Amarts and Set Function Processes

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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 o . Let

T C TN

Xn O

it follows that

IEXTI_ 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}_°}-< 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 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 n'

I 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 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 ~} , 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 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}
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_ ]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)

~

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

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})

~

~(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].

]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

_

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

... [ 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

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)


order complete

and c o u n t a b l y

order continuous.


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


(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 ~ +


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

~


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

AM-space. (b) Every

Every null

sequence

(order complete)

for some

(Stonian)

Furthermore, isomorphic property

(F)

is o r d e r bounded. is i s o m o r p h i c

space

K

is o r d e r c o n t i n u o u s

, for some index

o (P) if and only

in pa r t i c u l a r , dimension.

c

~

with unit

compact Hausdorff

an A M - s p a c e

to

in

AM-space

set

F

is a K B - s p a c e

C(K)

,

(Krein-Kakutani). if and only

if it is

, and an A M - s p a c e

if it is an o r d er c o m p l e t e

an A M - s p a c e

to

if and only

AM-space

has

w i t h unit;

if it has

finite

R e f e r e n c e s .

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Sur la valeur d'un p r o c e s s u s de fonctions d'ensembles. C.R. Acad.

[112]


(1979).

K.D. Schmidt: Espaces v e c t o r i e l s r~ticul~s, d ~ c o m p o s i t i o n s de Riesz, et c a r a c t ~ r i s a t i o n s de c e r t a i n s p r o c e s s u s de fonctions d'ensembles. C.R. Acad.

[113]

K.D.


(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]


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.


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]


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.


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]


1975-1976, Expos~ no. VIII, 6 p.



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


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]


(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).


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).


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.


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]


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]


(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]


Martingales,


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.


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 .


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.


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.


(1979).

A. Brunel and L. Sucheston [I]

Sur les amarts faibles ~ valeurs vectorielles. C.R. Acad.


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


(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.


Differentiation of measures. In: Measure Theory, Oberwolfach Lecture Notes in Mathematics, Berlin-Heidelberg-New

1975.

vol. 541, pp. 173-179.


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


(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.


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.


(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.


Characterizations

processes,

[4]

Soc. 67,


J. Functional [3]

(1976).

(see also: A. Bellow)

C.R. Acad.

[2]

Gebiete

Math.

C.R. Acad.

Egghe

verw.

Proc. Amer.

sous-martingales

L.


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


[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

[7]

(1982).

On sub- and s u p e r p r a m a r t s w i t h values in a B a n a c h lattice. In: M e a s u r e Theory, O b e r w o l f a c h

1981.

L e c t u r e N o t e s in Mathematics,

vol.

Berlin -Heidelberg -New

Springer

York:

945, pp. 352-365. 1982.

A. E n @ e l b e r t and H.J. E n g e l b e r t

[1]

Optimal stopping and almost sure c o n v e r g e n c e of r a n d o m sequences. 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 48, 309-325

[2]

(1979).

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Amarts and Set Function Processes

Markov Processes, Ray Processes and Right Processes

Genes and Cardiovascular Function

mRNA Formation and Function

Neurosteroids and Brain Function

Chromosomes: Organization and Function

Proteins: Structure and Function

Substance and Function

Mathematics, form and function

Chromosomes: Organization and Function

Mathematics: Form and Function

Exercise and Cognitive Function

Structure and Function

Operators and Function Theory

Nutrition and Immune Function

Chromosome Structure and Function

Psychopathology and Function

mRNA Formation and Function

Chromosomes: Organization and Function

Warrant and Proper Function

Mathematics: Form and Function

Chromatin: Structure and Function

Function Spaces and Applications

Protein stucture and function

Protein structure and function

Warrant and Proper Function

Protein structure and function

Chromosomes: Organization and Function

Algebra: Form and Function

Algebra: Form and Function

Supramolecular Structure and Function

Amarts and Set Function Processes

Markov Processes, Ray Processes and Right Processes

Genes and Cardiovascular Function

mRNA Formation and Function

Neurosteroids and Brain Function

Chromosomes: Organization and Function

Proteins: Structure and Function

Substance and Function

Mathematics, form and function

Chromosomes: Organization and Function

Mathematics: Form and Function

Exercise and Cognitive Function

Structure and Function

Operators and Function Theory

Nutrition and Immune Function

Chromosome Structure and Function

Psychopathology and Function

mRNA Formation and Function

Chromosomes: Organization and Function

Warrant and Proper Function

Mathematics: Form and Function

Chromatin: Structure and Function

Function Spaces and Applications

Protein stucture and function

Protein structure and function

Warrant and Proper Function

Protein structure and function

Chromosomes: Organization and Function

Algebra: Form and Function

Algebra: Form and Function

Supramolecular Structure and Function

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