s i d e r e d by Bru and H e i n i c h [25,26,28].
5.
F u r t h e r
a s p e c t s
of
amar
t
theory
.
In the literal sense, amarts should a s y m p t o t i c a l l y a p p r o a c h martingales. For a real or v e c t o r - v a l u e d set f u n c t i o n process
~
to be an amart,
this r e q u i r e m e n t means that there should exist a m a r t i n g a l e that the net
{ ~T - ~T
I ~ £ T }
~
such
c o n v e r g e s to zero in some sense.
The e x i s t e n c e of such a Riesz d e c o m p o s i t i o n is, of course, e q u i v a l e n t to the e x i s t e n c e of a limit m e a s u r e R ~ ,
~
such that the r e s t r i c t i o n s
T E T , have b o u n d e d v a r i a t i o n and the net
c o n v e r g e s to zero. Actually,
{ ~ T - RT~
I • £ T }
it is the type of c o n v e r g e n c e to zero of
these nets w h i c h d e t e r m i n e s the p r o p e r t i e s of the amart
~ . As in the
case of u n i f o r m amarts,
the type of c o n v e r g e n c e to zero of the nets
{ ~T-~r
{ ~ T - RT~
I T £T
}
and
I T £T
}
c a n n o t in general be
e x p r e s s e d by a c o n v e r g e n c e p r o p e r t y of the net Also, as in the c a s e of u n i f o r m w e a k amarts, measure
5
and the m a r t i n g a l e
~
{ BT (~)
I T C T }
the e x i s t e n c e of the limit
c a n n o t in g e n e r a l be d e d u c e d from a
d i f f e r e n c e property.
Quite generally,
the e x i s t e n c e of a limit m e a s u r e a l s o plays an e s s e n t i a l
role in the d e f i n i t i o n of the s t o c h a s t i c p r o c e s s e s in a B a n a c h space h a v i n g the R a d o n - N i k o d y m p r o p e r t y for w h i c h B e l l o w and Egghe
[18,19]
e s t a b l i s h e d p o i n t w i s e i n e q u a l i t i e s of the F a t o u - C h a c o n - E d g a r type; also B e l l o w
see
[15]. The stochastic p r o c e s s e s c o n s i d e r e d by these a u t h o r s
are d e f i n e d by the f o l l o w i n g p r o p e r t y of the induced set f u n c t i o n process: There exists an i n c r e a s i n g sequence
{ Tn E T(n)
stopping times such that the v a l u e
su~
exists a v e c t o r m e a s u r e
such that
holds for all
A £ F
~
; here
on Y
F
I n 6~
II ~rn 11(~) Y-lim
}
of b o u n d e d
is finite, and there (~Tn - RTn~) (A) = 0
is a H a u s d o r f f locally c o n v e x t o p o l o g y
which is w e a k e r than the n o r m t o p o l o g y and for w h i c h the unit ball is
212
closed. Evident examples of such a t o p o l o g y are the n o r m topology, the weak topology,
and, in the case of a dual B a n a c h space, the weak*
topology. C o m b i n i n g the i n e q u a l i t i e s of the F a t o u - C h a c o n - E d g a r type o b t a i n e d for the stochastic p r o c e s s e s d e s c r i b e d above w i t h a suitable d i f f e r e n c e p r o p e r t y p r o v i d e s a general concept for proving c o n v e r g e n c e theorems. This way, B e l l o w and Egghe o b t a i n e d a v a r i e t y of c o n v e r g e n c e theorems,
including those for u n i f o r m amarts and weak sequential amarts.
For related results,
see Edgar
For stochastic processes,
[54] and Egghe
[64].
there are v a r i o u s d i f f e r e n c e p r o p e r t i e s w h i c h
are r e l a t e d to d i f f e r e n t types of convergence. For example, a stochastic process
X
is a u n i f o r m a m a r t if and only if
lim suPT(T ) I 9 [ IX-
ETX o Jl dP
=
0
holds. This d i f f e r e n c e p r o p e r t y for u n i f o r m amarts is c l e a r l y r e l a t e d to L1-convergence.
Similarly,
one may formulate d i f f e r e n c e p r o p e r t i e s
for stochastic p r o c e s s e s w h i c h are r e l a t e d to c o n v e r g e n c e in p r o b a b i l i t y or to a.e. convergence.
A stochastic p r o c e s s
X
This leads to the f o l l o w i n g definitions:
is a p r a m a r t
(or amart in probability)
=
lim suPT(r ) P(IJ X T - E T x o Jl > e] holds for all
E 6 (0,~)
lim s U b ( n )
II
0
, and it is a mil
(or m a r t i n g a l e in the limit)
X n - EnX m ~I
a.e.
