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AO-A 136 383 
UNCLASSIFIED 


ON THE FATOU INEQUAL ITYIU) STANFORD UNIV CA DEPT OF 
STATISTICS A DVORETZKY OCT 83 TR-23 NOOO14-77-C-0306 


1 / f 




























A136383 


ON THE FATOU INEQUALITY 
by 

Aryeh Dvoretzky 
Hebrew University 


TECHNICAL REPORT NO. 23 
OCTOBER 1983 


PREPARED UNDER CONTRACT 
N00014-77-C-0306 (NR-042-373) 

FOR THE OFFICE OF NAVAL RESEARCH 


Reproduction in Whole or in Part is Permitted 
for any Purpose of the United States Government 

Approved for public release; distribution unlimited 


DEPARTMENT OF STATISTICS 
STANFORD UNIVERSITY 
STANFORD, CALIFORNIA 





83 12 


29 


024 


ON THE FATOU INEQUALITY 


by 

Aryeh Dvoretsky 
Hebrew University 


TECHNICAL REPORT NO. 23 
OCTOBER 1083 


PREPARED UNDER CONTRACT 
N00014-77-C-0306 (NR-042-373) 

FOR THE OFFICE OF NAVAL RESEARCH 


DEPARTMENT OF STATISTICS 
STANFORD UNIVERSITY 
STANFORD, CALIFORNIA 


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On The Fatou Inequality 


Aryeh Dvoretzky 



1. Introduction 


The classical Fatou inequality 


( 1 . 1 ) 


/ X n (a))daj < lin / X^CoOdu , 


) for non-negative measurable functions N 3^^is intimately connected with 
problems of convergence of random variables (r.v.)*^ 


iU Fa£r~ 




->The present paper focuses on the study of a modified form of -Hrrty 

t 

which has important applications in the theory of convergence of r.v. 

We formulate our results in the language of probability, (ft ,£.>?) 


is a probability space. (3 n ) ne # * s a non-decreasing sequence of sub-a- 

algebras of 3 . The sequences (X n > n€ tf of r.v. are adapted to (3 n > , 

l.e., X is 3 -measurable for all n . 
n n 

The stopping times associated with (3 n ) are the mappings t: ft +N 

such that {t*n} « {u>;t(w)*n} e 3 for all n. We put 3„ * (A;An {t»n} 

n t 

e 3 for all n} . X^ is the r.v. (X^)(u>) » X. * (u>) , it is 3 -measurable, 
n t t tvu>J t 

A bounded stopping time (b.s.t.) is a stopping time assuming only finitely 
many values. We donote by T the family of all bounded stopping times. 

We use the letter E to denote expectation. The variant of (1.1) 
studied in this paper is 


( 1 . 2 ) 


E limlx I s lira E|X | 

- n' —— 1 t 1 

teT 


t 


I 











(Throughout the paper lim, lim , etc. refer to the relevant index increasing 
to infinity). 

It is easy but not very interesting to establish the conditions for 
equality in (1.1). On the other hand, W. D. Sudderth [7] showed that if 
the stopping times in (1.2) are not restricted to be bounded, then the 
inequality reduced to an equality. (The proof in [7] relies on a martingale 
convergence theorem; the result can also be easily obtained from Lemma 4.1 
of the present paper). 

It is, however, the bounded stopping times that play a major part in 
various generalizations of the theory of martingales. The pioneering work 
in this connection is due to J. R. Baxter [2] and D. G. Austin, G. A. Edgar 
and A. Ionescu Tulcea (» A. Bellow) [1]. 

All stopping times considered throughout the paper are bounded. A 


detailed study of the excess of the right side of (1.2) over the left side 
is the core of the present paper. 


The Fatou discrepancy 


is the set function given by 


(1.3) lira Elxlr - E limlx |l A , 

where 1 A (as always in the paper) denotes the indicator of the set A . 

It plays a central role in the present study. 

For simplicity of statements we make the following assumptions: 

A1 . 3 * o(G) , i.e. , the a -algebra 3 is generated by the algebra G 

where G * U 3 . 

n eN n 

A2 . llm |x^l is almost surely (a.3.) finite, l.e. . P( lim |X^| < «°) ■ 1 . 

The second assumption is equivalent to the right side of (1.2) being 
o-finite over G , i.e., to the existence of sets A e G with 


2 







P(A) arbitrarily close to 1 for which E lim E tX 11 < <» . That this 

- * t A 

condition implies A2 follows from (1.2), the implication in the other 
direction is by a standard argument in the theory of stopping times (used 
also in the proof of Lemma 3.2). 

In section 2 , we consider the set function u*(A) * lim E||1^ , 

(Ae G) , show that it is finitely additive and introduce through y* a 
measure u on 3 . In the next section we define, for he G, a set 
function $(A) * U*(A) - Uq(A) * where \1q is the absolutely continuous 
part of u relative to P , and show (Theorem 3.1) that it is the Fatou 
discrepancy (1.3). The fact that 0 dwells on small sets (Lemma 3.1) is 
fundamental. 

