S.V. Astashkin and F.A. Sukochev- Series of Independent Random Variables in Rearrangement Invariant Spaces: An Operator Approach

8/3/2019 S.V. Astashkin and F.A. Sukochev- Series of Independent Random Variables in Rearrangement Invariant Spaces: An

1/32

ISRAEL JOURNAL OF MATH EMAT ICS 145 (2005) , 125-156

S ER IE S O F I N D E P E N D E N T R A N D O M V A R I A B LE SIN R E A R R A N G E M E N T I N V A R IA N T S PA C ES :

A N O P E R A T O R A P P R O A C H

B YS . V . A S T A S H K I N

Dep a r t men t o f M a t h em a t i c s , S a ma r a S t a t e U n i ve r s it ySamara , Russ ia

e-ma il : as tashkn@ ssu.sam ara.ru

A N D

F. A . SUKOCHEV*School o f In form at ics a nd Engineer ing , F l inder s U niver s i t y

Bedford Park, SA 5042 Aus tral iae-mai l : sukochev~in foeng . f l inder s .edu .au

AB S TR AC TT h i s p a p e r s t u d i e s s e ri e s o f i n d e p e n d e n t r a n d o m v a r i a b l es i n r e ar r a n g e -m e n t i n v a r i a n t s p a c es X o n [ 0 , 1 ]. P r i n c i p a l r e s u l t s o f t h e p a p e r c o n c e r ns u c h s e r ie s in O r l i cz s p a c e s e x p ( L p ) , 1 ~ p ~ c ~ a n d L o r e n t z s p a c e s A .O n e b y - p r o d u c t o f o u r m e t h o d s i s a n e w ( a n d s i m p l e r ) p r o o f o f a r e s u l td u e t o W . B . J o h n s o n a n d G . S c h e c h t m a n t h a t t h e a s s u m p t i o n L p C X ,p < o c i s s u f fi c ie n t t o g u a r a n t e e t h a t c o n v e r g e n c e o f s u c h s e r i e s i n X( u n d e r t h e s i d e c o n d i ti o n t h a t t h e s u m o f t h e m e a s u r e s o f t h e s u p p o r t so f al l i n d i v i d u a l t e r m s d o e s n o t e x c e e d 1 ) i s e q u i v a l e n t t o c o n v e r g e n c e i nX o f t h e s e r i e s o f d i s j o i n t c o p i es o f i n d i v i d u a l t e r m s . F u r t h e r m o r e , w ep r o v e t h e c o n v e r s e ( i n a c e r t a i n s e n s e ) t o t h a t r e s u l t .

* R e s e a r c h s u p p o r t e d b y t h e A u s t r a l i a n R e s e a r c h C o u n c i l .R e c e i v e d S e p t e m b e r 1 7, 2 0 0 3 a n d i n re v i s e d f o r m J u l y 1 5, 20 0 4

1 2 5


2/32

126 S .V . ASTA SHK IN AND F . A . SUKO CHE V Is r. J . M ath .1 . I n t r o d u c t i o nI t f o l lo w s f r o m t h e c l a s s ic a l K h i n t c h i n e I n e q u a l i t y t h a t f o r al l p E [1 , o c ) , t h eR a d e m a c h e r s y s t e m {rn}n=l, g i v e n b y r n ( t ) = s g n s i n ( 2 n T r t ) , t E [ 0 , 1 ) i n t h eL p - s p a c e s o n t h e i n t e r v a l [0 , 1] ( e q u i p p e d w i t h L e b e s g u e m e a s u r e A ) i s e q u i v a -l e n t t o t h e u n i t v e c t o r b a s i s {en}n=l o f 12, o r e q u i v a l e n t l y t o t h e s e q u e n c e o fd i s j o i n t t r a n s l a t e s r n ( t ) :- - r n ( t - n + 1 ) i n L 2 ( 0 , o c ) . S i m i l a r l y , i t f o l l o w s f r o ma r e m a r k a b l e i n e q u a l i t y d u e t o H . P . R o s e n t h a l [R ] f o r s e q u e n c e s {fn}n~__l o fi n d e p e n d e n t m e a n z e r o r a n d o m v a r ia b l es in L p [ O , 1 ], p _> 2 t h a t t h e m a p p i n gfk --+ fk , w h e r e ]k ( t ) := f k ( t - - k q - 1 )X [k_ l , k ) ( t ) , t E ~ e x t e n d s t o a n i s o m o r -p h i s m b e t w e e n t h e c l o s e d l i n e a r s p a n [fk]k__l ( t a k e n i n L p [ 0 , 1 ]) a n d t h e c l o s e dl i n e a r s p a n I f ] k = l ( t a k e n i n L p [0 , c o ) ~ L 2 [0 , o c ) ) . A n e x t e n s i o n o f R o s e n t h a r sI n e q u a l i t y t o L o r e n t z s p a c e s Lp,q , 2 < p < o c , 0 < q < o o is g i v e n i n [ C D ] . Af u r t h e r s i gn i fi c a n t g e n e r a li z a t io n t o t h e c l as s o f r e a r r a n g e m e n t i n v a r ia n t ( = r . i .)s p a c e s is d u e t o W . B . J o h n s o n a n d G . S c h e c h t m a n [ JS ] w h o i n t r o d u c e d r .i .s p a c e s ] Ix a n d Z x o n [ 0, o c ) l i n k e d w i t h a g i v e n r .i . s p a c e X o n [ 0, 1 ] a n d s h o w e d

k nh a t a n y f i ni te s e q u e n c e { f } k = l o f i n d e p e n d e n t m e a n z e r o ( re s p ec t iv e ly , p o s -i ti v e ) r a n d o m v a r i a b l e s in X is e q u i v a l e n t ( u n i f o r m l y i n n ) t o t h e s e q u e n c e o fi t s d i s j o i n t t r a n s l a t e s i n Y x ( r e s p e c t i v e l y , Z x ) , p r o v i d e d t h a t X c o n t a i n s a nL p - s p a c e fo r s o m e p < o c . A k e y t o o l in t h e i r p r o o f o f t h i s e q u i v a l e n c e i s t h ew e l l - k n o w n t a il p r o b a b i l i t y i n e q u a l i t y d u e to H o f f m a n n - J o r g e n s e n [ H -J ] (f o r i n-t e r e s t i n g s t r e n g t h e n i n g o f t h i s i n e q u a l i t y w e re f e r t o t h e r e c e n t p a p e r s [ H M ]a n d [ K N ] ) . I n p a r t i c u l a r , i t i m m e d i a t e l y f o l lo w s f r o m r e s u l t s o f [ JS ] t h a t i f t h e

k oos e q u e n c e { f ' } k = l o f i n d e p e n d e n t r a n d o m v a r i a b le s s a ti sf ie s , in a d d i t io n , t h ea s s u m p t i o n t h a t f o r a ll n E N

n

( 1.1 ) E A ( { I k ~ 0 } ) < 1,k--1

t h e n t h e c o r r e s p o n d e n c e f k ++ f k , k _> 1 b e t w e e n t h e s e q u e n c e { f '} k = l ko a n d~ k o o o ot h e d i s jo i n tl y s u p p o r t e d s e q u e n c e { f } k = l o f e q u i m e a s u r a b l e c o p ie s o f { f n } n = lk oo [fk]k=lx t e n d s t o a n i s o m o r p h i s m b e t w e e n t h e cl o s e d l in e a r s p a n s [ f ' ] k = l a n d - oo

i n X , p r o v i d e d t h a t X c o n t a i n s L ; [ 0 , 1] f o r s o m e p < o o .T h e m a i n q u e s t i o n s t u d i e d i n t h e p r e s e n t p a p e r i s t h e f o ll o w in g : f o r w h i c h

r . i . sp a ces X a n d Y o n [0, 1] d o e s t h e r e e x i s t a c o n s t a n t C = C ( X , Y ) > 0 s u c ht h a t f o r e v e r y s e q ue n c e { f k } ~ - - - - i C X o f i n d e p e n d e n t r a n d o m v a r ia b le s s a t i s fy i n g( 1 . 1 ) , i t f o l l o w s t h a t0 . 2 ) ~ f k v < _ C ~ f k ?k----1 k----1 X


3/32

V ol . 1 45 , 2 0 0 5 S E R I E S O F I N D E P E N D E N T R A N D O M V A R I A B L E S 1 27

W e r e c a ll t h a t a c o m p l e t e c h a r a c t e r i z a t i o n o f r e a r r a n g e m e n t i n v a r i a n t s p a c e sE o n [0 , 1] f o r w h i c h t h e K h i n t c h i n e I n e q u a l i t y h o l d s i s d u e t o V . A . R o d i n a n dE . M . S e m e n o v [ R S ] (s e e a l s o [L T ] , p p . 1 3 4 - 1 3 6 ) . C o n s i d e r t h e f a m i l y o f O r l i c zs p a c e s e x p ( L p ) = L N p , N p ( t ) : - - ex p l t l~ - 1 , t E I~, p >_ 1 . I t i s sh o w n in [RS]t h a t t h e c o r r e s p o n d e n c e r ~ ++ e n , n >_ 1 e x t e n d s t o a n i s o m o r p h i s m b e t w e e n

e c~t h e c l o s e d l i n e a r s p a n [r~ ]n ~_ _l i n E a n d [ ~ ] n = l i n 12 i f a n d o n l y i f E c o n t a i n st h e s e p a r a b l e p a r t o f t h e s p a c e e x p ( L 2 ) .

I t is p r o v e d h e r e t h a t i n t h e s e t t i n g t h a t X = Y , u n li k e th e s i t u a t i o n w i t ht h e K h i n t c h i n e I n e q u a l it y , t h e r e i s n o " m i n i m a l " r .i . s p a c e E s u c h t h a t f o r e v e r yr .i . s p a c e X D E t h e c o r r e s p o n d e n c e f k ++ / k , k _> 1 f o r a l l s e q u e n c e s { f k } k = lo f i n d e p e n d e n t r a n d o m v a r i a b l e s s a t is f y i n g ( 1. 1) e x t e n d s t o a n i s o m o r p h i s mk c ~k c~e t w e e n t h e c l o s e d l i n e a r s p a n s [ f ' ] k = l a n d [ f ' ] k = l i n X . I n f a c t , w e s h o w t h a ti f E i s s u c h a n r . i. s p a c e , t h e n i t c o n t a i n s a n L p - s p a c e f o r s o m e p < c o . T h i sr e s u l t i s t h e c o n v e r s e ( i n a c e r t a i n s e n s e ) t o [ J S ], T h e o r e m 1 .