=
0
P r a m a r t s were i n t r o d u c e d by M i l l e t and S u c h e s t o n had been i n t r o d u c e d earlier by Mucci
[94,95,96], w h i l e mils
[97,98,99]. Every u n i f o r m amart is
a pramart, and e v e r y p r a m a r t is a mil; moreover, fail to be a pramart,
if
there exist m i l s w h i c h
and there exist p r a m a r t s w h i c h fail to be a
u n i f o r m amart. For details,
see M i l l e t and S u c h e s t o n
[96].
It is i n t e r e s t i n g to note that no stopping times are n e e d e d in the d e f i n i t i o n of a mil. This is due to the fact that m i l s are d e f i n e d p o i n t w i s e and that the i d e n t i t y
(X r - ErXo) (co) holds for all
=
(Xn - EnXm) (0~)
~ 6 {~=n}n{o=m}
. However,
in the d e f i n i t i o n of a pramart.
stopping times are e s s e n t i a l
This will be clear from the f o l l o w i n g
if
213
remark:
A stochastic process
X
lim s U b ( n ) holds for all
is a ~ame w h i c h b e c o m e s fairer w i t h time if
P(II X n - EnX m II > e)
~ E (0,~)
i n t r o d u c e d by Blake a mil. Therefore,
=
0
. Games which b e c o m e fairer w i t h time were
[135]. Every game w h i c h b e c o m e s fairer w i t h time is
there exist games w h i c h b e c o m e fairer w i t h time w h i c h
fail to be a pramart.
In the real case, the fact that every a m a r t is a mil was first proven by Edgar and Sucheston o r i g i n a t e s from the Mucci
[61]; see also Blake
[21]. The interest in mils
(real) mil c o n v e r g e n c e t h e o r e m w h i c h is due to
[99] and g e n e r a l i z e s the a m a r t c o n v e r g e n c e theorem; a n o t h e r
c o n v e r g e n c e t h e o r e m for mils was given by Y a m a s a k i
[131]. U n l i k e amarts,
however, mils have u n s a t i s f a c t o r y stability properties. B e l l o w and D v o r e t z k y
It was shown by
[17] t h a t the class of all L 1 - b o u n d e d mils need not
form a vector lattice. Furthermore,
it was shown by Edgar and S u c h e s t o n
[61] that mils need not have a Riesz decomposition,
and that the optional
stopping t h e o r e m as well as the o p t i o n a l s a m p l i n g t h e o r e m may fail for mils. As to pramarts,
it seems to be u n k n o w n w h e t h e r or not the class of
all L l - b o u n d e d p r a m a r t s forms a v e c t o r lattice. However, M i l l e t and Sucheston Thus,
[96] p r o v e d that p r a m a r t s have the o p t i o n a l sampling property.
since p r a m a r t s g e n e r a l i z e amarts,
decomposition;
see Edgar and Sucheston
In the v e c t o r - v a l u e d case,
they need not possess a Riesz [61], or T h e o r e m 2.7.4.
it seems to be an open q u e s t i o n w h e t h e r or
not every L 1 - b o u n d e d mil in a Banach space h a v i n g the R a d o n - N i k o d y m p r o p e r t y c o n v e r g e s a.e. However,
e x t e n s i o n s of the u n i f o r m amart
c o n v e r g e n c e theorem were proven by M i l l e t and S u c h e s t o n of class
[95] for pramarts
(B), w h i c h is the c o n d i t i o n of T - b o u n d e d n e s s for s t o c h a s t i c
processes,
and by P e l i g r a d
lim s U b ( n )
I~
[103] for m i l s s a t i s f y i n g the c o n d i t i o n
II X n - E n X m II dP
=
0
Further c o n v e r g e n c e theorems for v e c t o r - v a l u e d p r a m a r t s and mils were o b t a i n e d by B e l l o w and Dvoretzky, Edgar
[54], Egghe
see B e l l o w and Egghe
[65], M i l l e t and S u c h e s t o n
[18,19], and by
[95], and Mucci
[98].
As a c o m m o n a b s t r a c t i o n of real p r a m a r t s and submartingales, M i l l e t and Sucheston
[96] also i n t r o d u c e d subpramarts.
Egghe
[67] and Slaby
[138]
214 studied
subpramarts
in a B a n a c h
real and v e c t o r - v a l u e d
lattice.
subpramarts,
There are also g e n e r a l i z a t i o n s
For a d e t a i l e d
see E g g h e
discussion
of
[68].
of amarts w h i c h
concern
the range of
these processes:
Amarts
in a F r ~ c h e t
nuclear Fr~chet
space w e r e
spaces
are c h a r a c t e r i z e d
to the c h a r a c t e r i z a t i o n Bellow
[7].
In
space,
as well
in a F r ~ c h e t
Multivalued The v a l u e s
space h a v i n g
amarts were
convex
embedding
convex [101]
sets).