Whereas in sections _2 and _3 we consider only the absolute values 

|X n | of the r.v. , we turn in 4^ to the r.v. themselves and their 

possible limits. We denote by 0 the set of r.v. which are pointwise 

limits of II and by 0 the subset of integrable r.v. in C . The main 

result here is (Theorem 4.1) lim E|X — Y | » <f>(ft) + p(Y,C) for all 

integrable r.v. Y where p(Y,C) is the L^ distance between Y and 0 . 

This yields (Corollary 4.1) a characterization of C (quite different from 

that given by A. Bellow [4] in terms of sub-martingales). We draw attention 

to a useful approximation result (Lemma 4.1). 

In section 5^ we consider simultaneous approximations. The principal 

result here (Theorem 5.1) is that we can associate with every YeC a 

sequence of bounded stopping times t (Y) such that X ( . Y a.s. and, 

n c n' Y ' 

moreover, if Y-Z is integrable then X - X , + Y - Z in L. norm. 

Results of this nature were proved in [ 5 ], [ 3 ] and [4] (indeed they are the 
key to the proof of the amart convergence theorem) . The main point here is 


3 


I 





not in weakening the assumptions and extending the conclusion but in having 





the same sequence t R (Y) for all Z . 

Section 6^ introduces a finitely additive real-valued set function $ 

on G having the property (Theorem 6.1) that there is associated with every 

Y € C a sequence of bounded stopping times t fl (Y) such that, if lim E | [ 

m 

< », we have E X (y) V ■* EYV + £ M(A.) for every simple r.v. 

c n v w 1 * 

m 


l 

i-1 


x iV 


(A. s G). In particular, E(X 


ti(Y) 


- \< 2 )> 


V -► E(Y - Z) V for 


all Y, Z? Z when i,j ® independently of one another (unlike the situa¬ 
tion in the preceding paragraph). 

Except in the last section, only real-valued r.v. are considered. In 
2 the results are extended to r.v. assuming values In a Banach space. Those 
of 6^carry through to finite dimensional Banach spaces;all other results 
remain valid (with minor variations) In infinitely dimensional Banach spaces. 

We use the letters t,s,T to denote bounded stopping times and X,Y,Z,V 
to represent random variables. 


2. The set function y* and the measure y . 

Definition 2.1 . We define a set function u*(A) on G by 


( 2 . 1 ) 


U*(A) - lim E|X t |l A , (A£ G) . 


4 


f 






/ « * 


We recall that (2.1) is equivalent to the following statement. There 
exists a sequence ^^^n€A , bounded stopping times increasing to 

infinity such that 


( 2 . 2 ) 


lim E|X |1 - u*( A) , 

c n A 


while for all sequences (s ) „ 

n neN 


of b.s.t. with s ® we have 
n 


lim E|X |1 > u*(A) . u* is a monotone set function from G to [o,®]. 

S A 

n 

When it is necessary to exhibit the dependence of u* on the sequence 

X we shall write p*(-,(X )) . Similarly for u,...,<j> which will be 
n n 

introduced later. 

Lemma 2.1 . u* i£ finitely additive on G , i»e. , 


(2.3) 


/(AuB) - M (A) + /(B) 


for disjoint A,B € G . 

Proof. Let t -► • be b.s.t. for which (2.2) holds with A replaced 
- n 

by A u B . Then 


(2.4) 


U*(A u B) 2 limE|X t |l A + lim E|X |l 

n n 

2 lim E|X fc |1 A + lim E|X £ |l fi - w*(A) + u*(B) • 


On the other hand, let Ac 3 k and t « satisfy (2.2) and s n -*■ « 

be b.s.t. satisfying E | X |l B ^u*(B). Assume further s n >k, t^k for 

n 

all n . Putting t * t for w £ A and t ■ s for u» / A 

n n n n 


5 






we have 


'.2.5) 


U*(A u B) S lim E|X |1 a d 
1 t 1 AuB 
n 


lim E|X t |1 A + lim E|X g |lg 
n n 

W*(A) + y*(B) . 


Together with (2,4) this yields (2.3). □ 

The following easy result will be useful. 

Lemma 2.2. If u*(A) < « and t n 00 are bounded stopping times 
satisfying ( 2 . 2 ) then 

(2.6) lim E|X |l = U *(B) , 

n B 

holds for every B € G with B c A . 

Proof . Otherwise, since y*(B) < y*(A) < 00 , we would have 

lim Ejx^ |lg > y*(B) . Taking C * A \ B we have, by definition, 
n 

lim E jx^ |l c ^ v* (C) . Combining the two relations we obtain 
n 

lim E||l A > u*(B) + u*(C) * u*(A) contradicting (2.6). □ 

n 

it 

U may not be a - additive on G (see Remark 2.1). We can, however, 
derive in the standard way a a - additive function. 

For every A € G we put 

(2.7) w( A) » inf) V v*( A >; Ac |j A , A e G,nefl 

I nkN n neN n *» e 



6 








1 


Since u* is monotone we may confine (2.7) to countable partitions of 

A into sets A e G . u* is a monotone function from G to [o,»]. 
n 

Clearly 

(2.8) u(A) < u*(A) , (A c G). 