W e s t u d y t h e a b o v e q u e s t i o n f o r t h e s e t t i n g s w h e n X = e x p ( L p ) , 1 < p _< c ca n d w h e n X i s a n a r b i t r a r y L o r e n t z s p a c e A [ 0 , 1]. I n t h e f i rs t c a s e , w e s h o w t h a tf o r a f i x e d p C [ 1, c c ] a n d X = e x p ( L p ) , t h e s e t o f a l l r .i . s p a c e s Y f o r w h i c h t h ei n e q u a l i t y ( 1 .2 ) h o l d s h a s a u n i q u e m i n i m a l e l e m e n t , w h i c h is t h e O r l ic z s p a c eL M q , w h e r e M q ( t ) : = e I tl ln l /~ ( e+ l tl ) - 1 , t E ] L a n d q = p / ( p - 1 ) . I n t h e s e c o n ds e t ti n g , w e g iv e a c o m p l e t e c h a r a c t e r i z a t i o n o f t h o s e c o n c a v e f u n c t i o n s s u c ht h a t t h e c o r r e s p o n d e n c e f k + + / k , k > 1 f o r a l l s e q u e n c e s { f k }k~_-i o f i n d e p e n d e n tr a n d o m v a r i a b l e s s a t i s f y i n g ( 1 . 1 ) e x t e n d s t o a n i s o m o r p h i s m b e t w e e n t h e c l o s e dl i n e a r s p a n s [ f k ] k = l a n d [)% ] k= 1 i n L o r e n t z s p a c e A [ 0 , 1 ].

O u r a p p r o a c h is b a s e d o n t h e s t u d y o f a c e r t a in l in e a r o p e r a t o r ]~ o n L I [ 0 , 1]a n d i s r e l a t e d t o t h e a p p r o a c h p r e v i o u s l y d e v e l o p e d b y M . S h. B r a v e r m a n [B r] inh is s t u d y o f t h e R o s e n t h a l I n e q u a l i t y i n r .i . s p a c e s , w h i c h , i n t u r n , w a s i n s p i r e db y e a r li e r i d e a s a n d p r o b a b i l i s t i c c o n s t r u c t i o n s o f V . M . K r u g l o v [K ]. W e s p e llo u t t h e s e c o n n e c t i o n s i n S e c t i o n 3 b e l o w , a f t e r i n t r o d u c i n g a l l n e c e s s a r y d e f i n i -t i o n s in S e c t io n 2 . W e s t u d y t h e m a i n q u e s t i o n i n t h e O r li c z s p a c e s e x p ( L p ) a n dt h e L o r e n t z s p a c e s A i n S e c t i o n s 4 a n d 5 r e s p e c ti v e l y . A s a b y - p r o d u c t o f t h ew o r k c a r r ie d o u t i n S e c t io n 5 fo r L o r e n t z s p a c e s , w e a ls o p r e s e n t t h e r e t h e c o n -v e r s e o f t h e m a i n r e s u l t f ro m [ JS ]. T h e l a t t e r a p p l i c a t i o n is p a r t l y b a s e d o n t h er e s u lt s o f S . M o n t g o m e r y - S m i t h a n d E . M . S e m e n o v [ M S] c o n c e r n in g r a n d o mr e a r r a n g e m e n t s , a l t h o u g h o u r e x p o s i t i o n is f a ir ly se l f- c o n t a in e d . I n S e c ti o n 6 ,w e s h o w a n e a s y w a y t o d i s c a r d t h e s i d e c o n d i t i o n ( 1 .1 ) a n d e x t e n d o u r r e s u l t st o a n a r b i t r a r y s e q u e n c e o f i n d e p e n d e n t r a n d o m v a r i a b le s . I n p a r t i c u l a r , i n t h es e t t i n g t h a t X = Y , w e r e c o v e r (a n d c o m p l e m e n t ) a n e ar l ie r re s u l t fr o m [ JS ]


4/32

1 28 s . v . A S T A S H K I N A N D F . A . S U K O C H E V I s r. J . M a t h .f o r t h e s p a c e Z x . O u r a p p r o a c h h e r e c o n s is t s in re d u c i n g t h e s t u d y o f a r b i t r a r ys e q u e n c e s o f i n d e p e n d e n t r a n d o m v a r i a b l e s t o t h o s e s a t i s fy i n g c o n d i t i o n ( 1 .1 ) .W e d e m o n s t r a t e t h e u t i l i t y o f o u r a p p r o a c h b y s tr e n g t h e n in g r e c e n t re s u l t s o fS . M o n t g o m e r y - S m i t h [M ] c o n c e r n i n g s p a c e s Z ~c , p E ( 1 , c o ] ( w h i c h g e n e r a l i z er e s u l t s o f [ J S ] f o r t h e s p a c e Z x ) . F i n a l l y , i n S e c t i o n 7 , w e p r e s e n t a n e c e s -s a r y c o n d i t i o n f o r a n a f f i r m a t i v e a n s w e r t o t h e m a i n q u e s t i o n i n th e c a s e w h e nX = Y .ACKNOWLEDGEMENT: W e t h a n k t h e r e f e r e e f o r u s e f u l c o m m e n t s , i n p a r t i c u l a rf o r t h e r e f e r e n c e [ K N ] .

S o m e o f t h e r e s u l t s o f t h i s p a p e r h a v e b e e n a n n o u n c e d i n [A S ].

2 . D e f i n i t i o n s a n d p r e l i m i n a r i e sW e d e n o t e b y S ( f t ) ( = S ( f~ , 7 ) )) t h e l i n e ar s p a c e o f a ll m e a s u r a b l e f in i te a . e . f u n c -t io n s o n a g iv e n m e a s u r e s p a c e ( f ~ , P ) e q u i p p e d w i t h t h e t o p o l o g y o fc o n v e r g e n c e l o c a l l y i n m e a s u r e .

A B a n a c h s p a c e ( E , [ [. liE ) o f r e a l -v a l u e d L e b e s g u e m e a s u r a b l e f u n c t i o n so n t h e i n t e r v a l [ 0 , a ) , 0 < a _~ c c ( w i t h i d e n t i f i c a t i o n A - a . e .) w i l l b e c a l l e dr e a r r a n g e m e n t i n v a r i a n t i f

( i) E i s a n i d e a l l a t t ic e , t h a t i s, i f y E E , a n d i f x i s a n y m e a s u r a b l e f u n c t i o no n [ O , a ) w i t h 0 _~ I xl < l Y l t h e n x E E a n d Ilxll E ~ IlYlIE;

( ii ) E i s r e a r r a n g e m e n t i n v a r i a n t i n t h e s e n s e t h a t i f y E E , a n d i f x is a n ym e a s u r a b l e f u n c t i o n o n [ 0 , c~) w i t h x * = y * , t h e n x E E a n d ] Ix iiE = IIY IIE .

H e r e , )~ d e n o t e s L e b e s g u e m e a s u r e a n d x * d e n o t e s t h e n o n - i n c re a s i n g , r ig h t -c o n t i n u o u s r e a r r a n g e m e n t o f x g iv e n b y

x * ( t ) = i n f { s _ > O : ~ ( { i x i > s } ) _ < t } , t > 0 .F o r b a s i c p r o p e r t i e s o f r e a r r a n g e m e n t i n v a r i a n t s p a c e s , w e r e f er t o t h e m o n o -g r a p h s [ B S ] , [ K P S ] , [ L T ] .

T h e K h t h e d u a l E o f a r e a r r a n g e m e n t i n v a r i an t s p a c e E o n t h e i n te r v a l[0 , a ) c o n s i s t s o f a l l m e a s u r a b l e f u n c t i o n s y f o r w h i c h

{ / 0 }IY]IE : = s u p tx(t )y(t )]d t: x E E , tIxIIE


5/32

Vol. 145, 2005 SER IESO F I N D E P E N D E N T R A N D O M V A R I A B L E S 12 9t h a t t h e n o r m I1" l i E o f t h e r e a r r a n g e m e n t i n v a r i a n t s p a c e E o n [0 ,(~ ) i s o r d e r -c o n t i n u o u s if a n d o n l y i f E i s s e p a r a b l e . T h e n a t u r a l e m b e d d i n g o f E i n t oi ts K S t h e b i d u a l E i s a s u r j e c t i v e i s o m e t r y i f a n d o n l y i f E h a s t h e F a t o up r o p e r t y , i . e . i f i t f o l l o w s f r o m { ] n } n > > _ lC _ E , ] E S [ 0 , c ~ ), f n - + f a .e . on [0 , c~ )a n d s u p n I l f n l l E < c ~ , t h a t f E a n d I l f l l E < _ l i m i n f n _ ~ I l f n l l E . S u c h s p a c e sa r e a l so c a ll ed m a x i m a l . S o m e w h a t w e a k e r t h a n t h e n o t i o n o f F a t o u p r o p e r t yo f a n r .i . s p a c e E i s t h e n o t i o n o f a F a t o u n o r m . I f E i s a r. i. B a n a c h f u n c t i o ns p a c e o n [ 0 , a ) , 0 < a


6/32

1 30 s .v . A S T A S H K IN A N D F . A . S U K O C H E V I sr . J . Math .Def in i t i on 3 .1 : A n r .i . s p a c e X i s s a i d t o h a v e t h e K r u g l o v p r o p e r t y ( X E K ) ,i f a n d o n l y i f

f E X = ~ 7 r ( f) E X .T h i s p r o p e r t y h a s b e e n s t u d i e d a n d e x t e n s i v e l y u s e d b y M . S h . B r a v e r m a n [B r] ,w h o n o t e d i n p a r t i c u l a r t h a t o n l y t h e im p l i c a t i o n f E X ~ 7 r ( f) E X i s n o n -t r i v i a l , s i n c e t h e i m p l i c a t i o n ~ r ( f ) E X ~ f E X i s a l w a y s s a t i s f i e d ( s e e [ Br ],p . 1 1 ) .