Finally, respect
theorem
Earlier,
are
let us r e m a r k
that a m a r t s
set.
a rich l i t e r a t u r e
In r e c e n t years,
aspects
amarts
Dam and N g u y e n
Duy Tien
[120],
of
it follows in a
in the case of c l o s e d b o u n d e d
martingales
had b e e n
approach,
studied by N e v e u
see C o s t ~
have also b e e n g e n e r a l i z e d
interest
directed
we have c o n f i n e d integers,
the final part of this volume.
has b e e n d e v o t e d
set.
of a m a r t t h e o r y may be f o u n d
[44].
convex
From a refinement
[43].
with
ourselves
but there also
on a m a r t s w h i c h are indexed by d i f f e r e n t
increasing
are i n d e x e d b y a g e n e r a l
Fr~chet
sequential
as strong a m a r t s
to a m a r t s w h i c h are indexed by the p o s i t i v e exists
similar g i v e n by
to be c l o s e d b o u n d e d
theoretic
In these notes,
[62,63],
property.
space.
(with unit,
multivalued
spaces
in a n u c l e a r
for w e a k
[105], g i v e n by S c h m i d t
for the m e a s u r e
to the index
supposed
may be c o n s i d e r e d
cone of an A M - s p a c e
and others;
theorem
sets in a B a n a c h
In
and a strong c o n v e r g e n c e
studied by Bui Khoi
amarts
Banach
strong a m a r t s
the R a d o n - N i k o d y m
of these p r o c e s s e s
that m u l t i v a l u e d generating
decomposition
for c e r t a i n
[62,63,66].
in terms of amarts,
dimensional
as a w e a k c o n v e r g e n c e
sets or c o m p a c t R~dstr~m's
of finite
[66], a Riesz
t h e o r e m are o b t a i n e d
s t u d i e d by E g g h e
References
sets.
to a m a r t s w h i c h
to p a p e r s
in the b i b l i o g r a p h y
on these
on a m a r t s
in
A p p e n d i x
In this appendix, Banach
A Banach
A Banach
lattice
to
0
every d o w n w a r d
B a n a c h
~
is
~
l a t t i c e s
some d e f i n i t i o n s and further
and by L i n d e n s t r a u s s
(countabl~)
(countable)
lattice
decreasing
For proofs
[109]
every n o n - e m p t y
to
we recall
lattices.
by Sch a e f e r
on
and p r o p e r t i e s
details,
we refer
and T z a f r i r i
order complete
majorized
set
A c ~
directed
family
in
to
~
of specific
to the books
[91].
if
sup A
exists
for
.
is c o u n t a b l ~ o r d e r c o n t i n u o u s
is n o r m c o n v e r g e n t
.
if every
sequence
in
0 , and it is order c o n t i n u o u s
with
infimum
0
if
is n o r m c o n v e r g e n t
0 .
For a B a n a c h
lattice
(a)
~
, the f o l l o w i n g
is order
(b)
is o r d e r c o m p l e t e
are equivalent:
continuous. and e v e r y
continuous
linear f o r m on
~
is
o r d e r continuous.
(c) (d)
is c o u n t a b l y
order complete
and c o u n t a b l y
order continuous.
is c o u n t a b l y
order complete
and no B a n a c h
sublattice
is v e c t o r
lattice
isomorphic
(e)
Under
evaluation,
(f)
Every
order bounded
~
to
1~
is i s o m o r p h i c increasing
of
. to an ideal
sequence
in
~
in
~"
.
is n o r m
convergent. (g)
Every order
A Banach
lattice
isomorphic Every
~
interval
in
has p r o p e r t y
~
lattice h a v i n g
compact.
(P) if it is, u n d e r evaluation,
to the range of a p o s i t i v e
Banach
is w e a k l y
property
contractive
projection
(P) is o r d e r complete.
in
~"
.
216
A Banach
lattice
is a K B - s p a c e
For a Banach
lattice
(a)
~
is a KB-space.
~
(b)
~
is o r d e r c o n t i n u o u s
if it is w e a k l y
, the f o l l o w i n g
(c)
No Banach
(d)
Under evaluation,
sublattice
(c)
Every norm bounded
~
complete.
are e q u i v a l e n t :
and has p r o p e r t y
of
~
sequentially
is v e c t o r
is isomorphic
increasing
(P).
lattice
isomorphic
to a b a n d
sequence
in
in
~
~"
to
c
o
.
is n o r m
convergent. Every B a n a c h
lattice h a v i n g
in particular, Furthermore,
every
is r e f l e x i v e isomorphic
A Banach li x + y
reflexive
a KB-space
is or d e r dentable,
see G h o u s s o u b
~
is a KB-space. property
and T a l a g r a n d
if no B a n a c h
is an A L - s p a c e
II x II + II y il
holds
For a B a n a c h
lattice
(a)
is i s o m o r p h i c
~
lattice
is a KB-space;
if and only if it
[78], and a K B - s p a c e
sublattice
of
~
is v e c t o r
lattice
11
lattice
II =
Banach
property
has the R a d o n - N i k o d y m
if and only
to
the R a d o n - N i k o d y m
~
if the identity
for all
x, y 6 ~ +
, the f o l l o w i n g
are e q u i v a l e n t
(as a t o p o l o g i c a l
vector
(Schlotterbeck) :
lattice)
to an
AL-space. (b)
Every positive
Every A L - s p a c e (Q,Z,~)
to
s e q u e n ce LI(Q,Z,~)
in
~
is a b s o l u t e l y
, for some m e a s u r e
summable.
space
(Kakutani).
