s. v. astashkin- extraction of subsystems "majorized" by the rademacher system

8/3/2019 S. V. Astashkin- Extraction of Subsystems "Majorized" by the Rademacher System

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M a t h e m a t i c a l N o t e s , V o l . 6 5 , N o . ~ , 1 9 9 9

E x t r a c t i o n o f S u b s y s t e m s " M a j o r i z e d " b y t h e R a d e m a c h e r S y s t e mS . V . A s t a s h k i n U D C 5 17 .9 82 .2 7

COA B S T R A C T . I n t h i s p a p e r i t is p r o v e d t h a t f r o m a n y u n i fo r m l y b o u n d e d o r t h o n o r m a l s y s t e m { f n ) n = t o fCOr a n d o m v a r i a b le s d e f i n e d o n t h e p r o b a b i l i t y s p a c e ( f l, E , P ) , o n e c an e x t r a c t a s u b s y s t e m { f n i } i = t m a j o r i s e d

i n d i s t r i b u t i o n b y t h e R a d e m a c h e r s y s t e m o n [0 , 1 ]. T h i s m e a n s t h a t

P ~ r a , / . , ( ~ ) > z < c t z [ 0 1 ] : y ~ a , , , ( t ) > ~ ,'i-=--1

w h e r e ( 7 > 0 i s i n d e p e n d e n t o f r n E I~ 1, a i E R ( i = 1 , . . . , m ) a n d z > 0 .K ~ .Y W OR D S: s y s t e m s o f r a n d o m v a r i a b l e s , p r o b a b i l i t y s p a c e , B a n a c h s p a c e , m a j o r i z a t l o n b y t h e R a d e m a c h e rs y s t e m , o r t h o n o r m a l s y s t e m o f r a n d o m v a r ia b l es , H o l m s t e d t ' s f o r m u l a , e q ui v a l e n ce p ri n c i p l e f o r d i s t r i b u t io nf u n c t i o n s .

I n t r o d u c t i o nS u p p o s e t h a t r i ( t ) = s i g n s in 2 1-1 ~r~ ( i = 1 , 2 , . . . ) i s t h e s y s t e m o f R a d e m a c h e r f u n c t i o n s o n [ 0, 1 ].

I n w h a t f o ll o w s , w e s h a l l s a y t h a t a s y s t e m ( f i } ~ = l o f r a n d o m v a r ia b le s d e f i n e d o n t h e p r o b a b i l i ty s p a c e( f t , ~ , P ) i s m a j o r i z e d i n d i s t r ib u t i o n b y t h e R a d e m a c h e r s y s t e m i f t h e r e e x i s t s a c o n s t a n t C > 0 s u c ht h a t f o r a r b i t r a r y m E N , a i E R ( i = 1 , . . . , m ) , a n d z > 0 w e h a v e

( f r o m n o w o n , le [ is t h e L e b e s g u e m e a s u r e o f t h e s e t e C [ 0 , 1 ]) .S u p p o s e t h a t {f~,},,~176 is a u n i f o r m l y b o u n d e d o r t h o n o r m a l s y s t e m o f r a n d o m v a r i a b l e s o n t h e p r o b a -

b i li t y s p a c e ( f t , ~ , P ) . I t is w e ll k n o w n ( s e e , f o r e x a m p l e , [ 1, p . 2 8 7 o f t h e R u s s i a n t r a n s l a t i o n ] ) t h a t f r o ms u c h a s y s t e m w e c a n e x t r a c t a s u b s y s t e m { / ,~ , ~ x t h a t i s a n S p - s y s t e m s i m u l t a n e o u s l y f o r a l l p < c o .T h i s m e a n s t h a t

~ _ ~ a d , , , < a" i=X P " i=1

w h e r e K p > 0 i s i n d e p e n d e n t o f m E N a n d a i E R ( i = 1 , . . . , m ) .T h e m a i n r e s u l t o f th i s p a p e r i s a s t r e n g t h e n i n g , i n a c e r ta i n s e n s e , o f t h e s t a t e d a s s e r t i o n . I t w ill

b e p r o v e d t h a t f r o m a n y u n i f o r m l y b o u n d e d o r t h o n o r m a i s y s t e m o f r a n d o m v a r ia b l es o n e c an e x t ra c t as u b s y s t e m t h a t is m a j o r i z e d i n d i s tr i b u ti o n b y t he R a d e m a c h e r s y s t e m .

T o p r o v e t h i s r e s u l t , w e n e e d s o m e a s s e r t i o n s t h a t a r e o f i n t e r e s t i n t h e m s e l v e s . F i r s t , l e t u s r ec a ll af e w d e f i n i t i o n s .