Lemma 2.3 . y is_ countably additive on G . 

Proof . Let A,B € G be disjoint and C » AuB . Since each partition 

(C ) „ of C corresponds to the partitions (A n C ) and (BnC ) of A 

n neiv n'n 

and B and vice versa the finite additivity of jj follows at once from that 

of u* . 

We have to show that if A^e G(n e N) are disjoint and A * UA^ e G 

then 

(2.9) y ( U A ) - l y(A ) . 

n n 

Since \i is finitely additive and monotone the left side of (2.9) is > the 

right side. Thus it remains to show that u(UA) $ I y(A ) and this only 

when the right side is finite. Let e > 0 be given and let, for every n , 

A 4 (j€#) be a partition of A into sets of G satisfying \ u*(A ) < 
n »J n n,j 

^(A^) + e/2 n . Then A^ ^j^ne/V), constitute a countable partition of A 

and \ u*(A j) < 1 u(A ) + G • D 

j »n n,j n n 

Since, by (2.8) and Assumption A2, u is a - finite on G it extends 
uniquely to a measure on 3 * a(G). We denote this measure by the same letter 






Definition 2.2 . y is the measure induced on 3 bjr (2.7). 

Remarks, 2.1 . y* need not be countably additive on G . Thus for 

Q * (0,1],P the lebesgue measure, X ® 2 n 1 /A 0 -ni and 3 - a(X-,...,X ), 

n lu»z J n l n 

we have y*(A ) = 0 for all n but y*(ft) = y*(UA ) * 1, where A * (2 n ,2 n+ ^] 
n n n 

2.2. As seen from the proof of Lenana 2.1, lim Elx |l A _ £ lim Elx |l A + 

r 9 - 1 t* AuB - 1 t‘ A 

lim E |X^. 1 l p for all disjoint measurable A and B (not necessarily in 

G) . Moreover, equality holds if either A € G or B e G . 

If, however, neither A nor B are in G there may occur a sharp 

inequality. Indeed, taking in the preceding example A = U(2~^ 2n ^',2 ^n-1).j 

and B as its complement, we have lim E X 1 * lim E X 1 = 0 but 

t A t B 

U*(n) = liffl E X » 1 • 

2.3. For measurable A l G we need not have u(A) £ lim E|x |l. 

t A 

as in (2.8). Indeed, modify the example in Remark 2.1 through replacing 
ft - (0,1] by ft - [0,1]. Then y({0}) - 1 . 

3. The set functions 4 ),A and the measures yg ,y^ • 

Since y is a-finite it can be decomposed according to Lebesgue. 

Definition 3.1 . y^ and y^ are , respectively , the absolutely con¬ 
tinuous and singular components of the measure y relative to P . 

y^ and y^ are, of course, defined for all measurable sets. The 
following definition Introduces a set function on G which occupies a 
central position in the present paper. 

Definition 3.2 . The Fatou discrepancy of the sequence of random 
variables (X ) relative to the q-fields (3 ) is given by 


8 







(3.1) 


(A c G) , 


♦ <A) - u*(A) - Uq (A) , 

where , in case y^(A) * °° , this is to be interpreted as sup{<J>(B) ;B c A, 
Be G, Uq(B) < »}. 

We put 


(3.2) A (A) - y*(A) - u(A) , (Ac C) , 

where again, if u(A) * «> , this is to be interpreted as above. Clearly 

(3.3) <p * A + v l • 

A and <f> are finitely additive (on their domain of definition G) . 

One could loosely describe A and y^ as the dissipative and singular 

components, respectively, of y* - 

It follows from Theorem 3.1 that y^ depends only on the sequence 

(X ) and not on the filtration (3 ) . It then follows from (3.1) that 
n n 

any two filtrations which yield the same value of y*(A) will give also 
the same value of 4>(A). Remark 3.1. will point out that this is not true 
for A (A) or y^(A) . This is the reason why it is not A or y^ but 
their sum (3.3) that figures prominently in our results. 

Theorem 3.1 . We have for all measurable sets 

(3.4) m q (A) - E lim|X n |l A , (Ac3) . 


9 







The following result is the key to the proof of Theorem 3.1. 



Proof . (This result is equivalent to the apparently more general 

statement: given B c G and e > 0 there exists a set Ac B, Ae 8 such 

that P(A) > P(B) - e and <A) < e.) Since by Assumption A 2_ f ft contains 

sets of G with probability arbitrarily close to 1 with finite u* 

it is enough to prove (3.5) for the case y*(ft) < 00 . Let A n (n € iV) be a 

. m 

partition of ft into sets of G for which Zy (A^) < u(ft) + e/2 . Then A((JA n ) 

* m m m 

U (U A ) - u(U A ) < e/2 forall me N and P(UA)>l-e/2 for large m . 

1 n 1 n 1 n 

Hence there exists A f € G with P(A f ) > 1 - c/2 and A(A r ) < c/2 . 