Furthermore,
every A L - s p a c e
Radon-Nikodym
property
for some index
set
A Banach
lattice
II =
contains
~
is an A M - s p a c e holds
an A L - s p a c e
to
has the II(F)
is r e f l e x i v e
, if and
(with unit)
for all
x, y 6 2 +
if the identity (and the unit ball
element).
For a B a n a c h
lattice
(a)
is isomorphic
~
and an A L - s p a c e
if it is i s o m o r p h i c
dimension.
11 x il v li y II
a largest
is a KB-space,
if and only
F ; in p a r t i c u l a r ,
only if it has finite
li x v y
summable
is isomorphic
~
, the f o l l o w i n g
are e q u i v a l e n t
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Sci. Paris S~rie A 289, 75-78
(1979).
Schmidt:
Sur la c o n v e r g e n c e d'une a m a r t i n g a l e born~e et un t h ~ o r ~ m e de Chatterji. C.R. Acad.
[114]
Sci. Paris S~rie A 289,
181-183
(1979).
K.D. Schmidt: On the value of a stopped set f u n c t i o n process. J. M u l t i v a r i a t e Anal.
[1,15]
I_~0, 123-134
(1980).
K.D. Schmidt: T h ~ o r ~ m e s de structure pour les a m a r t i n g a l e s en p r o c e s s u s de fonctions d ' e n s e m b l e s ~ v a l e u r s dans un espace de Banach. C.R. Acad.
[116]
K.D.
Sci. Paris S~rie A 290,
1069-1072
(1980).
Schmidt:
T h ~ o r ~ m e s de c o n v e r g e n c e pour les a m a r t i n g a l e s en p r o c e s s u s de fonctions d ' e n s e m b l e s ~ v a l e u r s dans un espace de Banach° C.R. Acad.
[1t7]
Sci. Paris S~rie A 290,
1103-1106
(1980).
K.D. Schmidt: On the c o n v e r g e n c e of a b o u n d e d amart and a c o n j e c t u r e of Chatterji. J. M u l t i v a r i a t e Anal.
I!I, 58-68
(1981).
230
[118]
K.D. Schmidt: Generalized martingales and set function processes. In: Methods of Operations Research, vol. 44, pp. 167-178. K6nigstein:
[119]
Atheneum 1981.
K.D. Schmidt: On the Jordan decomposition for vector measures. In: Probability in Banach Spaces IV. Lecture Notes in Mathematics, Berlin-Heidelberg-New
[120]
vol. 990, pp. 198-203.
York: Springer 1983.
K.D. Schmidt: On R~dstr~m's embedding theorem. In: Methods of Operations Research,
vol. 46, pp. 335-338.
K6nigstein: Atheneum 1983.
[121]
J.L. Snell: Applications of martingale system theorems. Trans. Amer. Math. Soc. 73, 293-312
[122]
(1952).
C. Stricker: Quasimartingales,
martingales
locales,
semimartingales et
filtration naturelle. Z. Wahrscheinlichkeitstheorie [123]
(1977).
L. Sucheston: Les amarts
(martingales asymptotiques).
In: S~minaire Mauray-Schwartz Palaiseau: [124]
verw. Gebiete 39, 55-63
1975-1976, Expos~ no. VIII, 6 p.
Ecole Polytechnique,
Centre de Math~matiques,
1976.
J. Szulga: On the submartingale characterization of Banach lattices isomorphic to 11 . Bull. Acad. Polon. 65-68
[125]
Sci. S~rie Sci. Math. Astronom.
Phys. 26,
(1978).
J. Szulga: Boundedness and convergence of Banach lattice valued submartingales. In: Probability Theory on Vector Spaces. Lecture Notes in Mathematics, vol. 656, pp. 251-256. Berlin-Heidelberg-New
York: Springer 1978.
231 [126]
J.
Szulga:
Regularity of Banach lattice valued martingales. Colloquium Math. 41, 303-312 [127]
(1979).
J. Szulga and W.A. Woyczynski: Convergence of submartingales Ann. Probability 4, 464-469
[128]
in Banach lattices.
(1976).
J.J. Uhl jr.: Martingales of vector valued set functions. Pacific J. Math. 30, 533-548
[129]
(1969).
J.J. Uhl jr.: Pettis mean convergence of vector-valued asymptotic martingales. Z. Wahrscheinlichkeitstheorie
[130]
verw. Gebiete 37, 291-295
(1977).