I f X i s a r a n d o m v a r i a b l e d e f i n e d o n t h e p r o b a b i l i t y s p a c e ( f t , E , P ) , t h e n n l x l ( z ) = P { w E f l :[ x @ ) l > z } a n d X * ( t ) ( t ~ [ 0, 1 ]) i s a n o n i n c r e a s in g l e f t - c o n t in u o u s r e a r r a n g e m e n t o f t h e f u n c t i o n I x I .A s is w e n k n o w n [ 2, p . 8 31 , I X I a n d x* a r e e q u i m e a s u r a b l e , i . e ., f o r a l l z > 0 w e h a v e n l x l ( z ) = n x . ( z ) .T h e B a n a c h s p a c e E o f r a n d o m v a r i a b le s X = X ( w ) o n ( f~ , Z , P ) i s ca l l ed s y m m e t r i c i f

1 ) f r o m o f t h e f a c t t h a t ] X [ < IY I a l m o s t e v e r y w h e r e a n d Y E E , i t f o l lo w s t h a t X E E a n dIlXll < IIYII;2 ) f r o m o f t h e f ac t t h a t n l x l ( z ) = = l r t ( z ) ( z > 0 ) a n d Y E E , i t f o ll o w s t h a t X E E a n d l lXl l = I l r l l .

T r a n s l a t e d f r o m M a t e m a t i c h e s k i e Z a m e t k l , Vo l . 6 5 , No . 4 , p p . 4 8 3 - 4 9 5 , Ap r i l , 1 9 9 9 .O r i g i n a l a r t i c l e s u b m i t t e d A p r i l 2 7, I9 9 8 .

0 0 0 1 - 4 3 4 6 / 9 9 / 6 5 3 4 - 0 4 0 7 5 2 2 . 0 0 ( ~) 19 99 K | u w e r A c a d e m i c / P | e n u m P u b l i s h e r s 40 7

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I n a d d i t i o n t o L p - s p a c e s ( 1 O : f n s ( [ X ( u ~ ) l ) d P ( w ) < l } ,t h e M a r c i n k i e w i c z sp a c e M ( ~ o) :

{ 1 for }I ~ l l M ( ~ ,) = su p ~ X * ( s ) d s : O < t _ < l ,

t h e L o r e n t z s p a c e A p (~ o):f l ~ x/p[[z lls *'P = { ] o ( X ' ( s ) ) " d~o(s)~ .

H e r e S ( t ) > 0 i s a c o n v e x c o n t i n u o u s f u n c t i o n o n [0 , o o ) a n d ~ o ( t) > 0 i s a c o n c a v e i n c r e a s i n g f u n c t i o no n ( 0 , 1 1

A n i m p o r t a n t r o l e i n t h e t h e o r y o f i n t e r p o l a t io n o f o p e r a t o r s i s p l a y e d b y t h e P o e t r e / c - f u n c t i o n a l ( s e e ,f o r e x a m p l e , [ 3 ] ) :/ C ( t , x ; E o , E l ) --- i n f { l l ~ o l l s o + t l l ~ , l l E , : z = z o + z x , ~ i ~ E i ) ,

w h e r e E 0 a n d E 1 a r e s y m m e t r i c s p a c es , z E E 0 + E 1 , t > 0 . I t is r e a d i ly s h o w n t h a t f o r a f ix e dz E E o + E 1 t h e / C - f u n c t i o n a l i s a c o n c a v e i n c r e a s i n g f u n c t i o n o f t [3 , p . 5 5 o f t h e R u s s i a n t r a n s l a t i o n ] .

I n w h a t f o l lo w s , t h e e x p r e s s i o n F 1 F ~ in d i c a t e s t h a t f o r s o m e C > 0 w e h a v e t h e i n e q u a l i t yO - 1 F ~ < F 1 < O F 2 ; m o r e o v e r , a s a r u le , t h e c o n s t a n t C is i n d e p e n d e n t o f a ll o r p a r t o f t h e a r g u m e n t so f F 1 a n d F 2 . I f 1 < p < e o a n d X i s a r a n d o m v a r ia b le o n ( f / ~ , P ) , t h e n

I J X l lp ~ { L lX (o ~ ) lP P ( o ~ ) x / p ',b u t i f a = ( a , ) i = l , t h e n oo ~ l /p

i = 1

~ 1 . T h e / C - f u n c t i o n a l o n t h e R a d e m a c h e r s u m sI n [41 ( s e e a l s o [ 51 ) t w a s s h o w n t h a t f or f u n c t i on s o f t h e f o r m

ooa ooz C t ) = ~ a , r , C t ) a = ( , ) , - - , ~ e 2 , ( 1 )

i = 1

t he f o l l ow i ng r e l a t i on i s va l i d :/C(t , z ; L ~ , G ) 1 1,~ 2) ,

w h e r e G i s t h e c l o s u r e o f L o o i n t h e 0 r l i c z s p a c e L N , N ( t ) = e 2 - 1 . W e o b t a i n a s i m i l a r r e l a t i o n f o rt h e E - f u n c t i o n a l / C (t , z ; L 1 , L o o ) .