Similarly, since 1 P and 3 = a(G) , there exists A 1 '€ G with 

P(A") > 1 - c/2 and ^(A") < e/2. A= A ! n A" satisfies (3.5). □ 

Lemma 3.2 . For every e > 0 and k e N there exists A £ G and a 
bounded stopping time t > k such that 

(3.6) P(A) > 1 - e , E|X |l < E l±mlX |l + e . 

t A — n a 

Proof . Let A be the set described in the preceding Lemma with e 
replaced by e/4 . We may assume u*(A) < 00 . Since is absolutely 

continuous and Uq(A) < 00 there exists 6 > 0 such that Uq(B) < e/4 


10 




for every B c A with P(B) < 6 . Then also u*(B) * Uq(B) + < e/ 2 . 

Let Y be a simple (i.e., assuming only finitely many values) G-mea¬ 
surable r.v. with P(lim X < Y < lim X + e/4) > 1-6. Y is 5 -measurable 

- n - n m 

for some m and we may take m> k. Let the b.s.t. s> m be such that 

P (| | < Y) > 1-6. Let C = {| X q | < Y} and D be its complement. Let 

the b.s.t. t > s be such that E l^ T i^oD < + e/4 < 3e/4 . 

Put t * s 1 + t] _ then 

C D 1 


E |X t l 1 A = E|X t |l AnC + E|X t |l AnD < E|Y|l AnC+ 3e/4 


< E lim|X n |l A + e . 


Proof of Theorem 3.1. As remarked in the introduction lim|X_|l A 
- -■— n a 

is finite a.s. and both sides of (3.4) are a-finite measures on 3 = a(G). 

It suffices therefore to prove the assertion for A e G with y*(A) < ® . 

Without loss of generality we may take A = Q and assume Uq(^) < 00 * 

E limIX I < « . 

-n 1 

Let e>0 be given, choose Ae G satisfying (3.5) and denote its 

complement by B . Then E lim jX | * E lim |X |l A + E lim |X |l„ S y*(A) + 

E lim|X n |l B * u q (A) + 4 >(A) + E lim| x n l ] B - U Q (n) + e + E Umlxjlp . Since 

lim |X n 1 is integrable letting e -► 0 we obtain 

(3.6) E lim|X | s u Q (ft) . 

For the set A of Lemma 3.2 we have u*(A) £ E lim ]X^l + e . Denoting 

by B the complement of A we have u^(n) * Uq(A) + Pq(B) 5 E 1 + G + 

Ug(B) . Letting e -► 0 we obtain the reverse of inequality (3.6). □ 


11 







( 1 . 2 ) 


Theorem 3.2 . The sign of equality holds in the Fatou inequality 

E limlx I 5 lim EIX I , 

-1 n ' - 1 t i 


if and only if * 0 . 

The following result will be useful in the next section. 

Lemma 3.3 . Let (X^), n e N be a sequence of random variables adapte d 

to ( 3 ) and£ut u*(•)■ u*(•,(X )) f *(.)-?(*,(X)) etc. If sup||x|-|x|| 

n n neN n “ 

is integrable then . 

Proof . Putting Y = sup||x^| - i^ n ll ve note that the set function 

v = u “ U satisfies |v(A)| < EY1 A for A e G . Since v is finitely 

additive it follows that it is countably additive and hence (extends to) a 

signed absolutely continous measure on 3 . From Definition 2.1 it follows 

at once that TT = u + v, thus A * A . Since v is absolutely continuous 

the singular parts of p and U are the same. Hence p^ = p^ . □ 

Remarks 3.1 . In the example considered in Remark 2.3 we have =* 1, 

A(fl) * 0 . If we replace 3 = a(X- ,...,X ) by 3 * a({0},X- ,... ,X ) we 

n l n n in 

will have p^(ft) * 0, A(Q) « 1 . 

3.2 . Let Q * [0,1], P be the lebesgue measure and 3 be the algebra 
~“ n 

2 n 

generated by ((i-l)/2 n , i/2 n ), i*l,...,2 n and let X * 2 n £ 1 where 

n i -1 A i,n 

A. * (i/2 n - l/2^ n , i/2 n ) . Then X -► 0 a.s. but p and p n coincide 
i, n n (j 

with the Lebesgue measure. Here, of course, X is not adapted to 3 , thus 

n n 

exhibiting the necessity of this requirement for the validity of our theorems. 


12 










3.3 . It follows from Theorem 3.1 and Lemma 2.2 that If u*(n) < « 

and the b.s.t. t - • satisfy E|X„ | u*(0) then E|X |l - E lim|X |l + 

n t n Si A n 

<(>(A) for every As G . Hence, even when “ 00 » there exists a 

sequence of b.s.t. t -*■ 00 such that E |X^ |l^ E lim|X n |l^ + $(A) holds 

n 

for every A £ G . 

4. The cluster set . 

Until now, we have considered only the absolute value |x^| of the 

r.v. X . From now on we shall be concerned with the r.v. themselves, 
n 

Definition 4.1 . The cluster set C = C(X n > of_ the sequence of random 

variables (X ) „ is the set of random variables X which are a.s. 

- n n cN -- 

pointwise limits of the random variables X^, i.e., which satisfy 

(4.1) P(lim|X n - X| - 0) - 1 . 