M.A. Woodbury: A decomposition theorem for finitely additive set functions. Bull. Amer. Math. Soc. 56,
[131]
171-172
(1950).
M. Yamasaki: Another convergence theorem for martingales TShoku Math. J. 33, 555-559
[132]
in the limit.
(1981).
C. Yoeurp: Compl~ments sur les temps locaux et les quasi-martingales. Ast~risque 52-53,
[133]
197-218
(1978).
K. Yosida: Vector lattices and additive set functions. Proc. Imp. Acad. Tokyo 17, 228-232
[134]
(1941).
K. Yosida and E. Hewitt: Finitely additive measures. Trans. Amer. Math. Soc. 72, 46-66
(1952).
2~
Additional [135]
references: L.H.
Blake:
A generalization of martingales and two consequent convergence theorems. Pacific J. Math. [136]
35, 279-283
(1970).
D.I. Cartwright: The order completeness of some spaces of vector-valued functions. Bull. Austral. Math. Soc. I!, 57-61
[137]
(1974).
M. Slaby: Convergence of submartingales and amarts in Banach lattices. Bull. Acad. Polon. Sci. S~rie Sci. Math.
[138]
30, 291-299
(1982).
M. Slaby: Convergence of positive subpramarts and pramarts in Banach spaces with unconditional bases. Bull. Pol. Acad. Sci. Math.
3_!1, 75-80
(1983).
I n d e x
.
absolutely additive
summing set
function
AL-space
216
AM-space
216
amart amart
operator
129
62,
125
89 in p r o b a b i l i t y
asymptotic
bounded
212
martingale
89
set
function
62,
bounded
set
function
process
bounded
stopping
cone
absolutely
time
theorem: 103
2.5
14.
-
amart
2.3
9.
-
martingale
80
3.3
9.
-
martingale
144
72
60
summing
convergence
125
operator
4.6
9.
-
order
amart
4.7
I.
-
order
potential
4.6
12.
-
positive
4.3
6.
-
positive
submartingale
4.5
5.
-
positive
weak
2.5
13.
-
potential
3.6
9.
-
strong
3.5
12.
-
uniform
3.5
11.
-
uniform
potential
3.6
8.
-
uniform
weak
204 208
hypomartingale
182
potential
102
amart
163
amart
155
amart
206
154 162
197
128
2S4
3.6.7.
-
u n i f o r m weak p o t e n t i a l
4.5.2.
-
weak a m a r t
3.6.11.
-
weak
196
sequential
amart
additive
measure,
countably
order complete
countably
order continuous
difference amart
4.6.2.
-
order a m a r t
Banach
lattice
Banach
-
semiamart
-
strong a m a r t
-
uniform amart
145 149 190
Doob p o t e n t i a l
82,
enveloppe
179
de Snell:
semiamart
107-108
game w h i c h b e c o m e s
fairer w i t h time
generalized
Radon-Nikodym
derivative
generalized
Radon-Nikodym
operator
hypomartingale
205
decomposition
207
KB-space
216
Krickeberg
decomposition
190
decomposition:
2.3.6.
-
martingale
77
3.3.6.
-
martingale
143
2.1.4
-
measure
3.1 .9.
-
vector measure
64
limit measure
martingale martingale maximal
215
106
Doob c o n d i t i o n
Lebesgue
215
lattice
200-201
2.6.2.
Jordan
65,
94
3.4.3.
-
vector measure
property:
-
2.6.3.
164
countably
2.5.5.
3.5.1.
162
75,
74,
133 142
142
in the l i m i t
212
ingquality:
2.5.12.
-
set f u n c t i o n
process
101
3.5.10.
-
set f u n c t i o n
process
154
213 68, 68,
134
134
133
2S5
measure mil
62
212
~-bounded ~-norm
o-lim
71, 137
set f u n c t i o n process 72
200
o p t i o n a ~ sampling theorem: 2.7.3.
amart
-
117
o p t i o n a l s t o p p i n g theorem: 2.7.2.
amart
-
115
order amart
200
o r d e r b o u n d e d set function
169 215
o r d e r c o m p l e t e B a n a c h lattice
215
order c o n t i n u o u s B a n a c h lattice order p o t e n t i a l
partition
203
59
p o s i t i v e set function potential pramart property
169
95 212 (P)
215
p u r e l y f i n i t e l y a d d i t i v e measure,
quasimartingale
84,
155
Radon-Nikodym operator regular o p e r a t o r
67, 134
170
r e p r e s e n t i n g linear o p e r a t o r restriction
127
70, 137
r e s t r i c t i o n map
70, 137
Riesz decomposition: 2.5.8.
-
amart
4.3.11.
-
negative submartingale
4.6.7.
-
order a m a r t
4.6.10.
-
positive hypomartingale
4.3.4.
-
positive submartingale
2.4.7.
-
quasimartingale
85
2.7.1.
