Proofs of generality of martingale convergence theore

Chatterji, S. D.

Welsh 

NASA Technical Reports Server

COMMENTS ON T H E  M A R T I N G A L E  C O N V E R G E N C E  T H E O R E M  
( P r e l i m i n a r y  R e p o r t )  
S .  D .  C h a t t e r j i  
T E C H N I C A L  R E P O R T  N O .  2 1  
P R E P A R E D  U N D E R  R E S E A R C H  G R A N T  N O .  N s G - 5 6 8  
( P R I N C I P A L  I N V E S T I G A T O R :  T .  N .  B H A R G A V A )  
F O R  
N A T I O N A L  A E R O N A U T I C S  and  S P A C E  A D M I N I S T R A T I O N  
To b e  p u b l i s h e d  i n  t h e  P r o c e e d i n g s  o f  a N A T O  
C o n f e r e n c e  on P r o b a b i l i t y  and S t a t i s t i c s  h e l d  
i n  G r e e c e ,  A u g u s t ,  1 9 6 6 .  
D E P A R T M E N T  OF M A T H E M A T I C S  
K E N T  S T A T E  U N I V E R S I T Y  
GPO PRICE s 
CFSTI PRICE(S) $ 
NC;7 10R75 K E N T ,  O H I O  
! 
IACCESSIUN NUMBER) 5 
67 8
L 
> (CODE) 
t '~PAGES) 19 
ICAFEGORY) k TUX OR AD NUMBER)  
Hard copy (HC) // &@ O c t o b e r  1966 ' 
Microfiche (MF) ,P& 
ff 653 July 65 
https://ntrs.nasa.gov/search.jsp?R=19670001552 2020-03-16T17:44:33+00:00Z
. 
Comments on t h e  Martingale Convergence Theorem 
S. D .  Chat ter j i*  
L e t  (O,B,P) be a p r o b a b i l i t y  s p a c e  and l e t  X b e  a Banach 
s p a c e .  A s equence  o f  X-valued B o c n n e r - i n t e g r a b l e  random v a r i -  
ables  fn  on Q w i l l  be  s a i d  t o  form a m a r t i n g a l e  w i t h  r e s p e c t  
t o  t h e  s u b a l g e b r a s  An*n=1,2,  ..., A n C A n + l  ( i n  s h o r t  { f n , A n )  
i s  a m a r t i n g a l e )  i f  
A, 
A, 
11 where E i s  t h e  c o n d i t i o n a l  e x p e c t a t i o n  o p e r a t o r  w i t h  r e s p e c t  
t o  t h e  a-algebra A n .  
d e f i n e d  f o r  a r b i t r a r y  Banach-space-valued i n t e g r a b l e  f u n c t i o n s .  
I n  t h e  f o l l o w i n g  i t  w i l l  be assumed tha t  t h e  a lgebra  A= f An n-1 
g e n e r a t e s  t h e  a-algebra 8. The g e n e r a l  c a s e  can  be  h a n d l e d  v i a .  
It i s  known t h a t  t hese  o p e r a t o r s  are w e l l -  
s t a n d a r d  r e d u c t i o n  t o  t h i s  c a s e .  
My main conce rn  w i l l  b e  p r o v i n g  a lmos t  everywhere  ( a . e . 1  
convergence  theorems f o r  m a r t i n g a l e s .  F o r  t h e  sake o f  b r e v i t y ,  
I s h a l l  l i m i t  m y s e l f  i n  t h i s  t a l k  t o  c o n s i d e r i n g  o n l y  t h e  
f o l l o w i n g  s t a t e m e n t s :  
A 
(S1) I f  f n = E  "f t h e n  l i m  fn  = f a . e .  ( s t r o n g  l i m i t  i n  X); n + m  
( S 2 )  I f  { f n , A n )  i s  a m a r t i n g a l e  and t h e  f n ' s  are  u n i f o r m l y  
i n t e g r a b l e  ( i . e .  lim n- 
u n i f o r m l y  i n  n > 1) t h e n  3 f m  such  t h a t  l i m  f n=fm a.e . ;  
f 11 fnII I{II f n I I >  N )  dP  = 0 
n- - 
( S 3 )  I f  { f n , A n )  i s  a m a r t i n g a l e  w i t h  s u  
t h e n  3fm such  t h a t  l i m  f n = f m  a . e .  