C is the subset of C consisting of the integrable random variables 
in the cluster set, i.e. , C * CnL^(ft,3,P) . 

C is a closed set in the metric space ■ L^(ft,3%P) . The 
distance from an integrable r.v. Y to C Is given by p(Y,C) » 
inf{E|Y-X|; X£ C} . Unless C is empty there exists X’ e C for which 
the infimum is achieved. 

If YsS(X) then, obviously, |Y | e cC |X |) . Conversely, if YeC(|x |) 

n n n 

there exists Z£ C(X n > with |z| - Y. Indeed, there exist b.s.t. t„ -► « 

for which |X t | -*• Y (this is a well-known result, see e.g. [1]; it also 

n _ 

follows from Lemma 4.1). We may take Z ■ Y on {lim X > 0} and Z « -Y 

Si 

otherwise. 


13 



It follows from this observation that if X 1 € C and E|x* - Y| « p(Y,C) 

then P(lim|X n -Y| - |x’-Y|) - 1. 

C(X ) is not empty since lim|x I c C(|x I) . Also, C is not 
n n n 

empty if and only if 


(4.2) E lim|xj < « . 

A special case of the following approximation result was used in the 
proof of Theorem 3.1. 

Lemma 4.1 . For every c > 0 there exists a set A £ G with P(A) > 1-e 
such that for every Y £ C there exist arbitrarily large bounding stopping 
times t satisfying 

(4.3) E|X t -Y|l A <e. 

If YeC there exist sequences of G- measurable set A^ and bounded 

stopping times t + ® such that X 1. Y in L.-norm . 

- lz —° - n -t A — 1 - 

n n 

Proof . The first part is a restatement of Leona 3.2 for the sequence 

(X - Y) . The second part follows upon denoting by t and A a b.s.t. and 
n n n 

set satisfying (4.3) with c * 1/n . 

Theorem 4.1 . I£ Y is an integrable random variable then 

(4.4) lim E|X t - Y| - $(«) + p(Y,C) . 


14 







Proof . By Lemma 3.3 the $ corresponding to the sequence of r.v. 

(X - Y) is the same as that corresponding to the sequence (X ) • 
n n 

Assume first C*0 , then we have by Theorem 3.1 lim E|X t ~ Y| » + 

E lim jx^- Y| and, as remarked above, p(Y,C) equals the second sunmand on 
the right. 

It remains to check that if (4.2) fails then lim E|X t * Y| > lim E|X^| - 
E|Y| * ~ . □ 

Corollary 4.1 . If Y is an integrable random variable then 

(4.5) lim E|X t - Yj > , 

and if 

(4.6) lira E|X - Y| < ® , 

equality occurs in (4.5 ) when and only when Y c C . 

The following is an immediate extension of Theorem 4.1. (We recall 
that Assumption A_2 implies C ^ 0) . 

Corollary 4.2 . Let Z e C and denote by C z the set of integrable 
random variables X for which Z + X € C . Then , for every integrable 
random variable Y we have 


(4.7) llmE|X t -(Z +Y)| - <K«, (X R - Z)> + p(Y,C z > . 

n 

The following result is useful. 


15 





Lemma 4.2 . If $(&) < 00 , Y is an integrable random variable and the 
bounded stopping times t^ -► « satisfy 

lim E|X fc - Y| - (Kfi) 
n 

then X -► Y in probability, 
n 

Proof. If the conclusion fails there exist 6 > 0 and a subsequence 

(s n ) of (t n ) such that E|X g - Y|l > 6 for all B with P(B) > 1-6 

n 

and all n . 

Take e < 5 , let A be the set described in Lemma 4.1 and let B be 

its complement. Then lim E|X - Y| > lim E|X - Y|l. + lim E|x - Y|l„ > 

s n s n A s n B 

6 + $(B) > + 6 - e > . □ 

Remarks 4,1 . Lemma 4.1 implies that every Ye C is the limit in pro¬ 
bability of a sequence X with t 00 • (This actually characterizes 

c n n 

C since a subsequence of X. will converge a.s. to Y .) This result 
about approximation in probability is well known and has been extensively 
used in the study of Amarts and related topics. It is explicitly stated 
and proved in [1] and is implicit and crucial in [2]. 

4.2* Notice that the set A in the approximation Lemma 4.1 does not 
depend on Y . 

4^3. If Z,Z f e C and E|Z f - Z| < « then, with the notation of Cor¬ 
ollary 4.2, C^t. ■ thus the right side of (4.7) does not change if Z 

is replaced by Z f . 


16 










5. Simultaneous approximations . 

If Y, Z e C and * 0 there exist sequences (s n >, (t R ) of b.s.t. 

such that X Y , X -► Z in L, norm and hence also X^ - X -► Y - Z 
s n c n 1 s i c j 

in L^ norm as i,j 00 independently of one another. 