-
quasimartingale
114
2.6.6.
-
semiamart
3.4.4.
-
strong amart
2.4.2.
-
submartingale
96 186
203
110 146 82
205 180
vector m e a s u r e
112, 165
236
4.3.15.
-
submartingale
2.4.2.
-
supermartingale
3.5.3.
-
uniform amart
3.6.6.
-
u n i f o r m weak a m a r t
semiamart
71,
simple
stopping
process
basis
basis
145 146 81,
supermartingale
179
81,
179
set f u n c t i o n
process
71,
137
72
topological
orthogonal
stochastic
uniform amart
u n i f o r m weak
system
basis
60
150
amart
157
u n i f o r m weak p o t e n t i a l
160
uniformly
l-continuous
universal
vector measure
variation
197
149
uniform potential
martingale
77,
143
59
127
vector measure
w-lim
60
59
potential
submartingale
trivial
136
60
strong a m a r t
T-norm
69,
59
time
T-bounded
160-161
126
stochastic
stochastic
strong
150
function
standard
82
137
semivariation set f u n c t i o n
188
125
157
weak
amart
weak
potential
157
weak
sequential
Yosida-Hewitt
~-continuous ~-singular
196 amart
163
decomposition
measure,
measure,
112,
165
vector measure
vector measure
64, 64,
132
132
Allan
Gut
A m a r t s
a
and
Klaus
D.
Schmidt:
-
b i b l i o g r a p h y
239
Amart theory has rapidly grown since its "foundation" J.R. Baxter
[I], R.V. Chacon
A. Ionescu Tulcea
[I], and D.G. Austin,
[I]. The principal
in 1974 by
G.A. Edgar,
and
purpose of the present b i b l i o g r a p h y
is to list the literature on amarts.
It also contains papers which led
to or were inspired by amart theory,
as well as a small number of papers
concerning
further generalizations
of m a r t i n g a l e s
whose relation to
amarts may be subject to further research.
K.A. Astbur[ [I]
Amarts
indexed by directed
sets.
Ann. P r o b a b i l i t y 6, 267-278 [2]
Order convergence
(1978).
of m a r t i n g a l e s
in terms of countably
additive and purely finitely additive martingales. Ann. P r o b a b i l i t y 9, 266-275 [3]
The order convergence
(1981).
of m a r t i n g a l e s
indexed by directed
sets. Trans. Amer.
D.G. Austin, [I]
Math.
G.A. Ed@ar,
Soc.
265, 495-510
(1981).
and A. Ionescu Tulcea
Pointwise c o n v e r g e n c e
in terms of expectations.
Z. W a h r s c h e i n l i c h k e i t s t h e o r i e
verw. Gebiete 3_~0, 17-26
J.R. Baxter [I]
[2]
Pointwise
in terms of weak convergence.
Proc. Amer.
Math.
Convergence
of stopped random variables.
Adv. Math.
Soc.
2!, 112-115
46, 395-398
(1974).
(1976).
A. Bellow
[i]
On v e c t o r - v a l u e d
asymptotic
Proc. Nat. Acad.
Sci. U.S.A.
martingales. 7_~3, 1798-1799
(1976).
(1974).
240 [2]
Stability properties of the class of asymptotic martingales. Bull. Amer. Math. Soc. 82, 338-340
[3]
(1976).
Several stability properties of the class of asymptotic martingales. z. Wahrscheinlichkeitstheorie
[4]
verw. Gebiete 37, 275-290
Les amarts uniformes. C.R. Acad. Sci. Paris S~rie A 284,
[5]
(1977).
1295-1298
(1977).
Uniform amarts: A class of asymptotic martingales for which strong almost sure convergence obtains. Z. Wahrscheinlichkeitstheorie
[6]
verw. Gebiete 41, 177-191
(1978).
Some aspects of the theory of vector-valued amarts. In: Vector Space Measures and Applications I. Lecture Notes in Mathematics,
vol. 644, pp. 57-67.
Berlin - H e i d e l b e r g - New York: Springer 1978. [7]
Submartingale characterization of measurable cluster points. In: Probability on Banach Spaces. Advances in Probability and Related Topics, vol. 4, pp. 69-80. New Y o r k - B a s e l :
[8]
Dekker 1978.
Sufficiently rich sets of stopping times, measurable cluster points and submartingales. In: S~minaire Mauray-Schwartz
1977-1978,
S~minaire sur la
G~omAtrie des Espaces de Banach, Appendice no. 1, 11 p. Palaiseau: [9]
Ecole Polytechnique,
Martingales,
Centre de Math~matiques,
amarts and related stopping time techniques.
In: Probability in Banach spaces III. Lecture Notes in Mathematics,
vol. 860, pp. 9-24.
Berlin - H e i d e l b e r g - New York: Springer 1981.