T h e o r e m 1 . The fo l lowing re la t ion i s va l id :

U( 2 )

wh e r e t h e c o n s ta n t s a r e i n d e p e n d e n t o f a = ( a i ) i = l a n d 0 < u

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P r o o f . F o r a = ( a , ) i = l e t2 w e s e t t a a ( t ) = K : ( t , a ; e l , e 2 ) . I n [ 6 ] i t w a s s h o w n t h a t f o r f u n c t i o n s o ft h e f o r m ( 1 ) t h e f o l lo w i n g i n e q u a l it i e s h o l d :

, ~ l . l ( ~ o ( t ) ) _ < 2e-e/~n l . l ( A - * ~ p ~ ( t ) ) A - l e - A '2 '

( 3 )( 4 )

w h e r e A > 1 is i n d e p e n d e n t o f a a n d t > 0 .I t f o ll o w s f r o m i n e q u a l i t y ( 3) b y t h e d e f i n i ti o n o f t h e r e a r r a n g e m e n t z * ( 3 e - t 2 /2 ) < 9 , ( t ) o r a f t e r t h es u b s t i t u t i o n s = 3e - t2 /2 t h a t

0 < s < l .H e n c e / o i "9 " ( s ) d . _ < ~

O n t h e o t h e r h a n d , i n e q u a l i ty ( 4) y ie l dst a ~ ( i n l P ! ) d s , O A - l ~ , ~ ( A - ' / 2 1 n l / 2 ( A - l s - X ) ) , 0 < s < A - x .S i n ce t h e f u n c t i o n ~ a ( u ) i s c o n c a v e , w e h a v e

T h e r e f o r e , f o r a l l 0 z* (s ) ds _> 3 - ' A -S /Z V . In*~ '- -t3 a t .T h u s w e h a v e z * ( s ) d s ~ lnX/~ 3 d s ,

9 O 0w h e r e t h e c o n s t a n t i s i n d e p e n d e n t o f a = ( a , ) i = l a n d 0 < u _< 1 ,L e t u s n o w p r o v e t h a t

F i r s t , s u p p o s e t h a t u = 3 - k , k E N . S i n c e ~ a ( t ) i s c o n c a v e , w e c a n w r i t e

( 5 )

( 6 )

o o 3 - i

: ~ f a ~ ' ( ln 1 / 2 3 )i = k - i - t ~ d s Z T a ( ~ 3 ) 3 - i 2 T ~ ( v / i ) 3 - i "i = k i = k ( 7 )A t t h e s a m e t i m e , w e h a v e

o o k 3 J

i = k j = 0 i - = - k 3 JT ~ ( v / i) 3 - i ~ T ~ ( v ~ 3 J / 2 ) 3 - k a J .

. i = 0

S i n ce t h e l a s t s u m i s g r e a t e r t h a n T ~ ( v ~ ) 3 - k a n d l es s t h a nc o

~ ( v ~ ) 3 - k Z 3 J/ 2+ k -k 3 i < ~ ( v / ' k ) 3 - k Z 31-J/2

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i t fo l low s t ha t f o r u = 3 - k , k E N , r e l a t i o n ( 7 ) im p l i e s ( 6 ) .I f u E ( 0 , 1] is a r b i t r a r y , c h o o s e k E N s o t h a t 3 - k < u < 3 - k + l .

a r e c o n c a v e , w e h a v e

[ ( ) /oOa h 1 1 /2 3 ds a-ka n d r e l a t i o n ( 6 ) i s p r o v e d .

I t fo l lo w s f r o m ( 5 ) a n d ( 6 ) t h a t

T h e n , s i n c e t h e f u n c t i o n s i n q u e s t i o n

I n v i e w o f o f t h e w e l l - k n o w n r e l a t i o n ( s ee , f o r e x a m p l e , [ 3, p . 1 42 o f t h e R u s s i a n t r a n s l a t i o n ] )/ /C ( u , ~ ; L ~ , L ~ ) _ _ _ ~ ' ( ~ ) d . ,J

t h e l a s t r e l a t i o n is e q u i v a l e n t t o (2 ) , a n d t h e t h e o r e m is p r o v e d . [ ]A s u f fi c ie n t ly g o o d a p p r o x i m a t i o n t o t h e f u n c t i o n ~ a ( t ) - ] C (t , a ; l l , s i s g i v e n b y t h e w e l l- k n o w n

H o l m s t e d t f o r m u l a [ 7 ] [ t2 l o o 1 / 2i= l i = [ t 2 ] - S r l

w h e r e ( a * ) i s a n o n l n c r e a s i n g r e a r r a n g e m e n t o f t h e s e q u e n c e ( ] a i [ ), a n d [z] i s t h e i n t e g r a l p a r t o f th en u m b e r z .