W e s h a l l n o w d e f i n e a n o p e r a t o r E o n S ( [ 0 , 1 ], )~ ) w h i c h i s c l o se l y l i n k e dw i t h t h e K r u g l o v p r o p e r ty . F r o m a t e c h n i c a l v i e w p o i n t, i t is m o r e c o n ve -n i e n t t o a s s u m e t h a t t h is o p e r a t o r t a k e s i t s v a lu e s i n S ( ~ , P ) , w h e r e ( ~ , P ) : =l-Ik ~_ _0 ([0 ,1 ], ) ~k ) ( h e r e , ) ~ k i s L e b e s g u e m e a s u r e o n [ 0 ,1 ] f o r e v e r y k >_ 0 ) . L e t{ E n } b e a s e q u e n c e o f p a i r w i s e d i s j o i n t s u b s e t s o f [0 , 1 ], m ( E n ) = 1 / ( e . n ! ),n E l~ l. F o r a g iv e n f E S ( [0 , 1] , A ) , w e se t

oo n

(3 .1 ) t : f ( w o , w l , w 2 , . . .) : : ~ E f ( w k ) ~ ( E . ( W o ) .n= l k= l

L e t a l s o 5: ( f / , P ) - ~ ( [0 , 1 ], )~) b e a m e a s u r e p r e s e r v i n g i s o m o r p h i s m . F o re v e r y g E S ( i ) , P ) , w e s e t T ( g ) ( x ) : = g ( 5 - 1 x ) , x E [ 0 ,1 ] . N o t e t h a t T i s ar e a r r a n g e m e n t - p r e s e r v i n g m a p p i n g b e t w e e n S ( f l , P ) a n d S ( [0 , 1], A ). W e s h a llb e m a i n l y i n t e r e s t e d i n th e o p e r a t o r T E a c t i n g o n S ( [0 , 1], A ) a n d b y a n a b u s e o fl a n g u a g e f r e q u e n t l y r e f e r t o t h e l a t t e r o p e r a t o r a s )E . F r o m a c e r t a i n v i e w p o i n t ,o u r m a i n o b j e c t o f s t u d y i n t h is p a p e r is th e d i s t r ib u t i o n o f K f ( fo r v a r io u sc l a ss e s o f m e a s u r a b l e f u n c t i o n s f S [ 0 , 1 ]) . T h e r e f o r e , i t w i ll b e c o n v e n i e n t toa d o p t t h e f o l l o w i n g n o t a t i o n . I f f S ( [ 0 , 1 ], A ) a n d { f n , k } ~ = l i s a s e q u e n c e o fm e a s u r a b l e f u n c t i o n s o n [0 , 1] s u c h t h a t :

( i) t h e s e q u e n c e ] n, 1, f n , : , . . . , f n ,n , ~E ,~ is a s e q u e n c e o f i n d e p e n d e n t r a n d o mv a r i a b l e s V n N ;

( i i) ~ 'S ,.k = ~ ' f , V n e N , k = l , 2 , . . . , n ,t h e n w e w r i te

oo n

( 3 . 1 ) ' x [ 0 , 1 ] .n = l k = l

I t is c l e a r t h a t t h e d i s t r i b u t i o n f u n c t i o n o f ] C f ( = T I C f ) i s t h e s a m e a s t h ed i s t r i b u t i o n f u n c t i o n o f I C ' f . F r e q u e n t ly , a g a i n b y a n a b u s e o f l a n g u a g e , w es h a l l a l s o r e f e r t o / C ' f a s )E l .

T h e m a i n o b j e c t iv e o f t h i s p a p e r is to s t u d y t h e a c t i o n o f t h e p o s i t iv e l i n e a ro p e r a t o r / C o n v a r i o u s c l a s se s o f r .i . s p a c e s X .


7/32

Vol. 145, 2005 SER IESO F I N D E P E N D E N T R A N D O M V A R I A B L E S 131O u r f ir s t re m a r k c o n c e r ni n g t h e o p e r a t o r E i m m e d i a t e l y f o ll o w s f r o m t h e

c l o s e d g r a p h t h e o r e m ( se e a l s o [B r ], L e m m a 1 , p p . 1 1 - 1 2 ) .L E M M A 3 . 2 : I f X a n d Y a r e r . i . s pac e s on [0, 1] an d E f E Y for e v e r y f E X ,t he n t h e r e e x i s t s C > 0 s uc h t ha t

II~flly ~ C I I f l l x .L e t f E S [ 0 , 1 ] a n d l e t E ' f b y d e fi n e d b y ( 3 .1 ) '. T h e d i s t r i b u t i o n f u n c t i o n o f

t h e r a n d o m v a r i a b l e E ' f i s g i v e n b y

s v(x) + S s ( x ) + 1 , x E ~ ,/=2

w h e r e ( ~ ' / ( x ) ) *t i s t h e / - f o l d c o n v o l u t i o n o f ~ S ( ' ) c o m p u t e d a t t h e p o i n t x .T h i s d i s t r i b u t i o n is a m i x t u r e o f t h e d i s c r e t e P o i s s o n d i s t r i b u t i o n w i t h

p a r a m e t e r 1 a n d a f a m i l y o f c o n v o l u t i o n s o f : T s's , w h i c h i s f r e q u e n t l y r e f e r r e dt o a s t h e g e n e r a l i z e d P o i s s o n d i s t r i b u t i o n ( s ee , e . g ., [L u ] , C h . 1 2 ) . D i r e c t c o m -p u t a t i o n s h o w s t h a t t h e c h a r a c t e r i s ti c f u n c t i o n ~c 'S o f / C ~ f i s g i v e n b y

( / ? )S K y ( t ) ) = C X : , / ( t ) = e x p (e t z - - 1)d .T f (x )Oo= e x p ( f ( t ) - 1 ) = G ( f ) ( t ) , t E ] ~T h i s r e m a r k t o g e t h e r w i t h D e f i n i ti o n 3 .1 j u s ti f ie s t h e f o l l o w i n g a s s e r ti o n .L E M M A 3 . 3 : I f X i s a n r . i. s p a c e o n [0, 1], t h e n t h e o p e r a t o r E m a p s Xb o u n d e d l y i n t o i t s e l f i f a n d o n l y i f X E K .

R e m a r k 3 . 4 : T h e s t a t e m e n t o f L e m m a 3 . 3 i n t e r m s o f o p e r a t o r b o u n d e d n e s sw i l l e n a b l e u s l a t e r t o a p p l y i n t e r p o l a t i o n t e c h n i q u e s T H E O R E M 3 .5 : L e t X C Y be r .i . spac es on [0, 1]. C ons i de r t he f o l l ow i ngcondi t ions :

k n( i ) t h e r e ex i s t s C > 0 such t h a t (1 .2 ) ho l ds f o r an a r b i t r ar y s e que nc e { f } k = lC X o f i nde p e nde n t r a ndo m v ar i ab le s s a t i s f y i ng (1 .1 ) ;

(ii) there ex i s t s C > 0 such t h a t (1 .2 ) ho l ds f o r an ar b i t r ar y s e qu e nc e { f k }~ - = lC X o f i nde p e nd e n t i de n t i c a l l y d i s t r i bu t e d r an dom v ar iab l es s a t i s f y i ng(1.1);

( i i i ) t h e o p e r a t o r ]C a c t s b o u n d e d l y f r o m X i n t o y x x ;( i i i ) ' t he ope r a t or 1C a c t s b o u n d e d l y fr o m X i n to Y .


8/32

1 32 S . V . A S T A S H K I N A N D F . A . S U K O C H E V I s r . J . M a t h .

T h e f o l l o w i n g i m p l i ca t i o n s h o l d : ( ii i) ' ~ ( i) .z----~ i i) . I f t h e s p a c e Y i s e q u i p p e dw i t h a F a t o u n o r m , t h e n ( i) ~ = = ~ (i i) ~ = ~ O ii) .Proof ." T h e i m p l i c a t io n ( i) = = * ( ii ) is o b v i o u s. A s s u m e n o w t h a t ( ii) h o l d s . F i xf E X a n d n E N a n d c h o o s e h C X s u c h t h a t 5Ch = 5 of a n d s u c h t h a t h

h na n d X [ 0,1 /n ] a r e i n d e p e n d e n t . S e t h n : = hX[o ,1 /n] , a n d l e t {X[o ,1 /n ] , ~ ,k }k = l b ea s e t o f ( n + 1 ) i n d e p e n d e n t r a n d o m v a r i a b l e s s u c h t h a t Y h ~,~ = ~ - h , f o r a l l1 < k < n . S i n c e t h e f u n c t i o n s I ~ = 1 h n ,k l a n d [h i h a v e t h e s a m e d i s t r i b u t io nf u n c t i o n , w e c o n c l u d e t h a t t h e f u n c t i o n s [ ~ 2 = t h n,k [ a n d I f[ a r e e q u i d i s t r i b u t e d ,a n d t h e r e fo r e , b y a s s u m p t i o n ,

(3 .2 ) L h ~ , k y -< C L ~tn,k x - - C I I f l l x .k = l k : l

A d i r e c t c o m p u t a t i o n s h o w s t h a t C h , ( t) = n - i C y ( t ) + (1 - - n - 1 ) f o r a l l t E If{.H e n c e , t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s u m H n : = ~ , ~ ' = t h n , k i s g i v e n b y

C H ~ ( t ) = ( n - t ( f ( t ) - - 1) + 1 ) n , Vt E I~ .S i n c e l i m n - ~ C H . ( t ) = e x p ( / ( t ) - 1 ) = ~ ( f ) ( t ) , f o r a ll t E II~ w e s e e t h a tH n c o n v e r g e s w e a k l y t o ~ f . C o m b i n i n g t h is w i t h ( 3 . 2) , w i t h [B r] P r o p o s i t i o n3 , p p . 3 - 4 a n d w i t h t h e f a ct t h a t t h e n a t u r a l e m b e d d i n g o f Y i n to y x x i s a ni s o m e t r y , w e c o n c l u d e t h a t I lK : l il y _< C l l f l l x . T h i s c o m p l e t e s t h e p r o o f o f t h ei m p l i c a t i o n ( i i ) = = * ( i i i ) .

