BY PAWE L HITCZENKO Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC , USA

Transcription

1 Absrac Tangen Sequences in Orlicz and Rearrangemen Invarian Spaces BY PAWE L HITCZENKO Deparmen of Mahemaics, Box 8205, Norh Carolina Sae Universiy, Raleigh, NC , USA AND STEPHEN J MONTGOMERY-SMITH Deparmen of Mahemaics, Universiy of Columbia Missouri, Columbia, MO 65211, USA Le (f n ) and (g n ) be wo sequences of random variables adaped o an increasing sequence of σ-algebras (F n ) such ha he condiional disribuions of f n and g n given F n 1 coincide Suppose furher ha he sequence (g n ) is condiionally independen Then i is known ha f k p C g k p, 1 p, where he number C is a universal consan The aim of his paper is o exend his resul o cerain classes of Orlicz and rearrangemen invarian spaces This paper includes fairly general echniques for obaining rearrangemen invarian inequaliies from Orlicz norm inequaliies 1 Inroducion Le (F n ) be an increasing sequence of σ-algebras on some probabiliy space (Ω, F, P ) We will assume ha F 0 = {, Ω} A sequence (f n ) of random variables is called (F n )- adaped if f n is F n -measurable for each n 1 In he sequel we will simply wrie adaped if here is no risk of confusion For any sequence (f n ) of random variables, we will wrie f = sup n f n and f n = max 1kn f k Throughou he paper all equaliies or inequaliies beween random variables are assumed o hold almos surely Given a σ-algebra A F and an inegrable random variable f, we will denoe he condiional expecaion of f given A by E A f If A = F k hen we will simply wrie E k f for E Fk f The condiional disribuion of a random variable f given A is denoed by L(f A) Thus, L(f A) = L(g A) means ha for each real number we have ha P (f > A) = P (g > A) The following definiion was inroduced by Kwapień and Woyczyński in a preprin of heir paper [10], which was disribued as early as 1986 We refer he reader o heir book [11] for more informaion on angen sequences Definiion Le (F n ) be an increasing sequence of σ-algebras on (Ω, F, P ) (a) Two adaped sequences (f n ) and (g n ) of random variables are angen if for each n 1 we have L(f n Fn 1 ) = L(g n Fn 1 ) (b) An adaped sequence (g n ) of random variables saisfies condiion (CI) if here exiss a σ-algebra G F such ha for each n 1 L(g n F n 1 ) = L(g n G), 1

2 and (g n ) is a sequence of G-condiionally independen random variables A sequence (f n ) is condiionally symmeric if (f n ) and ( f n ) are angen sequences of random variables Every sequence of random variables (f n ) admis (possibly on an enlarged probabiliy space) a angen sequence which saisfies condiion (CI) (cf eg [11 p 104]) Throughou his paper, such a sequence will be denoed by (f n ), and will be called a decoupled version of (f n ) I is useful o noe ha he σ-algebra G can be chosen so ha he random variables f n, n 1, are G-measurable In his paper, we will be ineresed in comparing Orlicz and rearrangemen invarian norms of sums of angen sequences A rearrangemen invarian space X is a space of random variables f equipped wih a complee quasi-norm X such ha eiher he condiions (i), (ii) and (iii) or (i), (ii) and (iii ) below hold: (i) if g # f # and f X, hen g X wih g X f X ; (ii) if f is simple wih finie suppor hen f X; (iii) f n X and f n 0 implies f n X 0 (iii ) f n X and 0 f n f and sup n f n X < imply f X wih f X = sup n f n X Here f # denoes he decreasing rearrangemen of f, ha is, f # (s) = sup{ : P ( f > ) > s } Examples of rearrangemen invarian spaces include Orlicz spaces and Lorenz spaces Le Φ : [0, ) [0, ) be an increasing funcion such ha Φ(0) = 0, and such ha here is a consan c > 0 such ha Φ(c) 2Φ() for all 0 (Funcions saisfying he laer condiion have been called dilaory in [14]; le us noe ha if Φ is convex, and Φ(0) = 0 hen his condiion is saisfied wih c = 2) Given such a funcion Φ we define he Orlicz norm of a random variable f o be f Φ = inf {λ > 0 : E(Φ( f /λ)) 1} We le L Φ = {f : f Φ < } Noe ha if Φ is convex, hen L Φ is a normed space However, we do no wish o resric ourselves o normed spaces Oher examples are he Lorenz spaces Given 0 < p, q, we define he space L o be hose random variables f for which he following quaniy is finie: f = ( q 1 1/q s (q/p) 1 f # (s) ds) q if q < p 0 sup s 1/p f # (s) if q = s>0 Noe ha L is no a normed space unless 1 q p Noe ha he L p spaces are special cases: L p = L Φ = L p,p, where Φ() = p We refer he reader o [13] for more deails abou hese spaces In he presen paper we will be ineresed in he dominaion of a rearrangemen invarian norm of a sum of an arbirary sequence of adaped random variables by he rearrangemen invarian norm of a sum of is decoupled version I is already known (see [4]) ha if Φ saisfies he 2 -condiion, ha is, here is a consan c > 0 such ha 2