T h u s w e o b t a i n t h e f o ll o w in g r e su l t .C o r o l l a r y 1 . T h e f o l l o w i n g r e l a t io n i s v a li d :

* ( a ' ) : .E i = I a i r i ( t ) , d t x ~ ~.= a i + h l l / 2 ~ i -[ 1 - ] + 1

OQw h e r e t h e c o n s t a n t s a r e i n d e p e n d e n t o f a = ( a i ) i = l E ~ 2 a n d 0 0 w e h a v e

l C ( u , z ; E o , E i ) = u l C ( 1 , z ; E 1 , E o ) ,i t fo l lo w s f r o m r e l a t i o n ( 2 ) t h a t

) c ( u , z ; L = , n l ) v o ( l n 1 /2 3 u ) , u > 1 ,o r e q u i v a l e n t l y ) c ( u , z ; L = , L 1 ) v a ( ln X / 2 ( i + u '~ )) , u > 1 .B u t i f 0 < u < 1 , t h e n i t f o ll o w s f r o m t h e d e f i n i ti o n o f t h e ) C - f u n ct i o n al , a s w e l l a s f r o m t h e i n e q u a l i t i e sHZ{{1 _ ~ [[Z[[o o, [Jail2 _< [[a[[1, a n d I12~[]1< ~ H all2

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R e m a r k 2 . I n e x a c t ly t h e s a m e w a y , f or f u n c t io n s o f t h e f o r m (1 ) w e c a n p r o v e t h a tl C ( u z ; L p , L o ~ ) a ; ~ l , t 2 ) , 0 0 , ( 9 ' )w h e r e 1 < p < o o ( n a t u r a l l y , t h e c o n s t a n t s o f t h e s e e q u i v a l e n c e s d e p e n d o n p ) .

T h i s i m p l i e s t h e f o l l o w i n g r e s u l t .C o r o l l a r y 2 . The fo l lowing equ iva lence i s va lid :

9 OOw h e r e t h e c o n s t a n t s a r e i n d e p e n d e n t o f a = ( a , ) i = l and 0 < u < 1 .w O n c o m p a r i n g d i s t r i b u t i o n f u n c t i o n s

I n w h a t f o l lo w s , w e s h a l l n e e d a v e r s i o n ( p r o v e d i n [ 9]) o f t h e e q u i v a l e n c e p r i n c i p l e f o r d i s t r i b u t i o nf u n c t i o n s .

I t s p r o o f is e x a c t l y t h e s a m e a s t h a t o f t h e p r i n c ip l e i ts e f f ( s ee [ 9]).P r o p o s i t i o n 1 . Supp ose tha t X > 0 and Y >_ 0 are two random var iab les de f ined , in genera l , on

9 O 0 .d i f fe ren t p robab i l i t y spaces , an d { X , } i = x an d { I ~ } i = I are indep ende n t "cop ies" o f X and Y , respec tive ly,( i . e . , n x , ( z ) = n x ( z ) a n d n ~ ( z ) = ~ r C z ) ) . W e ~ , s u m e ~ a t

I 1 , ~ = ~ 1 1 1 < e l l l m a x X ' I I 1 , ( 1 0 )- - . . . - - i= l , . . . , nI I m a x X i l[ 2 < C ~ I I m a x x , l l~ , ( 1 1 )i=1 , . . . , n - - i=1 , . . . , n

where C1 > 0 an d C2 > 0 are indep ende n t o f n E N .Th en there ez i s t s a C > 0 depend ing on ly on C1 and C2 such tha t fo r a l l z > 0

L e t u s s h o w t h a t c o n d i t i o n s ( 10 ) a n d ( 11 ) c a n b e s t a t e d u s i n g th e n o t i o n o f a E - f u n c t i o n a l .P r o p o s i t i o n 2 . W e a s s u m e t h at { X , } i = I i s a sequence o f indepe nden t "cop ies" o f a random var iab le

X >__ 0 on the p rob ab ility space (12, ~ , P) 1

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S i n c e t h e f u n c t i o n s X ( w x ) ( w a E l'l) a n d X * ( t ) ( t E [ 0 , I ] ) a r e e q u i m e a s u r a b l e , a n d s i n c e n x ( X ' ( t ) ) = tf o r m o s t - " t E [ 0 , 11 [ 2 , p . 83 1 , w e o b t a i n

I I X I g = n ( X * F ( t ) ( 1 - n x ( X ' ( t ) ) ) " - ~ d t = n ( X ' ) ' ( t ) ( 1 - t ) ' ~ - ~ d r .T h e l a s t e q u a l i t y i m p l i e s

H x I I , = I l x * l l ^ , , . . ) . ( 1 2 )w h e r e A p ( ~ , , ) i s t h e s y m m e t r i c L o r en t z sp a c e c o n s t r u c t e d f o r t h e f u n ct i o n ~ n ( t ) = 1 - (1 - t ) " .L e t u s s h o w t h a t f o r a n y n E N a n d t E [ 0, 1] w e h a v e

e - ' ~ , , ( t ) ~ , , (t ) _ ~ ( t ) , ( 1 3 )w h e r e I b , , ( t ) = r n : n { l , n t } .