The situation is quite different when > 0 . In the example of 

Remark 2.1, where C consists of the one r.v. Y = 0 , if t and s are 
any b.s.t. with min t > max s we have E|X fc - X g | >1 . Thus there can¬ 
not exist sequences (s ), (t ) satisfying E|X - X | -► E |Y - Y | * 0 as i,j 

n n s^ i j 

« independently of one another. If, however, we let j* j(i) then the 
approximation of Y-Z can be achieved. 

Theorem 5.1 . To every Ye C there exists an increasing sequence of 

bounded stopping times t n (Y) * n € N 9 satisfying X^ -► Y a.s. and such 

n 

that for any Y,Z with E |Y - Z | < 00 we have 


(5.1) 


X t (Y) “ X t (Z) Y ‘ Z * 
n n 


in - norm. 


Proof. Let A be a set having the properties described in Lemma 4.1 
- n 

2 

for c * 1/n . Let k be such that A z d. and s ^ k beab.s.t. 

n n k^ n n 

satisfying (4.3) for these A^ and e . We may take • Let (x^) 

be any increasing sequence of b.s.t. satisfying x^ ^ k^ . Put t^(Y) * 

3 (Y)l. + x 1_ where B is the complement of A . Then, for any 

n A n D n n 

n n 

Y,2,C we have ElX^, - - <Y - Z) | - E - Y - (X^, - Z> * 


EIY - Z11 <2/n +EY-Z1- -0 since Y-Z is integrable. 

B n B n 


0 


17 







A similar argument yields 


Theorem 5.2 . Lf C i£ not empty then for every Integrable Y there 
exists an increasing sequence of bounded stopping times t fl (Y) such that 
X t (Y) * — vtiere e C is nearest (in -me trie ) to Y, i .e. , 

E|Y- Y* I - p(Y,C) , and 

(5.2) lim E|X t (y) - X t (z) - (Y - Z) | < p(Y,C)+ p(Z,C) , 

for all integrable random variables Y,Z. 

The assertion implies that the limit in (5.2) exists. 

Remarks 5.1 . J. R. Baxter [3], following an extension of the Fatou inequality 

by It. V. Chacon [5], proved that if lim E | | < 00 and Y.ZcC there exist b.s.t. 

s n , t n + °° satisfying X g - ** Y - Z in L^-nonn , these sequences 

n n 

were, however, dependent on the pair Y,Z . (See also A. Bellow [4]). 

5.2 . If Y - Z is not integrable then, by Fatou, E |^ * x t I * 00 • 

5.3 . Theorems 5.1 and 5.2 can be extended to the situation considered 
in Corollary 4.2. 

6 . A signed set function ♦ . 

For simplicity of statements we assume in this section 

(6.1) ♦(ft) < - , 


18 



and put X ■ lim X 
— - n 


Definition 6.1 . Let , for A e G t 

(6.2) $ + (A) - sup{II5 E(X - X) + l. ; lim E|X,_ - X|l - $(A)} , 

c n ~ A c n A 

(i.e. , we consider only such sequences of bounded stopping times t R -► 00 
for which the condition holds ) . Let $ (A) - $(A) - $ (A) and 4>(A) ■ 

♦ + <A> - <j> (A). 

Clearly, 0 < ^ $ and |$| < <fi . 

Lemma 6.1 . <f> + , <p and $ are finitely additive on G . 

Proof . It suffices to prove this for <t> + . We remark that the sup in (6.2) 
is achieved. Let A, B be disjoint with A, B € 3^ . Let the increasing 
sequences of b.s.t. (s n > and (t n > satisfy s^ > k and E(X t - X) + 1 A ** 

<J> + (A), E|X - X|l A -► (A) and similarly for s^ and B . Let x^ » t^ 

n 

on A and ■ s on its complement. Then E|X t - x|l AuB ■* ♦(A) + <p<B) ■ 

n 

4>(AuB) while <fr + (A u B) > lim E |x - X|l AuB ■ 4> + (A) + , + (B). Starting 

n 

with b.s.t. satisfying E(X t - -^ +1 AuB ^ 4 > + (AuB) and EjX^ - —I ^AuB ^ 

n n n 

$(A u B) we obtain (see Remark 3.3) the opposite inequality. □ 

We need the following strengthened and extended version of Lemma 2.2. 

Lemma 6.2 . If the bounded stopping tiroes t -► • satisfy 

(6.3) E(X - X) + - 4> + («) . E|X - X| $(«) , 

c n n 


19 








then, for any sequence (A ) of sets with A e 3 (n e N) we have 
--- n- n t n 

(6.4) E(X - X)l. - $(A ) - 0 , EIX - X|l A ~ <f>(A ) ♦ 0 • 

C _ A Ii L A II 


In particular E(X t - X)1 A 1(A) for all AeG. 
n 

Proof . We recall that (6.1) is assumed. First we show that the second 

condition in (6.3) implies the second assertion in (6.4). Suppose 

lim(E|X^ - X11 A -(f) (A)) <0 , then there exists e > 0 and a subsequence 
1 — t — a n 

n n 

for which E|X - X|l A < 4 >(A ) - e . Without loss of generality we may 
t a n 

n n 

assume that this inequality holds for all n . Let be the complement 

of A . There exists s > t satisfying EIX - X| < <f>(B ) + e/2 . But 
n nn s — 1 n 

n 

then, putting t * t 1. + s 1- we have E|x - X| < <f>(A ) + 4 >(B ) - e/2 * 
n n A n is t — n n 

n n n 

<j>(Q) - e/2 which is impossible. If lim(E[X t - X|l A ~ <K A n )) > 0 then 

n n n 

lim E(|X t - x|l fi - <(>( B n )) < 0 which is again impossible. This establishes 
n n 

the second part of (6.4). 