E (  I (  f n I I )  < m n > P  - 
n- 
*Work p a r t i a l l y  s u p p o r t e d  by NASA Resea rch  Gran t  No. NsG-568 
a t  Kent S t a t e  U n i v e r s i t y .  
2 
I n  s e c t i o n  2 I s h a l l  prove t h a t  (S1) i s  a lways  t r u e .  I n  
t h i s  g e n e r a l i t y  t h e  r e s u l t  i s  proved  by o t h e r  methods i n  
C h a t t e r j i  ( 2 b )  and a l s o  i n  A . I .  and C . I .  T u l c e a  ( 6 ) .  The p r e s e n t  
p r o o f ,  p a r a l l e l l i n g  t h e  p r o o f  i n  t h e  s c a l a r - v a l u e d  c a s e  as i n  
B i l l i n g s l e y  (1) , i s  as s i m p l e  ( p o s s i b l y ,  some would wi sh  t o  s a y ,  
t r i v i a l )  as one c o u l d  wish  f o r .  
I n  s e c t i o n  3 I s h a l l  p rove  t h e  main theo rem o f  t h i s  p a p e r ,  
v i z . , t h a t  i f  X s a t i s f i e s  t h e  f o l l o w i n g  ( R N )  c o n d i t i o n  ( R N  f o r  
Radon-Nikodym) t h e n  ( S 3 )  ( and  hence t r i v i a l l y  ( S 2 ) )  i s  v a l i d  f o r  
a l l  m a r t i n g a l e s .  The c o n d i t i o n  r e f e r r e d  t o  i s :  
( R N ) :  Every a - a d d i t i v e  X-valued s e t  f u n c t i o n  1.1 on 8 o f  
bounded v a r i a t i o n  w i t h  t h e  p r o p e r t y  t h a t  V u ,  t h e  v a r i a t i o n  of  1.1, 
i s  a b s o l u t e l y  c o n t i n u o u s  w i t h  r e s p e c t  t o  P ( V , < < P )  can  be 
r e p r e s e n t e d  as t h e  i n d e f i n i t e  i n t e g r a l  ofan X-valued Bochner- 
i n t e g r a b l e  f u n c t i o n .  
The non-negat ive  measure V, i s  d e f i n e d  as f o l l o w s :  
Y 
The i m p l i c a t i o n  ( R N e ( S 3 )  (more p r e c i s e l y  X has RN) p r o p e r t y  
w i t h  r e s p e c t  t o  B and P i m p l i e s  ( S  ) )  i s  more g e n e r a l  t h a n  t h e  
s t a t e m e n t s  o b t a i n a b l e  f rom ( 6 ) .  
v a l i d  f o r  r e f l e x i v e  X o r  s e p a r a b l e  dua l  s p a c e s  X ,  s t a t e m e n t s  
e x p l i c i t l y  made i n  ( 6 ) .  F o r  r e f l e x i v e  X ,  ( S 2 )  (weaker t h a n  
( S 3 ) )  was p roved  by d i f f e r e n t  methods i n  ( 2 a , b )  and by S c a l o r a  ( 5 ) .  
That some c o n d i t i o n  on X i s  n e c e s s a r y  f o r  t h e  v a l i d i t y  o f  ( S 2 )  
or ( S  ) i s  d e m o n s t r a t e d  by t h e  counterexample  i n  ( 2 a ) .  Here a 3 
m a @ t i n g a l e  f n  i s  c o n s t r u c t e d ' w h i c h  takes v a l u e s  i n  L (0:l) and 
which does n o t  converge  i n  any s e n s e ,  xeak  o r  s t r o n g ,  anywhere,  
3 
It a l s o  f o l l o w s  t h a t  (S3)  i s  
1 
-. 3 
L 
a l t h o u g h  among o t h e r  t h i n g s ,  1 1  fnII 1 for a l l  n > 1. - 
I n  s e c t i o n  4 it i s  shown t h a t  t h e  ( R N )  c o n d i t i o n  i s  a l s o  
n e c e s s a r y  if 8 i s  s e p a r a b l e  ( g e n e r a t e d  by a denumerable  c l a s s  
o f  s u b s e t s ) .  
are e q u i v a l e n t  c o n d i t i o n s .  