S i n c e ~ o : ( t ) = e : ( t ) = i , w e c a n a s s u m e t h a t n _ > 2 . I f t > t / n , h e n ~ o ,, ( t) < _ i - - r a n d~ . ( t ) > p . ( 1 / n ) = 1 - ( 1 - i / n ) - > 1 - e - : _ > r ( c l e a r l y , ( 1 - I / n ) " t i / e ) . : n t h e c a s e0 < t __ 0 a nd Y > 0 a re two rando m var iab l e s de f i ned , i n 9enera l , on d i f f e ren t

p r o b a b i li t y s p a c e s . W e a s s u m e th a t

/ : I '' ( s ) d .

7/11

C o r o l l a r y 3 . Suppose that E i s a sym me tr ic space on [0 , 1 ] , z = z( t ) and y = y( t ) are measurable(on [ 0 , 1 ] ) /unct ions for which condi t ions ( 1 4 ) and ( 1 5 ) are valid. Th en i f z e E , then y E E andI Iu l IE < 6 ' I I~ I IE , where C > 0 depends only on Cx and C2.

P r o o f . U s i n g C o r o l l a ry 2 , w e f in d t h a t f o r a ll z > 0, ~ t , l ( ~ ) < _ c ' , ~ l ~ I ~ ,

o r e q u i v a l e n t l y y* (t) < C'~rc, z*(t) ( t E ( 0 , 11 ), w h e r e o ' r z ( t ) = z ( t / T ) i s t h e d i l a t a t i o n o p e r a t o r . S i n c e a~-is b o u n d e d i n a n y s y m m e t r i c s p ac e a n d I I ~ .I I E - . ~ < m ~ x { 1 , r } [ 2 , p . 133] , i t f oUows tha t I l Y l I ~ -< C I I ~ I I E ,w h e r e C = C'max{1,C'}. [ ]

R e m a r k 3 . I t is e ~ s l ly v er i fi e d t h a t c o n d i t i o n (1 5 ) i n C o r o l l a r y 3 i s e s s e n t i a l . M o r e o v e r , if E i s as y m m e t r i c s p ac e a n d n o t t h e in t e r po l a t io n s p a c e b e t w e e n th e s p ac e s L t a n d L ~ , t h e n w e c a n a lw a y sf in d a f u n c t i o n z E E a n d a l in e a r o p e r a t o r T b o u n d e d l y a c t i n g i n t o L t a n d L r s u c h t h a t y =T z ~ E [ 2, p p . 1 3 0 , 1 6 6] .

W e n o w a p p l y T h e o r e m 2 fo r t h e c a se i n w h i c h

1 5 , I , I( t ) = a ~ r , ( t , r ( ~ ) = a d ~ ( ~ , ,,~ e N , a~ e R ,i = 1w h e r e t h e r ~ (t ) a r e R a d e m a c h e r f u n c t io n s o n [0 , 1 ], a n d t h e y ~ ( ~ , ) a r e r a n d o m v a r i a b l e s o n s o m e p r o b a -b i l i t y s p a c e ( 12 , ~ . , P ) .

9 OOT h e o r e m 3 . Suppose t ha t { f i } i = l is an orthonormal system of random variables on the probabil i tyspace (1 2, ~ , e ) , I / d ~ ' ) l - < M , w e 12, i = 1 , 2 , . . . .The/ol lowing condi t ions are equivalent :.1 ) for an arbi t rary sequence a ( a , ) i = 1 E t 2 , t he / u nc t i on f = E~=I a i f i belongs t o L N , w here

N ( t ) = e t 2 - 1 ;9 "~ [0 , 1] such that) t h e r e ez i s t s a cons tan t C > 0 indepen dent o f r n E N , a = ( a , ) i = l , and t E

i ' ( ) ( )a i f i ( s ) d s < C l t / C i n l / 2 3~ , a ; s 1 6 3 ; (16)3 ) t he s y s t em ( f i } ~ = x i s major i zed in d i s t r ibut ion by the Rademacher sys tem.P r o o f . F i r s t , l e t u s p r o v e t h e i m p l i c a t i o n 1 ) ,~ 2 ). A s i s w e ll k n o w n [1 0], t h e O r l i c z s p a c e L N

c o i n c id e s w i t h t h e M a r c i n k i e w i c z s p a c e M ( ~ o ) , w h e r e ~ o(t) = t l og ~ /2 ( 2 / t ) . T h e r e f o r e , b y t h e d e f i n i ti o n o ft h e n o r m i n M ( ~ o ) , w e h a v e

ai f i s )d s < Ct t log~/2 -[ [Jail2, (17)w h e r e C t is i n d e p e n d e n t o f a = (ai)i~176E s and t E [0 , 1 ] .