A. Bellow and A. Dvoretzk~
[1]
A characterization of almost sure convergence. In: Probability in Banach Spaces II. Lecture Notes in Mathematics,
vol. 709, pp. 45-65.
Berlin - H e i d e l b e r g - N e w York: Springer 1979.
1978.
241
[2]
On m a r t i n g a l e s
in the limit.
Ann. Probability 8, 602-606
(1980).
A. Bellow and L. Egghe
[1]
[2]
In~galit~s
de Fatou g~n~ralis~es.
C.R. Acad.
Sci. Paris S~rie I 292, 847-850
Generalized Ann.
Y. Ben[amini [I]
(1981).
Fatou inequalities.
Inst. H. Poincar~
Section B I-8, 335-365
(1982).
and N. Ghoussoub
Une c a r a c t ~ r i s a t i o n C.R. Acad.
probabiliste
de 11 .
Sci. Paris S~rie A 286,
795-797
(1978).
L.H. Blake
[1]
A generalization
of martingales
and two consequent
convergence
theorems. Pacific J. Math. [2]
3_~5, 279-283
A note concerning
(1970).
the L l - c o n v e r g e n c e
of a class of games which
become fairer with time. G l a s g o w Math. J. [3]
1_~3, 39-41
Further results concerning
(1972). games which become
fairer with
time. J. London Math. [4]
Soc.
A note concerning
(2) 6, 311-316
(1973).
first order games which become fairer with
time. J. London Math.
[5]
(2) 9, 589-592
Every amart is a m a r t i n g a l e J. London Math.
[6]
Soc.
Soc.
Weak submartingales J. London Math.
Soc.
(1975).
in the limit.
(2) 18, 381-384
(1978).
in the limit. (2) I_~9, 573-575
(1979).
242
[7]
C o n v e r g e n t processes, martingale
projective
Glasgow Math. J. 20, 119-124
[8]
systems of measures
and
decompositions. (1979).
T e m p e r e d processes and a Riesz d e c o m p o s i t i o n martingales
for some
in the limit.
G l a s g o w Math.
J. 22, 9-17
(1981).
B. Bru and H. Heinich
[1]
Sur l'esp&rance C.R. Acad.
[2]
Sci. Paris S&rie A 288, 65-68
vectorielles. (1979).
adapt&es.
Sci. Paris S&rie A 288, 363-366
Sur l'esp~rance Ann.
[4]
al&atoires
Sur les suites de mesures v e c t o r i e l l e s C.R. Acad.
[3]
des variables
des variables
Inst. H. Poincar&
Sur l'esp&rance
al&atoires
(1979).
vectorielles.
Section B I-6, 177-196
des variables
al&atoires
(1980).
~ valeurs dans les
espaces de Banach r&ticul&s. Ann.
Inst. H. Poincar~
B. Bru, H. Heinich, [I]
Section B 16, 197-210
(1980).
and J.C. L o o t ~ i e t e r
Lois des grands nombres pour les variables &changeables. C.R. Acad.
Sci. Paris S&rie I 293,
485-488
(1981).
A . Brunel and U. Krengel
[1]
Parier avec un proph~te dans le cas d'un processus sous-additif. C.R. Acad.
Sci. Paris S&rie A 288, 57-60
(1979).
A. Brunel and L. Sucheston [I]
Sur les amarts faibles ~ valeurs vectorielles. C.R. Acad.
Sci. Paris S~rie A 282,
1011-1014
(1976).
243
[2]
Sur les amarts ~ valeurs vectorielles. C.R. Acad. Sci. Paris S~rie A 283, 1037-1040
[3]
(1976).
Une caract~risation probabiliste de la s~parabilit~ du dual d'un espace de Banach. C.R. Acad. Sci. Paris S~rie A 284, 1469-1472
(1977).
R.V. Chacon [I]
A "stopped" proof of convergence. Adv. Math. 14, 365-368
(1974).
R.V. Chacon and L. Sucheston [1]
On convergence of vector-valued asymptotic martingales. Z. Wahrscheinlichkeitstheorie
verw. Gebiete 33, 55-59
(1975).
S.D. Chatterji
[1]
Differentiation along algebras. Manuscripta Math. 4, 213-224
[2]
(1971).
Les martingales et leurs applications analytiques. In: Ecole d'Et~ de Probabilit~s: Lecture Notes in Mathematics, Berlin-Heidelberg-New
[3]
Processus Stochastiques.
vol. 307, pp. 27-164.
York: Springer 1973.
Differentiation of measures. In: Measure Theory, Oberwolfach Lecture Notes in Mathematics, Berlin-Heidelberg-New
1975.
vol. 541, pp. 173-179.
York: Springer 1976.
R. Chen
[1]
A generalization of a theorem of Chacon. Pacific J. Math. 64, 93-95
(1976).
244
[2]
A simple proof of a theorem of Chacon Proc. Amer. Math. Soc. 60, 273-275
[3]
(1976).