More p r e c i s e l y ,  i n  t h i s  c a s e  ( S 2 ) ,  ( S 3 )  and (RN) 
S e c t i o n  2 :  The main p r o b a b i l i s t i c  t o o l  i s  t h e  f o l l o w i n g  lemma: 
Lemma 1: For any m a r t i n g a l e  { f n , A n } ,  if A E A ~  and E > 0 t h e n  
7 0 I s' 1 1  f k l l > E 3  - E [ ~ ~ ~  
k>go - k>nb I l fklbpl '  
The lemma i s  known and an easy consequence o f  t h e  f a c t  t h a t  
1 1  f, 1 1  i s  a s u b m a r t i n g a l e .  
Theorem 1 (S1) : F o r  any space  X ,  l i m  E 
S k e t c h  o f  p r o o f :  I f  f i s  measurable  A =  An t h e n  ( S 1 )  i s  
A 
f = f a .e .  ( P ) .  
n- 
m 
n-1  
t r i v i a l  s i n c e  EA, f = f f o r  s u f f i c i e n t l y  l a r g e  n .  
For a g e n e r a l  f ,  lf, measurab le  i n  A such  t h a t  
E ( l l f  - f ,  1 1  < E .  
coup led  w i t h  l e m m a  1 l e a d s  us t o  (S1) q u i t e  smoo th ly .  
S e c t i o n  3: Given a m a r t i n g a l e  { f n , A n } ,  d e f i n e  t h e  s e t -  
f u n c t i o n s  on An as f o l l o w s :  
 AI = JAfn d P *  
The m a r t i n g a l e  p r o p e r t y  i s  e q u i v a l e n t  t o  t h e  p r o p e r t y  t h a t  
m 
l i m  p ( A ) = p ( A )  e x i s t s .  The s e t - f u n c t i o n  p on A i s  an  X-valued 
f i n i t e l y  a d d i t i v e  s e t - f u n c t i o n .  The s e t - f u n c t i o n  i s  o f  bounded 
n 
4 
. 
v a r i a t i o n  iff s u  JII fnII d P  < m. The main d i f f i c u l t y  i n  p r o v i n g  
n>B - 
m a r t i n g a l e  convergence  theorems i s  t h a t  IJ may n o t  be a - a d d i t i v e .  
The f o l l o w i n g  lemma g i v e s  a way o u t .  
Lemma 2 :  L e t  P be  a p r o b a b i l i t y  measure on an  a l g e b r a  A of 
s u b s e t s  of a s p a c e  n and p a f i n i t e l y  a d d i t i v e  X-valued s e t  
f u n c t i o n  o f  bounded v a r i a t i o n  on A .  Then 
where ,,a are  b o t h  o f  bounded v a r i a t i o n  and r7 i s  a f i n i t e l y  
a d d i t i v e  s e t - f u n c t i o n  such  t h a t  V, ( t h e  V a r i a t i o n  of r l )  1s 
s i n g u l a r  w i t h  r e s p e c t  t o  P ( i . e .  g i v e n  E, 6 > 0 , 3 A  E A ,  P(A)<E 
V, (A') < 6) and a i s  a a - a d d i t i v e  s e t - f u n c t i o n  such  t h a t  V, 
i s  a b s o l u t e l y  c o n t i n u o u s  w i t h  r e s p e c t  t o  P ( i . e .  g i v e n  E > O  
3 b o ,  P(A)<G*v~(A)< 
T h e  main i d e a  b e h i n d  t h e  p r o o f  o f  t h e  lemma w i l l  b e  s k e t c h e d .  
One t r a n s f e r s  P and 1.1 t o  t h e  space  (S,c,) where S i s  a t o t a l l y  
d i s c o n n e c t e d  compact Hausdorf f  s p a c e  and 1 i s  t h e  a l g e b r a  o f  
b e i n g  i somorph ic  t o  A .  It t u r n s  o u t  t h a t  c l o p e n  s e t s  i n  S ,  
P and 1.1 are  a - a d d i t i v e  on 1 and hence  can  be  e x t e n d e d  t o  t h e  
a - a l g e b r a  1 g e n e r a t e d  b y  ll. (These  are  s t a n d a r d  methods i n  
t h i s  s o r t  of work. See e . g . ( 3 ) , p p .  312-13).  On these  e x t e n d e d  
measures  on 1 
proved  by R i c k a r t  ( 4 )  and one r e t r a c e s  t h e  way back t h r o u g h  cl 
1 
4 
1 
2 
one a p p l i e s  t h e  Lebesgue decompos i t ion  theo rem as 
2 
t o  A t o  o b t a i n  t h e  decomposi t ion  i n d i c a t e d  i n  t h e  l e m m a .  