A s s u m e n o w t h a t ( ii i) h o ld s , i. e. t h a t t h e r e e x is t s C < o ~ s u c h t h a t [ [~ l lx _ ~ y _< C . W e s h a ll f i rs t s h o w t h e a s s e r t i o n ( i) u n d e r a n a d d i t i o n a l a s s u m p t i o n t h a t

k nh e s e q u e n c e { ~ ' } k = t i s s y m m e t r i c a l l y d i s t r i b u t e d . I n [ P r] , Y u . V . P r o k h o r o vp r o v e d t h a t i n t h i s c a s e, i f t h e s e q u e n c e { h k } ~= 1 c o n s i st s o f i n d e p e n d e n t r a n d o mv a r i a b l e s s u c h t h a t - ~h ~ = $ -,~ (I ~) f o r a l l k = 1 , 2 , . . . , n , t h e n

A >_X - - 8 1 hk > ~ , V X > 0 ." ' k = l ' k = l

I t t h e n f o ll o w s f r o m t h i s i n e q u a l i t y ( s e e e .g . [ K P S ] , I I . ( 4 . 1 7 ) ) t h a t

L 1 6 k ~ l h k Yv _


9/32

V o l . 1 4 5 , 2 0 0 5 S E R I E S O F I N D E P E N D E N T R A N D O M V A R I A B L E S 1 3 3

T h e r e f o r e ,n n

, ~ ( s ) ( t ) = I I ~ ( s ~ ) ( t ) = I I ~ ( t ) , t ~.k = l k-----1

In o t he r w ord s , 3c,~(S ) = f 'E~=~ hk a n d h e n c e , b y t h e a s s u m p t i o n , b y t h e e q u a l i t y 7r (/) = ~c (y ) ( se e t h e a r g u m e n t b e f o r e L e m m a 3 .3 ) a n d b y t h e i n e q u a l i t y a b o v e ,

~ ] k y 16 ~-~ Y 16C ~ - x ._ hk = 1611Efily _< 1 6 C I I fl i x = f kk~- -I k----1

S i n c e f k E X , k >_ 1, X C Y a n d Y h a s a F a t o u n o r m , w e c o n cl ud e t h a tII~_,k=l fk[iY


10/32

1 34 S . V . A S T A S H K I N A N D F . A . S U K O C H E V I sr . J . M a t h .

( r e s p e c t i v e l y , ]k(s) = ]~.(s) = hk(s) = 0 ) . C l e a r ly , e a c h o f t h e s e q u e n c e s{)~}~.=1 , {]~-}~=1 a n d { hk }~ -= l c o n s i s t s o f p a i r w i s e d i s j o i n t e l e m e n t s a n d ~ ' /k =~ ' f k , 5~/~. = ~ [ ~., 9~ h~ = Y h k f o r a l l k = 1 , 2 , . . . , n . I t f o l l o w s n o w f r o m t h es y m m e t r i z a t i o n i n e q u a l i t y ( s e e [ V T C ] , C h . 5, P r o p . 2 . 2 ) t h a t

A{ lhk l > x } < 2{ I fk I > x /2 } , x > O.H e n c e , i t fo l lo w s f r o m ( 3.4 ) t h a t , b y t h e f ir s t p a r t o f t h e p r o o f ,

n Y - < C ~ h k t < _ 4 C ~ ] k X3.5) ~ hkk----1 X k = l

O n t h e o t h e r h a n d , w e n o t e fi rs t t h a t b y ( 3 .4 ) w e h a v e I I ( ~ k = l f k ) X[O ,a]IIY =II ~-,~=1 k ll r . N o w , w r i t in g

( / o )k = f k - Eyk - - (y~ - - E f k ) E f := f ( x )d xa n d c o m b i n i n g [ B r], P r o p . 1 1, p . 6 w i t h ( 3 .4 ) , w e g e t t h a t f o r s o m e c o n s t a n tC ( Y ) w e h a v e

h k > c ( y )Y k = l k = l "()> C(Y) f k - - E f k Xsupp(Ek=n f k )k = l k = l r

~ C ( Y ) ( l ~ f k - E ~ -~ ,fk]']X [O ,a]H V)"k = l I IY k = lS in ce th e in eq u a l i ty ] ] f l ]Y -> ] IX [o ,1 ] [ r E [ f ] h o l d s i n e v e r y r .i . s p a c e Y ( s e e [K P S ]C h . 2 , T h e o r e m 4 .1 ) , w e in f e r f r o m t h e i n e q u a l i t y a b o v e a n d ( 3. 3) t h a t

h k y n t ] f k yC ( Y ) ~ f k ( 1 - H X [ o , , I [ I Y ~ C ( Y )k = l - - - - Y H X [ 0 ' I ] H Y ] ~ - 2 k = l "

T o g e t h e r w i t h ( 3 . 5 ) t h i s y i e l d s ( 1 . 2 ) w i t h t h e c o n s t a n t 8 C / C ( Y ) .F i n a ll y , l e t us c o n s i d e r t h e s e t t in g o f a n a r b i t r a r y s e q u e n c e o f i n d e p e n d e n t

r a n d o m v a r i a b l e s { fk}~ .=l C X s a t i s f y i n g c o n d i t i o n ( 1 .1 ) . W i t h o u t l o ss o f g e n -e r a li ty , w e m a y ( a n d s h al l) a s s u m e t h a t f o r a n y b a n d e a c h k = 1 , 2 , . . . , nw e h a v e A { f k = b } = 0 . S i n c e t h e s e q u e n c e { ]k }~=l s a t i s fi e s ( 1 . 1 ) , w e c a n s e l e c tn u m b e r s 0 = b o < b l < b 2 < - - . < bl = o c , w h e r e 1 : = [ l / a ] + 1 s u c h t h a t

nZ A { b k - 1 < f i < b k } < a , V k = l , 2 , . . . ,1 .i= l


11/32

V ol . 1 45 , 2 00 5 S E R I E S O F I N D E P E N D E N T R A N D O M V A R I A B L E S 1 3 5

S e t f i ,k (x) := f i(x)x{b~_l 0 . T h e ( L u x e m b u r g ) n o r m i n L ,~ = L [ 0 , a ) i s d e f i n e d b y

[ [ f [ [ = i n f { p > 0 : fo~b(lf(t)l)dt _ l , t el l~ ,k----0


12/32

136 S. V, ASTA SHKI N AND F. A. SUKO CHEV Isr . J . Math .

i s a n Or l ic z f unc t ion ( he r e , a s u sua l , [ l / p ] de no te s the in t e g r a l pa r t o f 1 / p ) . T h ec or r e spon d ing Or l i c z spa ce , L N p , i s f r e que n t ly de n o te d b y e xp( Lp) ( e xp( Loo) :=L ~ ) . I t f ol lows f rom the o r ig ina l pa p e r o f Kr ug lov [ K] tha t e xp( Lp) E K f o r a ll0 < p _< 1 and i t is not ed in [Br ] th a t th is i s no longer the case wh en p > 1 . Fo rc om ple te ne ss sa ke , we p r e se n t a s im ple p r oof o f Kr ug lov ' s r e su l t .PROPOSITION 4.1 : I f i s an O r l ic z f unc t ion such th a t f or som e cons ta n t B >_ 1 ,

( x + y ) < B ~ ( x ) ~ ( y ) , V x, y > O ,t h e n l C ( b o u n d e d l y ) m a p s L ~ i n t o i t s e l f .Proo f : W e s h a l l d e n o t e f ~ f ( x ) d xf o ~ ( b f ( x )D d x -< 1 (i.e. IIf[IL.


13/32

V ol. 14 5, 2 00 5 S E R IE S O F I N D E P E N D E N T R A N D O M V A R I A B L E S 1 37

W e r e m a r k t h a t t h e a s s e r t i o n o f P r o p o s i t i o n 4 .1 ( a n d t h u s t h a t o f C o r o l l a ry4 .2) r e m a in va l id ( w i th a na logou s p r oof ) i f t he Or l i c z f unc t io n s a t is f ie s thefo l lowing condi t ion:

( x + y ) < B ( ( x ) + ~ ( y ) ) , W , ~ > 0 .W e i n t r o d u c e n e x t t h e O r li cz f u n c ti o n s

M p ( t ) := e I t l l " l/ ' ( e+l t l ) - 1 , p > O, t E ~ .De no te by k9 the se t o f inc r e a s ing c onc a ve f unc t ions on [0 , ~ ) w i th ~ ( 0 ) =

( + 0 ) = 0 . I f E 9 , t h e n t h e M a r c in k i e w i cz s p a c e M e [ 0 , a ) c o n s is t s o f t h o s em e a sur a b le f unc t ions x f o r wh ic h

I lX l lM ~ [ O ,~ ) = = s u p 1 - 2 - - x * ( t ) d t < ~ .o < s < ~ (~)I t i s u se f u l t o no te th a t f o r e ve r y p > 0 , t he Or l ic z spa ce L M , ( respec t ive ly ,L g p ) c o inc ide s w i th the M a r c ink iewic z spa c e M C p ( r e spe c t ive ly , M C p) , wh e r e

t l n ( e / t ) ( r e spe c t ive ly ,p ( t ) := t l n l / p ( e / t ) ) , t > O .Cp( t ) := l n l / p ( l n ( e e / t ) )F or the r e a de r ' s c onve n ie nc e , we inc lude a sho r t p r oo f o f th i s obse r va t ionbe low. I n th i s p r oof , a s we ll as in f u r the r a r g um e n t s in th i s s e c t ion , we sha ll

f r e que n t ly use the e qu iva le n t e xpr e s s ions fo r the no r m s on the M a r c ink ie wic zspa c e s M e . a nd M , on th e in t e r va l [ 0, 1] wh ic h fo llow f r om [ KP S ] The or e mII.5.3,

tI L x l l . ~ s u p - - ~ *te (o ,1) p( t ) ( t ) , x e Mp an d(4.1) t

I l x l l M o , , sup x * ( t ) , x e M .r e ( 0 , 1 )LEMMA 4.3: T h e e q u a l i t i e s L M p = M ~ p a n d ( e xp( Lp) = ) L N p = M ~ p ( w i t hn o r m e q u i v a l e n c e ) h o l d f o r e ve r y p > 0 .P r o o f ' . W e sha ll p r ove on ly th e f i r s t e qua l ity , s inc e the p r o of o f the se c ond i ss im i la r ( a nd s im ple r ). F ix p > 0 . I t i s su f fi c ie n t to p r ove th a t t he f u nd a m e n ta lf u n c t io n s C L ~ , a n d ~ , , a r e e q u i v a le n t a n d t h a t t h e f u n c t io n ] p : t --+ ~ ( t ) / tbe longs to the Or l i c z spa c e L M ~ (see [KPS], Sec tion II .5.4 an d (4.1) abo ve) .S ince

1 l n l / P ( l n ( e e / t ) ) 1C L M p ( t ) - - M p , ( 1 / t ) , C M ~ , ( t ) = I n ( e / t ) ( = f p ( t ) ) ' t >O


14/32

1 38 S . V . A S T A S H K I N A N D F . A . S U K O C H E V I sr . J . M a t h .( s e e [BS] , Le m m a 4.8.17 a nd [KPS], T he or em I I .5 .7) , i t suff ices to check tha tth e f u n c t i o n s M p I ( 1 / . ) a nd f p ( . ) a r e e qu iva le n t in a ne ighb our h ood o f 0 . S inc ef o r e ve r y pos i t ive c we ha ve

ln l /P(e + c" ln(e / t )l i m l n l / V ( l n ( e ~ / t ) ) j ~ -- 1 ,t - ~ o l n l / V ( l n ( e e / t ) )we can se lec t 5 > 0 such th a t for a ll t E (0, 5) we have(4.2) M v (~ fp ( t) )


15/32

Vol. 145, 2005 SE RI ES OF

T o t h is e n d , n o t i n g t h a t f o r a ll

a n d t h u s

IND EPEN DEN T RANDOM VARIABLES 139

k E N

k ! t k = -el ( 1 + 1 )~ ( k + 1 )(k + 2 ) . . . (k + i )1(1 1 )< - 1 + +- e ~ ( k + i ) ( k + i + l )1 (_ < - 1 + _ < - ,e e1 1 2 1e k ~ . < _ t k < _ e k ~ , k E N .