The rest of the Lemma will follow if we establish E(X„ - X)*1 A - <J> + (A ) 

t — A n 

n n 

0 . Suppose, E(X - X) + 1 A > $ + (A ) + e then we can construct a b.s.t. 

c a n 

n n 

r for which E(X t - X) + > $ + (fl) + e/2 and E|X t - x| < E|X t - X|l A 
T n T n l n n 

+ $(B ) + 1/n which leads to a contradiction. The case lim (E(X - X) + 1 A " 

n _ c a 

n n 


$ + (A )) < 0 
n 


is treated by considering the sequence (B^) . 


□ 


Lemma 6.3 . If_ Y < C there exists an increasing sequence of bounded 

stopping times t^(Y) such that X^ ^ -► Y a.s. and 

n 


20 


I 







V W*'- 


(6.5) 


E(X t (Y) " Y)1 A - * (A) ’ 
n 


for every A e G . 

Proof. Let A n € G be such that P(A n > > 1 - 1/n and d(A n ) < 1/n 

(eee Lama 3.1). Let B n * G be a enbset of A^ and s„<»> be a b.e.t. 

each that P(B ) > 1 - 2/n 2 and E|X S (Y) - T|l < L/n Lem. 4.1). 

n n 

Let (t ) be a sequence of b.s.t. satisfying (6.3). We may assume that 
n 

(s (Y)) and (t ) are increasing and that A fl , B n £ 3 ( Y ) n * Put 

H tl tl It 

t (Y) -s n (Y)l B + t n l c where C n is the complement of B n . Then 
n n n n 

X t (Y) ■*> Y a.s. and 
n 

(6.6) E(X^ (Y) - Y)1 a - E(X g ^ (Y) - Y)!^ + E(Y-X)l AnCn 

+ E(X t - X)l An( , . 


The first two summands •* 0 . Also ^(AnC n ) “ ^(A) - |(AnB n ) , but ||(AnB n )| 

*(AnB ) s <t>(A ) . Thus ♦(AnC ) - $<A) and hence, by the preceding L'mma, 

Y n n 11 

the last summand in (6.6) -* $(A) . ^ 

An immediate consequence is 

Theorem 6.1 . If. lim E|xJ < - there exists for_e_very_ Y £ C an 
increasing sequence of bounded stopping times c n (^) such tha t^ 


(6.7) 


m 

EX m V + ETV + l X 6(A) , 
V Y) i-i 



m 

for every simple G- measurable random variable V - £ X,l. (A, € G. i«l,...m) 

i»l 1 1 1 

Corollary 6,1 . For all Y, Z e C and every simple G- measurable random 
variable V 

““ E< \ 1 (Y)- X t :| (Z)> V - E< ’ , - Z > V ' 

as i,j ■+• * independently of one another . 

Remarks 6.1 . There are, in general, many additive set functions $ 
for which Theorem 6.1 holds. E.g. if we replace lim by lim in (6.2) we 
obtain another, usually different , $ with the desired properties. 

6.2 . From the proof of Lenina 6.3 it is seen that the b.s.t. t (Y) 

n 

may be assumed to have simultaneously the properties described in Theorem 5.1 
and Theorem 6.1. 

6*3 . Theorem 6.1 is stated for the case that C i s not empty. If it 
is empty similar results hold for C z with ZeC . (See Corollary A.2 
and Remark A.3). 

7. Banach space valued random variables . 

In this section we consider vector r.v. Q + S where S is a fixed 
Banach space (not necessarily over the reals). 

Obviously nothing has to be changed in sections 2 and 3 beyond replacing 
the absolute value |*| by the norm | | • | | of S, Similarly C and C are 
defined by (A.l) with ||X n -X|| replacing |X r - X| . C is a closed set 
in the relevant 1^ space (of r.v. Y with E| |Y | | < <*») and p(Y t C) is 
defined as before. 


22 


i 







At this stage there does occur an important difference. The fact that 

C(||X ||) is not empty does not imply that C(X^) ^ 0. Indeed, every 

infinite dimensional Banach space contains points e^, tie N 9 of unit norm 

such that Me. - e, I I ^ 1 whenever i*1 . Then C(e ) = 0 whereas 
1 x j n 

C(||e ||) is the constant 1. This fact affects some of the results in 
sections 4 and 5. 

The approximation Lemma 4.1 reaiains valid (same proof). 