Some inequalities for randomly stopped variables with applications to pointwise convergence. Z. Wahrscheinlichkeitstheorie
verw. Gebiete 36, 75-83
(1976).
B.D. Choi
[1]
The RieSz decomposition of vector-valued uniform amarts for continuous parameter. Kyungpook Math. J. 18,
119-123
(1978).
B.D. Choi and L. Sucheston
[1]
Continuous parameter uniform amarts. In: Probability in Banach Spaces III. Lecture Notes in Mathematics, vol. 860, pp. 85-98. B e r l i n - Heidelberg - N e w York: Springer 1981.
Y.S. Chow
[1]
On the expected value of a stopped submartingale. Ann. Math. Statist. 38, 608-609
(1967).
B.K. Dam and N.D. Tien [I]
On the multivalued asymptotic martingales. Acta Math. Vietnam. 6, 77-87
W.J. Davis, N. Ghoussoub,
[1]
(1981).
and J. Lindenstrauss
A lattice renorming theorem and applications to vector-valued processes. Trans. Amer. Math. Soc. 263, 531-540
(1981}.
245
W.J. Davis and W.B. Johnson
[1]
Weakly c o n v e r g e n t
sequences of Banach space valued random
variables. In: Banach Spaces of Analytic
Functions.
Lecture Notes in Mathematics,
vol. 604, pp.
Berlin - H e i d e l b e r g - N e w
Springer
York:
29-31.
1977.
L.E. Dubins and D.A. F r e e d m a n [I]
On the expected value of a stopped martingale. Ann. Math.
A. Dvoretzky
[1]
Statist.
[1]
Generalizations
(see also:
Soc. 82, 347-349
of martingales.
P r o b a b i l i t y 2, 193-194
(1977).
D.G. Austin)
Inst. H. Poincar~
A s p l u n d operators
Section B 15,
Additive
197-203
(1979).
and a.e. convergence.
J. M u l t i v a r i a t e Anal. 10, 460-466 [3]
(1976).
U n i f o r m semiamarts. Ann.
[2]
(1966).
On stopping time directed convergence.
Adv. AppI.
G.A. Edgar
1505-1509
(see also: A. Bellow)
Bull. Amer. Math.
[2]
37,
(1980).
amarts.
Ann. P r o b a b i l i t y 10,
199-206
(1982).
G.A. Edgar and L. Sucheston [I]
Les amarts: C.R. Acad.
Une classe de m a r t i n g a l e s
asymptotiques.
Sci. Paris S~rie A 282, 715-718
(1976).
246
[2]
Amarts:
A class of asymptotic martingales.
A. Discrete
parameter. J. M u l t i v a r i a t e Anal. 6,
[3]
Amarts:
193-221
A class of a s y m p t o t i c
(1976).
martingales.
B. Continuous
parameter. J. M u l t i v a r i a t e Anal. 6, 572-591 [4]
The Riesz d e c o m p o s i t i o n Bull. Amer. Math.
[5]
Soc.
for v e c t o r - v a l u e d 82, 632-634
The Riesz d e c o m p o s i t i o n
On v e c t o r - v a l u e d
[8]
[I]
in the limit and amarts. 315-320
39, 213-216
(1977).
(1977).
de lois des grands nombres par les
Caract~risations
descendantes.
de la nucl~arit~
(1981).
of n u c l e a r i t y
Anal. 35, 207-214
Some new C h a c o n - E d g a r - t y p e Ann.
A new c h a r a c t e r i z a t i o n in L(LI,X)
Simon Stevin 54,
(1978).
in Fr~chet spaces. (1980).
inequalities
and characterizations
Inst. H. Poincar~
operator
dans les espaces de Fr~chet.
Sci. Paris S~rie A 287, 9-11
Characterizations
processes,
[4]
Soc. 67,
Sci. Paris S~rie I 292, 967-969
J. Functional [3]
(1976).
(see also: A. Bellow)
C.R. Acad.
[2]
Gebiete
Math.
C.R. Acad.
Egghe
verw.
Proc. Amer.
sous-martingales
L.
verw. Gebiete 36, 85-92
Martingales
D~monstrations
amarts.
amarts and dimension of Banach spaces.
Z. W a h r s c h e i n l i c h k e i t s t h e o r i e [7]
amarts.
(1976).
for v e c t o r - v a l u e d
Z. W a h r s c h e i n l i c h k e i t s t h e o r i e
[6]
(1976).
for stochastic
of Vitali-conditions.
Section B 16, 327-337
(1980).
of Banach spaces X for which every
is c o m p l e t e l y continuous.
135-149
(1980).
247
[5]
Strong c o n v e r g e n c e of p r a m a r t s in B a n a c h spaces. C a n a d i a n J. Math. 33, 357-361
[6]
(1981).
W e a k and strong c o n v e r g e n c e of amarts in F r ~ c h e t spaces. J. M u l t i v a r i a t e Anal. 12, 291-305
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