C o m b i n i n g S t i r l i n g ' s f o r m u l a k ! ~,, v / - ~ k k e - k w i t h t h e i n e q u a l i t y a b o v e , i tfo llows th a t fo r a l l su f f ic i en t ly l a rge k , we hav e

k - k < _ t k < t k - 1 l ~ 2 ~f k L , } _ < C ~ 2 -~ fk c"- k=l k= l

U s i n g t h e T a y l o r d e c o m p o s i t i o n o f t h e f u n c t i o n M l ( t ) = ( t + 1) t - 1 a t th en e i g h b o u r h o o d o f 0 , it is n o t h a r d t o s e e t h a t t h e l ef t h a n d s i d e o f t h e p r e c e d i n gine qu al i ty i s eq uiv a len t to I1 }--~k=lfk l lnM1 (see a lso [A]) .U s i n g C o r o l l a ry 4 .2 , w e s h a ll n o w s h o w h o w t h e r e s u l t o f T h e o r e m 4 .4 c a n b ee x t e n d e d t o a l l v a lu e s o f p E ( 1, c o ) .


16/32

140 S.V . ASTASH KIN AND F. A. SUKOC HEV Isr . J . Math.THEOREM 4.5: T h e set Yp, 1 < p < c~ o r d e r e d b y i n c l u s i o n h a s a u n i q u em i n i m a l e l e m e n t , L M ~ , w h e r e l i p + 1 / q = 1 .P r o o ~ I t fo l lo w s f r o m P r o p o s i t i o n 4 .1 a n d T h e o r e m 4 . 4 t h a t t h e o p e r a t o rm a p s b o u n d e d l y L c i n t o L M 1 a n d e x p ( L 1 ) ( = L N I ) i n to it s e l f. H e n c e , u s in gth e r e a l m e t h o d o f i n t e r p o l a t i o n ( se e , e . g ., [ L T]), w e in f e r t h a t

]C: (L ~ , L N1 )O , o~ -+ (L M~ , L N~ )O, oo , 0 < ~ < 1 .B y L e m m a 4 .3 , LN ~ = M e , , LM ~ ---- M ~ a n d i t i s w e l l k n o w n th a t t h e s p a c e L ~i s t h e M a r c i n k ie w i c z s p a c e M i d , w h e r e i d ( t ) = t for a ll t >__ 0. T he re fo re , us in gt h e k n o w n d e s c r ip t i on o f t h e s p ac e s o b t a i n e d b y t h e r e a l m e t h o d o f i n t e r p o l a t i o nin Marc ink iewicz coup les ( see , e .g . , [O] , Ex . 7 .1 .3 , p . 428 ) we ob ta in

( L ~ , L N 1 ) e , ~ = L Y e _ l , ( L M 1 , L N 1 ) e ,~ = LM (I_~)_I S e t t i n g p = 8 - 1 , w e i m m e d i a t e l y in f e r t h a t ]C m a p s L N p i n to L M , . I t n o w f o l lo w sf r o m ( 4.1 ) t h a t i n o r d e r t o c o m p l e t e t h e p r o o f o f T h e o r e m 4 .5 , i t is s uf f ic i e n t t os h o w t h a t f o r s o m e s c a l a r C ( p o s si b l y d e p e n d e n t o n p ) a n d a l l s u f fi c i en t l y s m a l lt > 0 we have

ho (t) 0 w e h a v e

~ { t : g o > 7 - / c } > _ ~ {ho > 7-} .Fu r th er , a d i r ec t ver i f ic a t ion show s th a t A{ho > 7-} < e -~ tn l /q~ - fo r a l l su f f i -c ie n t ly la rge 7-, hen ce i t i s su f f ic ien t to p rov e th a t A{~go > 7 -/3} > e - r l n l / q r ,o r e q u iv a l e n t l y ,(4.3) A{]Cgo > 7"} >_ e -3r Inl/qTfor all sufficiently arge 7-. We have

A(go > 7-} ---- ex p( 1 - 7--P) >_ e - r p .F i x n E N a n d l e t g l , g 2 , . . . , g n b e i n d e p e n d e n t c o p ie s o f g o. S e t t i n g g : =m i n ( g l , g 2 , . . . , g n ) w e h a v e

n

~ { g > 7 -} = ~ ( n i % l { g i > ~ } ) = I I ~ { g i > 7 -} > e - n r pi - - - - 1


17/32

Vol. 145, 2005 SERIES OF INDE PEND ENT RAN DOM VARIABLES 141H e n c e , u s i n g t h e d e f i n i t io n o f t h e o p e r a t o r ] C, w e o b t a i n

A{~:g0 > ~} =( o o n }

A E E g n , k X E , , > Tx n.~l k----1

> _ A { ~ F t m i n { g n , 1 , . . . , g n , n } X E , , > 7 " }oo

= ~ A { m i n g n,k > T / n } A ( E n )k = l , 2 , . . . , no o o o

> E e _ ~ ( , / . F A ( E . ) = I_ E e _ . , - , ~ . ~ l- - e n ! "n = l n . . = -1

N o t e t h a t i f n > v ln - 1 / P ( 1 + v ) , t h e n ( e q u i v a l e n t l y ) ~ - I n l - U P ( 1 + v ) >_ n l - P 7 p.T h e r e f o r e ,

(4 .4)A{~go :> T} > e x p ( - - T ln l - U P ( 1 + r ) ) ~ E

n > T l n - 1 / P ( l + r )1e x p ( - v I n 1 - 1/ p T) e[2T In -1 / p r ] !"

1n !

U s i n g S t i r li n g ' s f o r m u l a , w e s e e t h a t f o r s u f fi c ie n t ly l a r g e T1 > e x p ( - - 2 T In - U p T ln (2T In -U p T)) _> e -2T In l - ' /P T .e [2v In -1 /p T]! - -

C o m b i n i n g t h i s e s ti m a t e w i t h ( 4 .4 ), w e o b t a i n ( 4.3 ) . T h i s c o m p l e t e s t h e p r o o fo f T h e o r e m 4 .5 . |

COROLLARY 4 .6 : L e t p E (1 , c~] an d q = p / ( p - 1 ). T h e r e e x i s t s C p > 0 s u c ht h a t f o r a n y f i n i te s e q u e n c e o f i n d e p e n d e n t r a n d o m v a r ia b le s { f k }~-=1 C ex p( L p)s a t i s f y i n g ( 1.1 ) , w e h a v e

n n< cp Z h- - L M q k = l I l e x p ( L p )M o r e o v e r , i f a n r .i. s p a c e Y w i t h a F a t o u n o r m i s s u c h t h a t f o r a ll s e q u e n c e s

k n e x p ( L p ) a b o v e w ef~" }k-----X C a s h a v e

I I l l y < e l l e x , L . > k : l


18/32

1 4 2 S . V . A S T A S H K I N A N D F . A . S U K O C H E V I sr . J . M a t h .

then necessarily , L C Y .M q - -Proof: I f p = o c ( r e s p e c t iv e l y , p < o c ) , t h e n c o m b i n e T h e o r e m s 3 . 5 a n d 4 . 4( r e s p e c t i v e l y , 4 . 5 ) a n d u s e t h e f a c t t h a t L i s t h e s m a l l e s t a m o n g a ll r .i . s p a c e sY s a t i s f y i n g Y _D LMq. I

5 . T h e o p e r a t o r K i n L o r e n t z s p a c e sR e c a l l t h a t i s t h e s e t o f a l l i n c r e a s i n g c o n c a v e f u n c t i o n s o n [0 , o o ) w i t h g ; (0 ) = ( + 0 ) = 0 . I f E ~ , t h e n t h e L o r e n t z s p a c e ( A [ 0 , a ) , ]I " ]]i~ [0 ,~ )) o n [ 0 , a ) i st h e s p a c e o f a ll m e a s u r a b l e f u n c t i o n s x o n [0 , a ) f o r w h i c h

I l x l h , , [ 0 , . ) : = / / * ( t ) d ( t ) < ~ .0

T h e s p a c e A [ 0 , a ) , 1 _4 a < c o is a l w a y s s e p a r a b l e .T h e f o ll o w in g th e o r e m d e s c r ib e s L o r e n t z s p a c e s A o n w h i c h t h e o p e r a t o r / C

a c t s b o u n d e d l y i n t e r m s o f t h e f u n c t i o n E ~ .T H E O R E M 5 . 1 : L e t ~b E 62. T h e opera tor ]~ maps the Loren t z s p a c e A ~ in toi t se l f ( i .e . A E K ) i f a n d o n l y i f t h e r e ex i s t s C > 0 such tha t

oo k( 5 . ~ ) F _ , ~ ( ~ ) . < _ c ( ~ ) , ~ ~ ( o , ~ ] .k= l, n E o oProof: L e t f : = X ( o,u ] u E ( 0 , 1 ] a n d l e t t h e s e q u e n c e s { fn ,k}k=l , {X ~ } n = l ,

n l~ s a t i s f y t h e c o n d i t i o n s ( i) a n d ( ii ) f r o m ( 3 . 1 )' . T h e n , t h e f u n c t i o n f ~ : =~ . = 1 f i ~ , k h a s a b i n o m i a l d i s t r i b u t i o n w i t h p a r a m e t e r u , i . e .

A { f n = k } = C k n u k ( 1 - u ) n - k , k = 0 , 1 , . . . , n , n e N ,w h e r e C k = n ! / k ! (n - k )! . T h e r e f o r e , w e h a v e ( s e e ( 3 . 1 )' )

(x )

~ : f ( s ) = E k ~ A ~ ( s ) ,k----1

w h e r ec ~ k ) ' u k ( i ~ .(Ak)= ~ C~ uk(1-u)n-~(E~ )= _1 ~! _~)~_~

n=k e n=k k ! ( n i_ 1 u k ~ - , ( 1 - - u ) n _ l U k ( 1 - - U ) k ~ - ~ ( l - - u ) ~Z - , - / _ . ~ f( 1 - u)kk~ ~ - - ~ ~ ~ - ~ - ~ ~=0n~.k

u k~ e - - ' a k !"