With t h e  h e l p  o f  l e m m a  2 ,  I s h a l l  now p rove  t h e  main theo rem 
o f  t h i s  t a l k .  
Theorem 2 :  If X s a t i s f i e s  t h e  ( R N )  p r o p e r t y  w i t h  r e s p e c t  t o  P 
on 8 t h e n  any m a r t i n g a l e  { f n ,  A n )  w i t h  s u p  JIlfnII dP  conve rges ;  n.2 1 
5 . *  
L 
i . e .  Ifw such  t h a t  l i m  fn  = f, a .e .  ( P ) .  
n- 
S k e t c h  o f  t h e  P r o o f :  
1-1 r e s t r i c t e d  t o  An i s  an i n t e g r a l .  
w i t h  r e s p e c t  t o  P ,  i s  also a n  i n t e g r a l  s i n c e  X has t h e  (RN) 
p r o p e r t y .  
L e t  u be as b e f o r e  and o ,  u as i n  l e m m a  2 ,  
u ,  b e i n g  a b s o l u t e l y  c o n t i n u o u s  
L e t  a ( A )  = I h dP ,  A E A,  
A 
and  U ( A )  = u n ( A )  = hn d P , A  E An 
A 
I 
n A 
C l e a r l y  hn =E h .  
Hence TI r e s t r i c t e d  t o  An i s  also an i n t e g r a l ,  1 . e .  
T I ( A )  = on(A) = lAgn  dp, A E An 
I n  o t h e r  words, 'n - gn + hn, 
where gn ,h  
Moreover hn i s  E 
a re  a l s o  m a r t i n g a l e s  w i t h  r e s p e c t  to A n .  n 
An h and hence  theorem 1 e n s u r e s  t h e  convergence  
o f  hn t o  a l i m i t .  
I s h a l l  now show t h a t  l i m  gn = 0 a . e .  ( p ) .  
Given 1 > E ,  6 > 0 ,  f i n d  A E A ( a n d  hence  A E A 
such  t h a t  P ( A ' )  t V, ( A )  < %. 
f o r  Some n o )  
"0 
b Now 
T h i s  i s  c l e a r l y  
S e c t i o n  4 :  
nough t o  show t h a  
I gn 11 dP (by l e m m a  1) 
- - + - < 6 .  6 E  6 
2 2  
l i m  gn = 0 a . e .  ( P ) .  
I n  t h i s  s e c t i o n ,  t h e  main t h i n g  i s  t h e  f o l l o w i n g  
lemma for r e a l - v a l u e d  s u b m a r t i n g a l e s  : 
Lema 3: ~ If {gn,  A n )  i s  a p o s i t i v e  s u b m a r t i n g a l e  w i t h  s u p  E (g,) < w n7 - 
6 
W 
A E A =  l A n y  
n- n = l  
such  t h a t  p ( A )  = 
i s  a a - a d d i t i v e  P-cont inuous s e t - f u n c t i o n  ; t h e n  t h e  grits are  
u n i f o r m l y  i n t e g r a b l e .  
S k e t c h  of P r o o f :  If gn 2 0 i s  a m a r t i n g a l e  t h e n  i t  i s  easy .  
I n  g e n e r a l  3 h n  a m a r t i n g a l e  such  t h a t  0 - < g, 5 hn and 
s u c h  t h a t  {h,} i n d u c e s  t h e  same 1.1. Hence t h e  lemma. 