S u p p o s e t h a t a ~ = tai~ I ( k = 1 , 2 , . . . ) i s a n o n i n c r e a s i n g r e a r r a n g e m e n t o f t h e s e q u e n c e (la ~l)~ % a. I nv ie w o f i n e q u a l i t y ( 1 7) a n d o f t h e a s s u m p t i o n s o f t h e t h e o r e m , f o r a n a r b i t r a r y j E N w e o b t a i n. )f O 2 - J ( E a t / i ) ( . . q ) d , . q < ~ f 0 2 ' ( E a t . f t , ) ( 8 ) d s + f o 2 "I a i , f i . ( s ) d sk = j + l= l k = l

< - 2 - J M E I a i 'l + C 1 2 -j l~ E la ~'lek = l k = j + l< - 2 - J m a x ( M , 2 C l ' ( ~ - ~ a * k + V ~ ( ~ ( a ~ ) 2 ) 1 / 2 )

- - k = l k = j + l

8/11

w h e r e t h e l a s t i n e q u a l i t y f ol lo w s f ro m H o l m s t e d t ' s f o r m u l a ( 8 ).T h u s r e l a t i o n ( 1 6 ) i s s a t i s fi e d fo r t = 2 - j ( j E N ) . S i n c e t h e f u n c t i o n s i n q u e s t i o n a r e c o n c a v e , t h is

r e l a t i o n i s a l s o v a l i d f o r a l l t E [ 0 , 1 ] .T h e i m p l i c a t i o n 2 ) = = ~ 3 ) f ol lo w s f r o m T h e o r e m 1 , C o r o l la r y 2 , a n d T h e o r e m 2 .F i n a l ly , w e a s s u m e t h a t c o n d i t i o n 3 ) is s at i sf i ed . I t is w e ll k n o w n ( se e , f o r e x a m p l e , [1 1 , p . 3 4 2 o f t h e

R u s s i a n t r a n s l a t io n ] ) t h a t t h e f u n c t i o n g = Y ~ I a i r i b e l o n g s t o L N i f a ( a , ) i= 1 E s B y a s s u m p t i o n ,t h e fo l l o w i n g i n e q u a l i t y i s v a l id fo r t h e f u n c t i o n f = ~ ' ] ~ a i f i :

( $ )l f l ( Z ) < C n l g IT h e r e f o r e , s i n c e t h e s p a c e L N is s y m m e t r i c , w e f i nd t h a t f E L N a s i n t h e p r o o f o f C o r o l l a ry 3 , a n d t h et h e o r e m i s t h e r e b y p r o v e d . I-3

W e n e e d t h e f o l l o w i n g n o t a t i o n ( f or d e t a i l s, s e e [1 2, C h a p . 1 ]) . S u p p o s e t h a t ( f~ , E , P ) i s a p r o b a b i l i t ys p a c e , f it is a ~ , - s u b a l g e b r a o f t h e a - a l g e b r a ~ 3. I f f : f~ --+ R i s a r a n d o m v a r i a b l e , t h e n E [ f lg ~ ] is 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 f f w i t h r e s p e c t to ~ t . I f f it i s t h e ~ r - su b a l g e br a g e n e r a t e d b y t h e r a n d o m v a r ia b l esg l , g 2 , - . - , g n , t h e n t h e c o n d i t i o n a l e x p e c t a ti o n w i t h re s p e ct t o t h e m w ill b e d e n o t e d b y E [ f l g ~ , . . . , g a ].I n p a r t i c u l a r ,

E [ f ] = J ~ f (o J ) d P ( w ) .L e t u s r e c a l l t h a t t h e s e q u e n c e { f n } ~ = ] o f r a n d o m v a r ia b l e s o n t h e p r o b a b i l i t y s p a c e (1 2, E , P ) i s c a ll e d

m u l t i p l i c a t i v e i f f o r a n y k E N a n d n x < n 2 < - . - < n kE [ f , , , f , , , . - ' I , , , ] = O .