19/32

V ol. 14 5, 2 00 5 S E R I E S O F I N D E P E N D E N T R A N D O M V A R I A B L E S 14 3

H e n c e , t h e f u n c t i o n K : f h a s P o i s s o n d i s t r ib u t i o n w i t h p a r a m e t e r u , w h i c hc o i nc ide s w i t h t h e d i s t r i bu t i on o f t he fun c t i on

oo

h ( s ) := E x(0,~k](s), s e [0,1],k = lw h e r e Vk : = e -~ ~-~i~=k(ui/i!) , k E N. T herefo re ,( 5 . 2 ) I t ~ : f l l A v , = I l h l t A , = d e ( s ) = T k ) .

k-~l - -S i n c e K : i s a bo un de d pos i t i ve ope ra t o r f rom A i n t o t he s pa c e S [0 , 1 ], a nd s inc et h e e x t r e m e p o i n t s o f t h e p o s i ti v e p a r t o f t h e u n i t s p h e r e o f t h e s p a c e A a r eg i ve n by a ll no rm a l i z e d i nd i c a t o r f unc t i ons o f m e a s u ra b l e s ubs e t s o f [0 , 1 ], i t i ss u ff ic i en t t o ve r i fy t he bo und e dn e s s o f K o n A o n t he s e t o f a l l s uc h func t i ons( s e e C oro l l a ry 1 t o Le mma 5 . 2 i n [K P S ] , C ha p t e r I 1 . 5 ) . Therefore , i t fo l lowsf rom (5.2 ) t ha t K a c t s bou nd e d l y o n t he s pa c e A i f a nd on l y i f t he r e e x i s tsC > 0 s uc h t ha t

o0

(5 .3 ) E ( T k )


20/32

1 4 4 S . V . A S T A S H K I N A N D F . A . S U K O C H E V I sr . J . M a t h .

T h e e s t i m a t e ( 5. 4) n o w f o l lo w s e x a c t l y a s i n t h e p r o o f o f T h e o r e m 5 .1 .T h e c o n d i t i o n ( 5 .1 ) f o r c o n c a v e f u n c t i o n s ~b e q2 a p p e a r s a l s o i n [ M S ] ( s e e E q .

( 23 ) th e r e ) , w h i c h s t u d i e s r a n d o m r e a r r a n g e m e n t s i n r .i . s p a c e s . T h e f o ll o w i n gt e c h n i c a l e s t i m a t e i s e s t a b l i s h e d i n [M S ] , L e m m a 1 1.L E M M A 5 . 3 ( [M S ] ) : I f e i s s u c h t h a t ~ b ( 1 ) = 1 a n d ( t ) _< a t l i p f o r e dlt E [0, 1] a n d s o m e p , a [1 , oo ) , t h e n

1 o o ( U k )( 5 . 5 ) s u p ~ 0 s u c h t h a t f o r a n a r b i t r a r y se q u e n c e { f ' } k = l C Xo f i n d e p e n d e n t r a n d o m v a r ia b l es s a t i s f y i n g ( 1 .1 ) w e h a v e

( 5 . 6 ) ~ -~ fk x < _ C ~ ] k x "k----1 k----1

P r o o f ( i) B y t h e d e f i n i t i o n o f X , w e h a v e f o r e v e r y x X { / o I }lx l lxx : = s u p I x ( t ) y ( t) l d t : Y X , HY l I x < 1 .

T h i s c a n a ls o b e i n t e r p r e t e d a s( 5 . 7 ) X = h A c k , I Ix l lx ~ = s u p l l x l l A , , , ,

w h e r e t h e i n t e r s e c t io n a n d s u p r e m u m a r e t a k e n o v e r a ll % ~P s u c h t h a tC y ( t ) = y * ( s ) d s , t [ 0 , 1 ] , y X , I l Y l IN < 1 .

F o l l o w i n g [M S ] , f o r e v e r y s u c h % @ w e s e to ~ ( t ) . - ~ ( t ) + tI l y ll a + 1 ( y x x , t [ 0 , 1 ] ) .

C l e a r l y , 0 y @ a n d 0 u ( 1 ) = 1 f o r a ll y X . S i n c e ( s e e [ K P S ] , ( 1 1 . 4 . 6 ) )1I ly ll l + 1


21/32

V ol. 1 45 , 2 00 5 S E R I E S O F I N D E P E N D E N T R A N D O M V A R I A B L E S 1 45

w e d e d u c e t h a t( 5 . 8 ) I l X l l A ~ < ( x 0 ) + 1 ) I I x l I A , ~ , X X , y X , I l V l l x x < 1 .O n t h e o t h e r h a n d , f o r a ll x 's a s a b o v e , i t f ol lo w s f r o m ( 5. 7) t h a t

1 1I l x l lA ~ < I lx l lA ~ + I l x l l i < I I x l I A ~ + CXX, ,O- ' - - - - - -~IIxlIx (1 + ~ ) l l x l l x x ~ .C o m b i n i n g t h i s e s t i m a t e w i t h ( 5 . 7 ) a n d ( 5 . 8 ) , w e s e e t h a t

X = M A 0 y , I l x l l x x s u p I l x l l A ~ y ,y c x ,llyllx _ 1,a ( 0 , 1] a n d a l l t [0, 1].

U s i n g R e m a r k 5 . 2 a n d L e m m a 5 . 5 , i t i s n o w p o s s i b l e t o p r o v e t h e c o n v e r s e ,i n a c e r t a in s e n s e , t o t h e m a in r e s u l t o f [ J S ] ( f o r n o r m e d r . i . s p a c e s ) .


22/32

146 S. V. ASTASHKIN AND F. A. SUKOCHEV Isr. J. Math.COROLLARY 5 . 6 : I f an r. i. space E is such that for every max im al r . i. spaceX D E there ex is ts C > 0 such tha t (5.6) holds for an arbi trary sequence

k nf ' } k - - 1 C X of indep end ent ran dom variables sa t i s fy ing (1 .1), then E containsa n L ; - s p a c e f o r s o m e p E [1 , oo ) .Proof'. A c c o r d i n g t o [ K P S ] , T h e o r e m I I .5 .7 , E C_ M t/~ , w h e r e CE i s t h ef u n d a m e n t a l f u n c t i o n o f E . T h e r e f o r e , f or e v e r y f u n c t i o n E s u c h t h a t 0 , w e h a v e E C M t/cz~(t) C_ M t/( t ) . H e n c e ,b y t h e a s s u m p t i o n a n d T h e o r e m 3 .5 , t h e o p e r a t o r ~ b o u n d e d l y m a p s Mt/ ( t )i n t o i ts e l f a n d t h i s i m p l i e s ( s e e R e m a r k 5 . 2 ) t h a t c o n d i t i o n ( 5 .4 ) h o l d s fo r e v e r yf u n c t i o n a s a b o v e . B y L e m m a 5 .5 t h i s g u a r a n t e e s t h a t C E ( t ) _~ CE(1)at a f o rs o m e a _> 1 , a E ( 0 , 1 ] a n d a l l t E [ 0, 1 ]. T h e l a t t e r f a c t a n d [ K P S ] , T h e o r e mI I . 5 . 5 g u a r a n t e e t h a t E _ A V ~ _3 A ~ a n d s i n c e A t~ c o n t a i n s L p [ 0 , 1 ], p > 1 / a ,w e a r e d o n e . ICOROLLARY 5.7: I f ~ E ff2 i s s u c h tha t for every a E (0 , 1 ]

s u p t - a ~ ( t ) = o o,rE(0,1]the n th ere exists E such th at ~_ C~b f o r som e C > 0 suc h tha t the ope ra tor]C is no t bo und ed on an y r .i . space X wi th th e fund am enta l func t ion Cx = .

T h e p r e c e d i n g r e s u l t s s h o w t h a t i t i s n o t p o s s i b l e t o a n s w e r t h e m a i n q u e s t i o n( se e S e c t i o n 1 ) i n t e r m s a n a l o g o u s t o th e R o d i n - S e m e n o v c h a r a c t e r i z a t i o n o f r .i .s p a c e s s a t is f y i n g t h e K h i n t c h i n e I n e q u a l it y . I t s h o u l d b e a l so m e n t i o n e d t h a tt h e r e a r e m a n y r . i. s p a c e s w h i c h h a v e t h e K r u g l o v p r o p e r t y a n d w h i c h d o n o tn e c e s s a r i l y c o n t a i n s o m e L p - s p a c e , 1 < p < e c ( s ee , e .g . , C o r o l l a r y 4 . 2 ). T h en e x t p r o p o s i t i o n p r e s e n t s a ge n e ra l m e t h o d t o e x h ib i t s u c h e x a m p l e s a m o n gM a r c i n k i e w i c z s p a c e s .P R O P O S I T I O N 5 . 8 : For every quas i -concave non-negat ive func t ion p on [0, 1],le t ~ ( t ) : = p ( t l n e / t ) . I f ~ ( t ) : = t / ( t ) , t E [ 0 , 1 ] , t h e n M ~ E K .Proof'. W e d e n o t e b y l~o(1/p) t h e s p a c e o f a ll t w o - s i d e d s c a l a r s e q u e n c e s a =

oo{ a k } k = - o o s u c h t h a t{ ak ~oo I]llall~(llp) ~ Jk=-o oII~ < oo.