Theorem 3: If I3 i s  s e p a r a b l e  t h e n  ( S 2 e ( R N ) .  Hence i n  t h i s  c a s e  
( s ~ ) ~ ( s ~ % R N )  
S k e t c h  o f  P r o o f :  
0-algebra g e n e r a t e d  by A1,A2 ,.. . , A n .  
s a t i s f y i n g  t h e  c o n d i t i o n s  i n  ( R N ) ,  t h e  m a r t i n g a l e  { f n y  A n )  
i n d u c e d  by 1.1 i s  such  t h a t  11 fnII s a t i s f i e s  t h e  c o n d i t i o n s  o f  
lemma 3. 
Le t  B be g e n e r a t e d  by A 1 , A 2 , .  . . and An e q u a l  t o  t h e  
Given a s e t - f u n c t i o n  u on 8 
Hence ( S 2 )  i m p l i e s  t h e  convergence  o f  f n  t o  f w .  From 
h e r e  o n i t  i s  t r i v i a l  t o  show t h a t  v i s  t h e  i n d e f i n i t e  i n t e g r a l  
o f  f m .  
Note :  I n  t h e  r e a l - v a l u e d  c a s e  t h e  g e n e r a l  m a r t i n g a l e  convergence  
theorem S 3  can  be  deduced r a p i d l y  from S1 by t h e  f o l l o w i n g  
sequence  of  a rgumen t s :  
( I )  (S,)c/(S2) because  fn  u n i f o r m l y  i n t e g r a b l e  i m p l i e s  
t h a t  3 n  s u c h  t h a t  f + f w e a k l y  i n  L1 f o r  some f .  
EAnf w e a k l y .  But EAnf = f n  f o r  Hence E A n f  
l a rge  \. 
K nk 
k + z  nk nk 
Hence EAnf = f n ,  e t c .  
(11) e v e r y  u n i f o r m l y  i n t e g r a b l e  s u b m a r t i n g a l e  c o n v e r g e s :  
t h i s  f o l l o w s  f rom ( I )  v i a  t h e  Doob-decomposition f o r  sub- 
mart i n g a l e s  . 
(111) Every p o s i t i v e  m a r t i n g a l e  fn  conve rges  s i n c e  e-fn i s  
a u n i f o r m l y  bounded s e m i m a r t i n g a l e .  
7 
(IV) An a r b i t r a r y  m a r t i n g a l e  fn w i t h  s u p  E fi < m converges  
because  i t  i s  t h e  d i f f e r e n c e  o f  two p o s i t i v e  m a r t i n g a l e s  and 
(111). 
From here t h e  same theorem f o r  s u b m a r t i n g a l e s  can  also b e  e a s i l y  
o b t a i n e d .  
a 
Refe rences  
111 B i l l i n g s l e y ,  P . ,  E r g o d i c  Theory and I n f o r m a t i o n ,  John  Wiley 
and Sons I n c .  (1965). 
[ 2 1  C h a t t e r j i ,  S. D . ,  ( a )  M a r t i n g a l e s  o f  Banach-Valued Random 
V a r i a b l e s .  B u l l .  American Mathematics  S O C .  V o l .  66 
(1960)  pp .  395-3980 (b) A 1 dote  on t h e  Convergence of 
Banach-Space Valued M a r t i n g a l e s .  Ma themat i ca l  Annalen 
Vol. 153  ( 1 9 6 4 )  p p .  142-1119. 
[3 ]  Dunford,  N. and Schwar tz  J .  T . ,  L i n e a r  O p e r a t o r s ,  P a r t  I ,  
I n t e r s c i e n c e ,  N .  Y .  ( 1 9 5 8 ) .  
[ 4 ]  R i c k a r t ,  C .  E., Decomposi t ion of  A d d i t i v e  S e t  F u n c t i o n s ,  
Duke Math. J r .  V o l .  1 0  ( 1 9 4 3 )  pp .  653-665.  
[ 5 ]  S c a l o r a ,  F. S., A b s t r a c t  M a r t i n g a l e  Convergence Theorems,  
P a c i f i c  Jr. Math. V o l .  I1 (1961) pp.  347-374. 
[ 6 ]  T u l c e a ,  A .  I. and T u l c e a  C .  I . ,  A b s t r a c t  E r g o d i c  Theorems,  
T r a n s .  Am. Math. S O C . ,  Vol .  1 0 7  ( 1 9 6 3 )  pp .  1 0 7 - 1 2 4 .  


Comments on the Martingale Convergence Theorem Technical Report No. 21

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server