B y T h e o r e m 3 a n d C o r o l l a r y 3 f r o m [ 1 3], w e o b t a i n t h e f o l l o w i n g r e s u l t.C o r o l l a r y 4 . I f { f i } i ~ l i s a u n i f o r m l y b o u nd e d m u l t i p l ic a t i ve o r t h o n o r m a l s y s t e m o n (12 , ~3, P) , t h e n

i t c a n b e m a j o r i z e d i n d i s t r i b u t i o n b y t h e R a d e m a c h e r s y s t e m .R e m a r k 4 . I n [13] i t w a s s h o w n t h a t u n d e r th e c o n d i t io n s o f C o ro L la ry 4 t h e v e c t o r ( f ~ , f 2 , . - . , f ~ ) h a st h e s a m e d i s t r i b u t i o n a s t h e c o n d i t i o n a l e x p e c t a t io n E [ (g l , g 2 , . . . , g ,,)]9~ ] o f t h e v e c t o r ( g z, g 2 , . . . , g ~ ) ,

w h i c h h a s th e s a m e d i s t r i b u t io n a s t h e v e c to r ( r l , r 2 , . . . , r , ~ ) . T h i s d o es n o t i m p l y th e a s se r ti o n o fC o r o l la r y 4 , s i n c e t h e o p e r a t o r o f 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 c a n e v e n b e u n b o u n d e d i n a s y m m e t r i cs p a c e ff i t i s n o t a n i n t e r p o l a t i o n s p a c e b e t w e e n L x a n d L o o ( s e e, fo r e x a m p l e , [ 1 4, p p . 1 2 8 - 1 2 9 ]) .

w I s o l a t i o n o f t h e e x p o n e n t i a l l y i n t e g r a b l e s u b s y s t e m sI n t h e p r o o f o f t h e f o ll o w i n g as s e r t io n a n i d e a f r o m [1 5, L e m m a A ] is u s e d .L e m m a . S u p p o s e t h a t { f , ~ } ~ = l i s a s e q u e n ce o f r a n d o m v a r i a b l es d e f i n e d o n t h e p r o b a b i l i ty s p a ce

( 12 , V . , p ) , I f ~ ( ' ~ ) l < M , w E 1 2 , n E N , an d f ,~ --+ 0 w e ak l y i n L2( 12 ) . Th e n t h e r e e z i s t ( f , ~ , } ~ l C{ f , ~ } ~ = l a n d a s e q u e n c e o f r a n d o m v a r i a b le s { g k } F = l s u c h t h a t1 ) I g~@ ) l < 2 M , to E f ~ , k E N ;

2 ) E [ g k l g ~ , . . . , g k - 1 ] = 0 a l m o s t e v e r y w h e r e o n 1 2 ;3) l l f , , , - gk l l k < _ 2 - k , k E N .P r o o f . L e t u s p r o v e t h is a s s e r t i o n b y i n d u c t i o n . S i n c e t h e f,~ t e n d t o z er o w e a k l y in L 2 (~ '/ ), w e c a n

f in d a n n l s u c h t h a t1_ < g . ( 1 8 )

T h e r e e x is t s a f i n i t e - v a l u e d f u n c t i o n h i f o r w h i c h I h l ( w ) l < M a n d1h l ] _ < ( 1 9 )

41 4

9/11

W e s e t g l = h , - E [ h l ] . T h e n E [ g ,] = 0 a n d I g , ( ~ ) l < _ 2 M . M o r e o v e r , it fo l lo w s f r o m i n e q u a l i ti e s ( 1 8)a n d ( 1 9 ) t h a t

I I f , ~ l - g x l l x _ < E [ I f , , , , - h ~ l ) + E [ I h ~ - g x l ] = E [ I f , ~ , - h l l ] + I E [ h x ] l1< 2 E [ l f , , , - h , I ] + I E [ f , , , ] l 1 a n d t h a t t h e o b t a i n e d v al u e s f , ~ , , . . . , f , ~ , _ l 1 < n l < " ." < n ~ - x a n dg l , . . - , g J ,- 1 a r e s u c h t h a t f o r a l l s = 1 , . . . , k - 1a ) I g , ( ~ ) l < 2 M ;

- ( ~ " - ( ~ " " (~ ( i # j ) a n d U i e i) g o i s a c o n s t a n t o n t h e s e t s c i C 1 2, i 1 , 9 , t s , gi" " I l e j = ~ _ ( s ) = 1 2 ;_ (o -1 )- ( ' ) l ie e n t i r e l y i n t h e s e t s o f c o n s t a n c y e i , i 1 , . . . o f t h e f u n c t i o n g , - 1 ;o r e o v e r , e i = , Z s - - 1 ,c ) E [ g , l g l , . . . , g , - l ] = 0 a l m o s t e v e r y w h e r e o n a ;d ) I I f , , . - goll0 < 2 - ~ 9

T h e n w e o b t a i n t h e n e x t p a i r o f v a l u e s f ,~ , a n d g k s a t i s f y i n g t h e s a m e c o n d i t i o n s . S i n c e f,~ --+ 0w e a k l y i n L 2 ( 1 2 ) , t h e r e e x i s t s a n n k > n k - 1 s u c h t h a t f o r a l l i = 1 , . . . , i k - 1 w e h a v e

0 the followin# relation is valid:

P (,.,eI~: a ~ i, ~ ,( u ., ) > z _ .-- '

i = 1 '

I n c o n c l u s i o n , t h e a u t h o r w i s h e s t o e x p r e s s h is g r a t i t u d e t o V . F . G a p o s h k i n f o r a d v i c e o n q u e s t i o n sr e l a t e d t o t h e l a s t p a r t o f t h i s p a p e r .