S i nc e , b y P r o p o s i t i o n 4 .1 , t h e o p e r a t o r / C is b o u n d e d o n L 1 a n d LN1, w e i n f e rK K" ) l ~ ( i / p )h a t i t is a l so b o u n d e d o n t h e s p a c e (L1,LN1) l~(1/p) , w h e r e ( ., i s h e


23/32

Vol. 145, 2005 SERIES OF IN DE PE ND EN T RAN DOM VARIABLE S 147

f u n c t o r o f t h e r e a l in t e r p o l a t i o n m e t h o d , g e n e r a t e d b y t h e p a r a m e t e r lo0 ( l / p )( s e e , e . g ., [ O ] S e c t i o n 7 .1 , p . 4 2 1 ) . C o n s i d e r i n g t h e s p a c e s L 1 a n d LN~ a sM a r c i n k i e w i c z s p a c e s a n d a p p l y i n g [O ], E x . 7 .1 .3 , p . 42 8 ( c o m p a r e w i t h t h ep r o o f o f T h e o r e m 4 .5 ), w e o b t a i n (L1, LN1 )loo(1/p)K = M & |

6 . C o m p a r i n g i n d e p e n d e n t s u m s t o d is j o i n t s u m s i n t h e g e n e r a l c a s eH e r e w e c o n s i d e r t h e m a i n q u e s t i o n ( s e e S e c t i o n 1 ) i n t h e s e t t i n g w h e n t h ec o n d i t i o n ( 1 .1 ) is n o lo n g e r a s s u m e d . W e sh a l l s h o w t h a t t h e c o n d i t i o n X E Kr e m a i n s s u f f ic i e n t f o r a ( m o d i f ie d ) i n e q u a l i t y b e t w e e n t h e s u m s o f i n d e p e n d e n tr a n d o m v a r i a b l e s a n d t h e i r d i s jo i n t c o p i e s t o h o l d . F o r a n a r b i t r a r y r . i. s p a c eX o n [0 , 1] a n d a n a r b i t r a r y p [1 , ~ ] , w e d e f i n e a f u n c t i o n s p a c e Z ~ o n [0 , o c )b yZ ~ : : { f L a [ O , ~ ) + L ~ [ O , ~ ) : I l f l l ~ : = Hf*x[o,1]llx -4-IIf*x[1,~)llp < ~ } .

C l e a r l y , I I is a q u a s i - n o r m . I t is e a s y t o s e e t h a t Z ] e q u i p p e d w i t h t h ee q u i v a l e n t n o r m

IIflIz~ := I I f* x [o ,1 ] i Ix + IIflt(Ll+L p)(o,oo), f ZPxi s a n r .i . s p a c e o n [0 , c ~ ) . I n d e e d , t h e e q u i v a l e n c e o f t h e q u a s i - n o r m II " I 1 ~ a n dt h e n o r m I 1 " ]lZx f o l l o w s f r o m t h e w e l l - k n o w n f o r m u l a

f 0 'l f i l ( L , + L , , ) ( o , o o ) x f* (x )d x -4- ( ( f* (x ) )P dx ) 1 /p , f ( L 1 + L p ) ( 0 , c ~ ) ,w h e r e t h e s e c o n d s u m m a n d v a n is h e s w h e n p - - ~ a n d t h e e q u i v al e n c e c o n s t a n t sd o n o t d e p e n d o n p E [ 1, o c ) ( s e e [ B L ]) .

T h e s p a c e ( Z x , I1" I Izx) : = ( Z ~ , II " I I z~ ) w a s i n t r o d u c e d i n [J S] a n d o u r f i r s tr e s u l t i n th i s s e c t i o n c o m p l e m e n t s [J S] T h e o r e m 1 (s e e i n e q u a l i t y ( 4) t h e r e ) .T H E O R E M 6 . 1 : L e t X a n d Y be r .i . spac es on [0 , 1] such th a t X C_ Y . I f e i ther( i) the opera tor ]C ac ts bo un ded ly f rom X in to y x x and Y has Fa tou norm , o r( ii ) the op erator JC ac ts bou nd edly f rom X in to Y , then there ex is t s C1 > 0 suchtha t f o r e v e r y se que nc e { g i } , ~ l C X , n E N , o f inde p e nde n t ran dom v ariab le s,the fo l lowing inequal i ty ho lds:(6 .1 ) ~ g i Y K - c 1 ~ Zxi i=l /=1


24/32

1 48 S . V . A S T A S H K I N A N D F . A . S U K O C H E V I sr . J . M a t h .Proof: W i t h o u t l o ss o f g e n e r a l i t y w e m a y ( a n d s h a l l ) a s s u m e t h a t g i _> 0 a n dt h a t A { g i = v } = 0 f o r a l l T E l ~ a n d a l l i = 1 , 2 , . . . , n . F i x 0 = t ~ < t ~ - i< . ." < t l < t o = c c s u c h t h a t

nZ A {t j < g i < t j -1 } = 1 ,i= 1 j = 1 , 2 , . . . , n .

F o r t h e s equence {giX{gi>h}}inl c o n d i t i o n ( 1 .1 ) i s s a t is f ie d , h e n c e b y T h e o -r e m 3 . 5

( 6 . 2 )E g i x { g ~ > t ~l t ll

: C ( Z g i ) ' ~ ( [ O , 1 ] Xi= 1C n_< ~ i i= 1 II Z x

S i m i l a r l y , c o n d i t i o n ( 1 . 1 ) is a l s o s a t is f i e d fo r s e q u e n c e s {g ix l t j


25/32

Vol. 145, 2005 SERIES OF INDE PE ND ENT RAN DOM VARIABLES 149

W e c a n n o w c o m p l e m e n t T h e o r e m 3 . 5 , C o r o l l a r y 4 . 2 , C o r o l l a r y 4 . 6 a n dThe ore m 5 . 1 a s fo l l ow s .COROLLARY 6 . 2 : T h e following condition is equivalent to conditions ( i ) -( i i i ) inT h e o r e m 3.5:

(iv) the inequality (6.1) holds for e v e r y sequence {gi}n=l C X, n e N, ofindependent random variables.COROLLARY 6.3: / f (~ is an Orlicz [unction such that for some constant B > 1,

( x + y ) < V x , y > 0 ,

then there exists C > 0 such that for any finite sequence of independent randomk nariables {f }k=l C L~ we have

~ l f k L ,


26/32

150 S .V . ASTA SHK IN AND F . A . SUKO CHE V Is r . J . M ath.h nP r o o f : I t fo l lo w s f r o m [ C D S ], L e m m a 2 .3 t h a t f o r a n y s e q u e n c e { ~ } i = 1 o f

n o n - n e g a t i v e f u n c t i o n s f r o m S ( 0 , 1 ), w e h a v e

) , f ( n ) .fZk ( s ) ds O .k = l k = l "I t is e a s y t o i n f er f ro m t h e w e l l - k n o w n d e s c r i p t io n o f i n t e r p o l a t i o n s p a c e s f o ra c o u p l e ( L 1 , L o o ) ( s e e, e . g ., [ K P S ] , T h e o r e m 1 1 . 4 .3 ) t h a t t h e s p a c e Z x i s a ni n t e r p o l a t i o n s p a c e fo r t h e B a n a c h c o u p l e ( L l ( 0 , o o ) , L o o (0 , o ~ ) ). W i t h o u t l o sso f g e n e r a l it y , w e m a y ( a n d s h a ll ) a s s u m e t h a t t h e i n t e r p o l a t i o n c o n s t a n t o f Z xi s e q u a l t o 1 . T h e r e f o r e , i t fo l lo w s f r o m t h e i n e q u a l i t y a b o v e , o u r a s s u m p t i o n so n f k ' s a n d g k ' s a n d [ K P S ] , T h e o r e m 1 1 . 4 . 3 t h a t

k = l IIZx k=lA c o m b i n a t i o n o f t h i s i n e q u a l i t y w i t h T h e o r e m 6 .1 c o m p l e t e s t h e p r o o f . I

I n th e r e m a i n d e r o f t h is s e c t io n , w e sh a ll s h o w h o w o u r m e t h o d s m a y b e u s e dt o c o m p l e m e n t r e s u l t s f r o m [ M ] .

F o l l o w i n g [ L T ] p . 4 6 , w e d e f i n e t h e s p a c e X ( l p ) a s t h e s e t o f a ll s e q u e n c e sf = { f k ( x ) } ~ = l , f k X , k _> 1 s u c h t h a t

Ilfl lx-uZD) := sup I / k l p < on _ X

( w i t h a n o b v i o u s m o d i f i c a t io n f o r p = o o ) . T h e c l o s e d s u b s p a c e o f X ( l p ) g e n e r -a t e d b y a ll e v e n t u a l l y v a n i s h in g s e q u e n c e s f X ( I p ) i s d e n o t e d b y X ( I p ) .

B e f o r e p r o c e e d i n g , w e r e c al l t h e f o ll o w i n g c o n s t r u c t i o n d u e t o A . P . C a l d e r o n[C ]. L e t X o a n d X 1 b e t w o B a n a c h l a t ti c e s o f m e a s u r a b l e f u n c t i o n s o n t h e s a m e

3 ( 1 - 0 ' / 7m e a s u r e s p a c e ( M , m ) a n d le t 19 ( 0 , 1 ) . T h e s p a c e ~ o ~ 1 c o n s i s ts o f a llm e a s u r a b l e f u n c t i o n s f o n ( M , m ) s u c h t h a t f o r s o m e ,k > 0 a n d f~ X ~ w i t hI I / i l lx , _ 1, i = 0 , 1 ,

[f(x)l


27/32

Vol. 145, 2005 SERIES OF IN DEP EN DEN T RAN DOM VARIABL ES 151

f u n c t io n s o n s o m e m e a s u r e s p a c e ( M ' , m ' ) , t h e n a n y p o s it iv e o p e r a t o r A f r omS ( M , m ) i n to S ( J ~ ' , m ' ) , w h i c h a c ts b o u n d e d l y fr o m t h e c o u p le ( X 0 , X a ) i n tothe coup le (Y0, Y1) , a l so map s b oun ded ly X I - e x e 1 in to Y o l - e Y 1 and , in add i t ion ,

1-6IlA]lx~-Oxf_.+v~-~vo < [[Al[xo__+vo[IA[[__+v fo r a l l 0 (0 ,1 ) . Th e p roof o ft h e l a t t e r c l a im follo w s b y i n sp ec t io n o f t h e s t an d a rd a rg u m en t s f ro m [LT ],Proposi t ion 1 .d .2( i ) , p . 43 .