3 Φ(2) c Φ() for all 0, hen here is a consan C Φ such ha for every adaped sequence (f n ) of random variables one has: f i Φ C Φ f i Φ (11) Building on some special siuaions considered by Klass [4, Theorem 31] and Kwapień [9], Hiczenko [6] began o invesigae how he consan C Φ depends upon Φ He showed ha here is a universal consan C > 0 such ha f i p C f i p 1 p (12) In his presen paper, we will show among oher hings, ha inequaliy (11) holds wih C Φ uniformly bounded, a leas for cerain classes of Orlicz funcions Our firs heorem exends a resul of Klass who proved (11) for randomly sopped sums of independen random variables Le us define classes of Orlicz funcions Following Klass [8], for q > 0 we define he class F q as he class of all funcions Φ such ha: (i) Φ : [0, ) [0, ), Φ(0) = 0, (ii) Φ is nondecreasing and coninuous, and (iii) Φ saisfies he growh condiion: Φ(cx) c q Φ(x) for all x 0, c 2 For p > 0, we define he class G p as he class of all funcions Φ such ha (i) Φ : [0, ) [0, ), Φ(0) = 0, (ii) Φ is nondecreasing and coninuous, and (iii) Φ saisfies he growh condiion: Φ(cx) c p Φ(x) for all x 0, c 2 Then we obain he following resuls Theorem 11 There is a universal consan C > 0 such ha if Φ F q for some q > 0, hen for every adaped sequence (f n ) of random variables one has: ( ) ( EΦ fi C 1+q ) EΦ f i This inequaliy had already been obained by Klass [8] in he special case ha f k = I(τ k)ξ k, where (ξ k ) is a sequence of independen random variables and τ is a sopping ime More precisely, Klass proved his resul for Banach space valued random variables (ξ k ) (wih absolue value replaced by norm) To discuss Banach space valued random variables one needs o adjus noaion; for a random variable Y and a σ-algebra A, we use L(Y A) o denoe he regular version of he condiional disribuion of Y given A, ha is, L(Y A) = L(Z A) means ha for every Borel subse A of he Banach space, we have P (Y A A) = P (Z A A) Recall ha he exisence of he regular versions of he condiional disribuions is guaraneed, as long as our random variables ake values in a separable Banach space As i urns ou, in our generaliy, he inequaliy of Theorem 11 need no hold (wih any consan), unless some exra condiions are imposed on he geomery of he underlying Banach space (see eg [3]) Since i is unclear a his ime for which Banach spaces he inequaliy ( E fk p) 1/p cp ( E f k p) 1/p holds (even if he consan c p is allowed o depend on p), we confine our discussion o real valued random variables 3