R e f e r e n c e s1 . S . K a c z m a r s a n d H . S t e i n h a u s , TheoEie dex Orthogonalrelhen, W a x s a w - L v o v ( 1 9 3 5 ).2 . C . G . K r e i n , Y u . I. P e t u n i n , a n d E . M . S e m e n o v , Interpolation o[Linear Operators [ i n R u s s i a n ] , N a u k a , M o s c o w ( 1 9 7 8 ) .3 . J . B e r g h a n d J . L ~ f s t r S m , Interpolation Spaces. An Introduction, S p r i n g e r - V e r l a g , B e r l i n - H e i d e l b e r g - N e w Y o r k ( 1 97 6 ).4 . S . V . A s t a s h k i n , ~ O n t h e i n t e r p o l x t l o n o f s u b s p a c e s o f s y m m e t r i c s p a c e s g e n e ra t e d b y t h e R a d e m a c h e r s y s t e m , " Izv.RAEN. Set. MMMIU, 1 , No . 1 , 18 -35 (1997) .5 . S . V . A s t a s h k i n , ~ S e r i e s i n t h e R a ( l e m ~ g : he r s y s t e m t h a t a x e " c lo s e t o " L o o , " F u n k t s i o n a l . Anal. i Pri]ozhen. [FunctionalAnal. Appl.] ( t o a p p e a r ) .6 . S . M o n t g o m e r y - S m l t h , W T h e d i s t r i b u t i o n o f R a d e m a c h e r s u m s ," Proc. Amer. Math. Soc., 1 0 9 , N o . 2 , 5 1 7 - 5 2 2 ( 1 9 9 0 ) .7 . T . H o l m s t e d t , ~ I n t e r p o l a t io n o f q u a s i - n o r m e d s p a ce s ," Math. Scand., 26 , 177-199 . (1970) .8 . S . J. S z a r e k , " O n t h e b e s t c o n s t a n t s i n t h e K h l n c h ; - e i n e q u a l lt y , " Stuclla Math., 5 8 , 1 9 7 - 2 0 8 ( 1 9 7 6 ) .9 . N . H . A s m a r a n d S . M o n t g o m e r y - S m l t h , ~ O n t h e d i s t r i b u t i o n o f S i d o n s er i e s ," A r k . M a t . , 3 1 , N o . 1 , 1 3 -2 6 ( 1 9 9 3 ).

1O . Y ~. B . R u t i t s l d l , " O n - o m e ,- I,~ -,~ s o f m e a s u r a b l e f u n c t i o n s , " Uspekld Mat. Nauk [ R u s s / a n Ma~h. Surveys], 2 0 , N o . 4 ,2 0 5 - 2 0 8 ( 1 9 6 5 ) .1 1. A . Z y g m t m d , T~gonometrir Series, V o L 1 , C a m b r i d g e U n i v . P r e s s , C a m b r i d g e ( 19 5 9 ).

1 2 . R . E l l l o t t , Stochastic Calculus and Applic_~ons, S p r i n g e r - V e r l a g , H e i d e l b e r g ( 1 9 8 3 ) .1 3. J . J a k u b o w s l d a n d S . K w a p i e n , ~ O n m u l t l p ll c a .t l v e s y s t e m s o f f u n c t i o n s , " Bull. Acad. Polos. Scl. Set. Sci. Math., 2 7 ,N o . 9 , 6 8 9 - 6 9 4 ( 1 9 7 9 ) .1 4 . J . L i n d e n s t r a u s s a n d L . T s a f 6 . r i , C l a s s / c a l B a n a c .h S p a c e s , V o l. 2, S p r i n g e r , B e r l i n ( 1 9 7 9 ) .1 5. V . F . G a p o s h k i n , ~ C o n v e r g e n c e a n d l i m i t i n g t h e o r e m s f or s u b s e q u e n c e s o f r a n d o m v a l u e s ," T e o r . Veroyatnost. i P r h n e n e a .[Theory Probab. Appl.], 1T, No. 3 , 401--423 (1972).1 6. V . F . G a p o s h k i n , " G a p s e r i e s a n d i n d e p e n d e n t fu n c t i o n s ," UspeichiM a t . Nauk [Russian Math. Surveys], 21 , No . 6 , 3 -82

(1966) .SAMARA ST AT E UNIVE RSIT YB-mail address: a s t n s h k n O s s u . s a m a x a . r u

417

s. v. astashkin- extraction of subsystems "majorized" by the rademacher system

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