T h e fo ll ow i n g t h eo rem i s p ro v ed in [M] , i n t h e sp ec ia l c a se t h a t X = Y an dL q C _ X fo r so me q < o o. A t t h e sa me t i me t h e r e su l t o f [M] is co n ce rn ed w i t hseq u en ces o f r an d o m v a r iab l e s in a g en e ra l sy m m et r i c seq u en ce sp ace , w h e rea sw e co n s i d e r h e r e t h e ca se o f / p - sp aces o n l y .THEOREM 6 .7 : L e t X a n d Y b e r .i . s p a c e s on [0, 1] suc h th a t X C_ Y a n dl e t p [1,c]. I f e it h e r ( i) t h e o p e r a t o r lC a c t s b o u n d e d l y f r o m X i n t o Y a n d Y h a s F a t o u n o r m , o r (ii) t h e o p e r a t o r E a c t s b o u n d e d l y f r o m X i n t o Y ,t h e n t h e r e e x i s t s C > 0 w h i c h d e p e n d s o n X an d Y o n l y su c h t h a t f o r e v e r y

C X , n N , o f i n d e p e n d e n t fu n c t i o n s , t h e f o l lo w i n gs e q u e n c e g = { 9 i } i = li n e q u a l i t y h o l d s:

c6. 1) ' Ilgllv(t,,) -< ffi -Proof' . L e t T b e t h e r e a r r a n g e m e n t - p r e s e r v i n g m a p p i n g b e t w e e n S ( ~ , P ) a n dS([0 , 1 ], A) in t rod uce d in S ec t ion 3 . W e def ine the pos i t ive l inear m app ing Qfrom S (0, c~) in to S(f~, p)u{o} by set t in g

, ~)( Q f ( w o , w l . . . ) : = {fk( k)}k=o, f S(O, oc) ,w h e r e f k ( W k ) : = f ( w a + k ) , k > O . T h e p ro o f o f T h e o rem 6 .7 w i ll b e co m p l e t ed a sso o n a s w e sho w t h a t t h e p o s i ti v e l in ea r o p e ra t o r Q ' := T Q i s a bounded l inearo p e ra t o r f ro m Z p i n to Y ( l p ) . T h e k e y o b s e rv a ti o ns a r e t h a t t h e o p e r a t o r Q 'a c ts b o u n d e d l y f r o m Z ~ = Z x in to Y ( l a ) a n d f ro m Z ~ i nto Y ( l ~ ) . In d eed , t h ef ir s t o b se rv a t io n fo llo w s i mm ed i a t e l y f ro m T h eo rem 6 .1 ( if o n e t ak es i n t o acco u n tt h a t f o r ev e ry seq u en ce g = {g i } in l C X , n N , o f i n d ep e n d en t f u n c t i o n s i n X ,th e se que nce Ig l := {Igil}in--1 C X is again a s equ ence of ind ep en de nt func t ionsa n d t h a t II ~-~in_-I]9 il ]l zx = II ~ i~ 1 9 i l l z x ) . Th e second observa t ion follows f roma c o m b i n a t i o n o f th e eq u iv a l ences

f ! ool f ll z x I I I l z x I I f * x t o , x l l l x v y (w h e re t h e eq u i va l en ce co n s t an t s d o n o t d ep en d o n t h e r .i . sp ace X an d f Z ~ )


28/32

152 S.V . ASTA SHKIN AND F. A. SUK OCH EV Isr. J. M ath.w i t h t he i ne qua l i t i e s

x [ 0 , i ] > < m a x I f k l > < x [ o ,1 ] > w > ,- - k - - - -1 ,2 , . . . , nk nhe re { f } k= l C X , n E N , i s a s e que nc e o f i nde pe nde n t r a nd om va r i a b l e s i n

X a nd ] : = Z~ = I ]k ( s e e [H M] , P ropos i t i on 2 . 1 ) .I t f o ll ow s t ha t

Q ' : ( Z ~ i ) i - 6 ( Z ~ ) 0 --~ ( y ( ~ l ) ) l - ( Y ( l ~ ) ) ~a nd i t s no rm i s un i fo rm l y bou nd e d w i t h r e s pe c t t o 0 E (0 , 1 ). Th e p ro o f i sc om pl e t e by no t i ng t ha t f o r al l 0 E (0 , 1 ) w e ha ve

1(6.3) ZPx C_ ( Z ~ ) I - e ( Z ~ ) , ( Y ( l l ) ) l - e ( Y ( l ~ ) ) e C Y ( lp ) , p - 1 - 0 "To s e e t he f i rs t e mb e dd i n g a b ove , fi x g = g* E Z~ , Hgl[z~ = 1 a nd s e t

gl :-- gx[0,1] + gPx[1,oo), g~ : = gx[0,1] + X[1,oo)"Clearly, g = ( g l ) l- ( g o o ) a nd i t i s a s t r a i gh t fo rw a rd ve r i f i c a t i on t ha t gi E Z i xand Ugil lz~ -< C , i = 1 ,00 , wh ere C > 0 do es no t depe nd on p . Th e secon dem bed din g in (6.3) (in fac t , equ a l i ty) i s show n in [Bu] , Th eo rem 3 . |R e m a r k 6.8: T h e a s s e rt io n e s t a b li s h e d i n T h e o r e m 6 .7 fo l lo w s f ro m t h e b o u n d -e dne s s o f a c e r t a i n l ine a r op e ra t o r f rom Z~( in t o Y ( l p ) , w hi c h i s a c ons e que nc eo f t h e b o u n d e d n e s s o f t h is o p e r a t o r f r o m t h e c o u p l e ( Z ~ , Z ~ ) i n to t h e c o u p l e(Y( /1) , Y ( l c ~ ) ) . B y us i ng C a l de ro n -Lo z a no vs k i i ' s c on s t ruc t i on ( s e e , e .g . , [O ] ,S e c t i on 8 . 2 a nd a l s o [B u] ) , i t i s pos s i b l e t o e x t e nd t h i s r e s u l t t o more ge ne ra ls pa c e s tha n Z p a nd Y ( I p ) , b u t w e h a v e n o t p u r s u e d t h is s u b j e c t i n t h e p r e s e n tp a p e r .7 . F i n a l r e m a r k sI t f o l low s f rom C oro l l a r ie s 5 . 6 a nd 5 . 7 t ha t t he a s s um pt i on e xp (L1) C_ X i s no ts u f fi c ie n t f o r a n r.i. s pa c e X t o ha ve t he K ru g l ov p rope r t y . W e sha l l p r e s e n t ac o n c r e t e e x a m p l e o f a L o r e n t z s p a c e A w i t h o u t t h e K r u g l o v p r o p e r t y s a t i sf y i n ge x p ( L 1 ) C A .PROPOSITION 7.1: I f a ~ ( t ) := In -1 7 " ln- Z ( In e-), t E [0,1], /~ > 1, t h e ne x p ( L 1 ) C A ~ , b u t t h e l a t t e r s p a c e d o e s n o t h a v e t h e K r u g l o v p r o p e r t y .P r o o f : I t i s s u f fi c ie n t t o ve r i fy t ha t t he func t i on a ~ doe s no t s a t i s fy c ond i t i on(5 .1) for ev ery /~ > 1 . Such a ver i f i ca t ion is s t ra igh t forw ard b u t t echnica l , andw e om i t de t a i l e d c a l c u l a ti ons . |


29/32

Vol. 145, 2005 SERIES OF INDEPEN DEN T RAN DOM VAR IABLES 153W e pre s e n t now s ome ne c e s s a ry c ond i t i on fo r a r . i . s pa c e X t o ha ve t he

K r u g l o v p r o p e r t y .F o r n E N , w e de n o t e t he n t h r e pe a t e d l oga r i t hm by ln ,~ a nd s e t

On (u) := ex p( uln n(c n + u) ) - 1 , lnn Cn = 1 .Th e O r l i cz s pa c e L . c o i nc i de s w i t h t he M a rc i nk ie w i c z s pa c e M~ . , w he re

u l n ( e l u )~ n ( u ) := l n n + l ( e e , l u ) , u e (0 ,1] .Le t X be a n r .i. s pa c e . Th e a s s u m pt i on e xp (L1 ) C_ X s e e ms t o be ne c e s s a ry

f o r t h e o p e r a t o r K t o a c t b o u n d e d l y o n X . T h e f o l lo w i n g r e s u l t i s a s t e p t o w a r d sp rov i ng t h i s c on j e c t u re .THEOREM 7.2 : I f t h e o p e r a t o r E a c t s b o u n d e d l y o n t h e r. i. s p a c e X , t he nL,r, , C X for every n E N .P r o o f S i nc e L ~ C_ X , i t f o ll ow s f rom Th e ore m 4 . 4 t ha t

l n ( e / . )f l ( ' ) .- - i n ( i n ( e e l . ) )

E X ,

N o t i n g t h a t A { f l > 1"} x e - r l n ( l + r ) a nd a rgu i ng a s i n t he p ro o f o f Th e ore m4.5 , we ge t

ooA{K:f l > T} > C ~ _ e -'rln(l-{-v/n) --

- h i "n : l

I f n > T l n - - i ( T / e ) , t he n T l n (1 + T / n ) < w i n ( In (v ) ) a nd t he re fo re( 7 . 1 ) > _ > l / n ! .n>[2~ . - ~ (v/e)]U s i ng S t i r l ing ' s f o rmu l a fo r s u f f ic i e n tl y l a rge r ' s w e e s t i ma t e

[2 r l n -1 ~ ]! _> ex p (_ 2 T ln _ 1 T ln(2 Tln _ 1 _T)) _> e_ 3Le eApplying (7 .1) , we obta in for a l l suf f i c ien t ly l a rge r

(7.2) A { ~ f l > T} > C e-rln(ln(r))e-3r > e -4 t In ( In ( r ) ) .N ot e t ha t f o r t he M a rc ink i e w i c z s pa c e M ~ . , n >_ 1 t he a na l ogu e o f fo rmu l a (4 .1 )holds . Th erefore , to prov e th a t M~2 i s con ta ined in X, i t is suf f ic ien t to show


30/32

154 S.V . ASTASHK IN AND F. A. SUKOC HEV Isr. J . Math.th a t th e fu nc t ion f2 ( t ) := t --+ ~2 ( t ) / t b e lo n g s t o X . I n p a r t i c u l a r , i t is s u f f ic i e n tt o v e r i f y t h a t 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 h t h a t f o r a l l s u f f i c i e n t ly l a r g e T ,(7.3) > < cA{ I, >D ir e c t c a l c u l a t i o n s n o w s h o w t h a t f o r s u f f ic i e n t l y l a r g e ~- w e h a v e A { f2 > 7 } _ 2) s p a n n e d b y s eq u e n c e s o findependent random var iables , I s rael Journal of Mathematics 8 (1970) , 273-303.

[ RS ] V . A . Ro d in a n d E . M . S e m e n ov , R a d e m a c h e r series in in s y mme tr ic s pac e s ,Analysis M ath em atica 1 (1975), 207-222.


32/32

15 6

[VTC]S. V. ASTASH KIN AND F. A. SUKO CHEV Isr. J . Math.

N . Va k h a n ia , V . Ta r ie l a d z e a n d S . Ch o b a n y a n , Probabi l i ty Dis tr ibut ions inBanach Spaces, Na uka , Moscow, 1985 (Ru ss ian) .

Documents

S.V. Astashkin and F.A. Sukochev- Series of Independent Random Variables in Rearrangement Invariant Spaces: An Operator Approach