4 Corollary 12 Given numbers p 0 > 0 and r 1, here is a consan c p0,r such ha if p p 0, and if Φ G p F rp, hen f i Φ c p0,r f i Φ The nex sep is o exend hese resuls o rearrangemen invarian spaces This will be accomplished hrough a raher general mehod of obaining rearrangemen invarian norm inequaliies from Orlicz norm inequaliies We believe ha his echnique will prove useful in oher conexs as well In paricular, we would like o menion ha his mehod could be used o deduce maringale inequaliies obained by Johnson and Schechman [7] from he corresponding inequaliies for Orlicz funcions Corresponding o he noions of Orlicz spaces lying in G p F q, we have he following noion We say ha a rearrangemen invarian space is an inerpolaion space for (L p, L q ) (in shor, a (p, q)-inerpolaion space) if here is a consan c > 0 such ha for every operaor T : L p L q L p L q for which T Lp L p 1 and T Lq L q 1 we have ha T X X c However, his noion is no quie wha we need Define K (f, ) = inf{ f p + f q : f + f = f # } We will say ha a rearrangemen invarian space X is a (p, q)-k-inerpolaion space if here is a consan c such ha whenever f and g are such ha K (f, ) K (g, ) ( > 0), and g X, hen f X and f X c g X The (p, q)-k-inerpolaion consan of X, denoed by C (X), is he infimum of c ha work for all funcions f and g I is quie easily seen ha every (p, q)-k-inerpolaion space is a (p, q)-inerpolaion space I is also known ha if 1 p, q, hen every normed (p, q)-inerpolaion space is a (p, q)-k-inerpolaion space (see [1]) We are able o esablish he following mehod for obaining rearrangemen invarian inequaliies from Orlicz inequaliies Theorem 13 Suppose ha f Φ g Φ for all Φ F q G p, where 0 < p < q < If X is a (p, q)-k-inerpolaion space, hen f X 2 2+1/p C (X) g X Corollary 14 Given numbers p 0 > 0 and r 1, here is a consan c p0,r such ha if p p 0, and if X is a (p, pr)-k-inerpolaion space, hen f i X c p0,rc p,pr (X) f i X In paricular, we are able o obain he following resul for Lorenz spaces Noe ha if one is ineresed in normed Lorenz spaces, hen p 0 below can be aken o be 1, and he resuling inequaliy exends (12) Corollary 15 Given a number p 0 > 0, here is a consan c p0 such ha if p, q p 0, hen f i c p0 f i 4

5 2 Inequaliies for Orlicz funcions We begin wih a proof of Theorem 11 Since our proof is based on well undersood echniques we will be somewha skechy and we refer he reader o [6] for deails ha are no explained here Throughou his secion we le (M n ) be a maringale wih difference sequence ( k ) Since F q1 F q2 whenever q 1 q 2, we can assume wihou loss of generaliy ha q 1 Our deparing poin is he following resul which can be found in he jus menioned paper (Theorem 51 and he beginning of he proof of Lemma 23) Lemma 21 Le 1 q <, and le ( k ) be a condiionally symmeric maringale difference sequence, and ( k ) is decoupled version Se T n,q (M) = (E n k=1 k q G) 1/q Then here exis δ 1 > 0, β > 1 + δ 1 and ɛ wih 0 < ɛ 1/2 such ha for every λ > 0 we have P (M βλ, (Tq (M) ) < δ 1 λ) ɛ q P (M λ) From his, we obain Lemma 22 Le ( k ) be as above, and assume ha w n is a F n 1 -measurable random variable such ha n w n for each n 1 Se N n = n k=1 k (n 1) Suppose ha δ 1, β, and ɛ are as in Lemma 21 Then, here exis δ > 0, δ 2 > 0 and 0 < α 1/2 such ha for every λ > 0 we have P (M βλ, N < δλ) ɛ q P (M λ) + P (w δ 2 λ) + (1 α q )P (M βλ) Proof: We have ha P (M βλ, N < δλ) P (M βλ, T q (M) < δ 1 λ, w < δ 2 λ) + P (w δ 2 λ) +P (M βλ, T q (M) δ 1 λ, w < δ 2 λ, N < δλ) (21) Suppose δ 2 δ 1 Then, in view of Lemma 21, for he firs probabiliy on he righ-hand side of (21) we have ha P (M βλ, T q (M) < δ 1 λ, w < δ 2 λ) P (M βλ, T q (M) w < δ 1 λ) ɛ q P (M λ) I remains o esimae he las probabiliy in (21) Since M, w and T q (M) are G- measurable, by condiioning on G, we see ha he las probabiliy in (21) is equal o: { } E I(M βλ, Tq (M) δ 1 λ, w < δ 2 λ)p (N < δλ G) By Kolmogorov s converse inequaliy (see eg [12, Remark 615, p 161]) for all sequences of independen and symmeric random variables (ξ k ) and for all > 0 we have ha P (S ) 1 ( 2 q 1 22q ( q + E(ξ ) q ) ) E(S ) q, 5