Theorem 4.1 has to be modified, but the Corollary 4.1 is not affected. 
Thus we have 

Theorem 7.1 . For all integrable random variables Y we have 

(7.1) *(8) < lim E | |X - Y| | < <KQ) + p(Y,C) , 

and if $(&) < <» the first inequality becomes an equality when , and only 

when , Y £ C # 

Proof, The first inequality (7.1) follows from <J>(*,(X )) * $(•*(X_ - Y)) 

n n 

The second inequality has to be proved only when C ^ 0 . If Xe C then 

lim E| |X t - Y| | < lira E( | | X fc - X | | + ||X-Y||) = lim E| lx - X| I + p(X,Y) but, 

by Lemma 4.1, lim E||X t - X|| < $(n) . 

It remains to prove that if <t>(Q) < ® then equality on the left implies 
Ye C. If Y i C then there exists c *> 0 such that eMx -YMl A > e 
for every A with P(A) > 1 - e and large t . Let A be the set described 
in Lemma 3.1 with e replaced by c/2 and denote by B its complement. 

Then Um E| |x - Y| | > lim E | |X - Y| 11 + lim E | |x - Y| |l > 0(B) + e > 

♦ (fl) + e/2. □ 


23 






Corollary 4.2 has to be modified in the same way as Theorem 4.1. 

Lemma 4.2 remains valid. 

The results on simultaneous approximation (Theorems 5.1 and 5.2) remain 
unchanged. 

The few reformulations of the results of sections 4 and 5 which were 
necessary in the general case are not needed when S is finite dimensional. 

It is not difficult to prove that for finite dimensional Banach spaces 
S , if Y € C(| |X ||) then there exists Z e C(X^) with | |Z|| = Y . 

Moreover, if S is a finite dimensional Banach space the results of 
section 6 also carry through. 

Theorem 7.2 . Let S be a finite dimensional Banach space and X^ 
be S-valued random variables . If lim E | | X | | < 00 then C(X^) _is not 
empty . Furthermore , there exists a finitely additive S -valued function 
<j> with domain G and for every Y € C there exists _a sequence of increas¬ 
ing bounded stopping times t_(Y) such that (6.7) holds for every scalar 

- n ——--- 

m 

valued simple random variable V = £ X. 1 A (A^ e G, i=l,...,m). 

i-1 1 A i 1 

The proof being similar to that of Theorem 6.1 we just show how to 

construct a suitable <j> . For brevity we do this for two-dimensional S 

* 11 

over the reals. Let (e. ,e 0 ) be a basis of S and X = X e. + X e 0 

1’ 2 n n 1 n 2 

with real X ? , x" . Let X f e. + X"e n € C be such that | |X f e. + X"e 0 | | = 
n n 1 2 11 1 2 ’ 1 

lim E | | X | | « X . Define, for A e G , 

4^(A) - supUiin E|x* - X* |l A ; Urn E| |X fc - X| jl A - <t»(A)} , 

n n 

_ M fl 

then let ^(A) * sup{lim|X t - X |; 1,11} where I is the condition used 

n » t 

in defining and II is the condition lim E |X^ - X 11^ * 4>^(A) . (If 


24 






S is stricly convex this step is not necessary since then ^ is 

+ - 1 • + 

determined by <p and <p^ .) Define <J>^(A) * sup{lim E(X t - X ) ; 1,11} 

n 

and similarly ^(A) . Let ^ = 2<J>+ - ^ (i-1,2) . $ = 4 ^ + $ e 2 

will have the required properties. 

Remark 7.1 . Theorem 7.2 fails in infinitely dimensional S , even 

if we add the requirement that C ^ 0 . Indeed, let e^ be points in S 

with | |ej | =1 and | |e^ - e_. | | > 1 for i * j . Let ft = [0,1 ] , P be 

the Lebesgue measure, X = 2 n e l rrk ~-n, and 5 * a(X, t ...,X ) . 

n n [0,2 ] n 1* n 

Then C is not empty but EX^, t •+ » , has no limit point in the 

norm topology. It is possible to obtain results similar to the above only 

if one either looks at weaker forms of convergence or imposes restrictions 

on the sequence X . 


25 


I 






References 


1. D. G. Austin, G. A. Edgar and A, Ionescu Tulcea: Pointvise convergence 
in terms of expectations. Z. f, Wahrscheinlichkeitsth. u. verw. Geb. 30 , 
17-26 (1974). 

2. J. R. Baxter: Pointwise in terms of weak convergence. Proc. American 
Math. Soc. 46, 395-398 (1975). 

3. J, R. Baxter: Convergence of stopped random variables. Advances in 
Math. 21, 112-115 (1976). 

4. A. Bellow: Submartingale characterization of measurable cluster points. 
Probability in Banach spaces. Advances in Probability and Related Topics 
4, 69-80 (1978). 

5. R. V. Chacon: A ’’stopped" proof of convergence. Advances in Math. 14 , 
365-368 (1974). 

6. P. Fatou: Series trigonomdtriques et series de Taylor. Acta Math. 30, 
335-400 (1906). 

7. W. D. Sudderth: A "Fatou Equation" for randomly stopped variables. 

Ann. Math. Stat. 42, 2143-2146 (1971). 


26 





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On the Fatou Inequality 


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Aryeh Dvoretzky 


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Department of Statistics 

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