6 where S n = n k=1 ξ k Applying his resul condiionally on G, we obain ha P (N < δλ G) 1 1 ( 2 q 1 22q ((δλ) q + E G ( ) q ) ) E G (N ) q (22) Also, if w n < δ 2 λ, hen n < δ 2 λ, and since w n is F n 1 -measurable, and he condiional disribuions of n and n coincide, i follows ha n < δ 2 λ Therefore, on he se we have {T q (M) δ 1 λ, w < δ 2 λ}, (δλ) q + E G ( ) q E G (N ) q (δλ)q + (δ 2 λ) q (δ 1 λ) q, so ha he condiional probabiliy in (22) does no exceed Choosing δ = δ 2 = δ 1 /12, we obain ha whenever α 1/6 Therefore, 1 1 ( 2 q 1 4q (δ q + δ q 2 ) ) 1 1 ( 2 q 1 4q (δ q + δ q 2 ) ) = 1 1 (1 2 q 2 ) 3 q 1 α q, { } E I(M βλ, T q (M) δ 1 λ, w < δ 2 λ)p (N < δλ G) δ q 1 δ q 1 (1 α q )EI(M βλ, T q (M) δ 1 λ, w < δ 2 λ) (1 α q )P (M βλ) This complees he proof of Lemma 22 Now we are ready o complee he proof of Theorem 11 By an argumen similar o one used in [6, proof of Lemma 21], i follows ha in order o prove EΦ( f k ) c q EΦ( f k ), (23) i suffices o esablish (23) for (f k ) = ( k ), a condiionally symmeric maringale difference sequence By a rouine applicaion of Davis decomposiion (cf eg [2] and references herein), we may also assume ha n w n, where w n is a F n 1 -measurable random variable, and ha w 2 The laer inequaliy, ogeher wih he inequaliy P (f ) 2P (g ) valid for all angen sequences (f k ) and (g k ) (cf [4] or [11, Theorem 521 (i)]), implies ha P (w ) P ( /2) 2P ( /2) 2P (N /4) (24) 6

7 By Lemma 22, we have ha so ha Consequenly, Therefore, P (M βλ) P (N δλ) + P (w δ 2 λ) + (1 α q )P (M βλ), ( α β α q P (M βλ) P (N δλ) + P (w δ 2 λ) α q EΦ(M /β) EΦ(N /δ) + EΦ(w /δ 2 ) ) qeφ(m ( α ) qeφ(βm ) = /β) β By (24) we have ha ( α ) qβ p EΦ(M /β) EΦ(N /δ) + EΦ(w /δ 2 ) β δ q EΦ(N ) + δ q 2 EΦ(w ) EΦ(w ) 2EΦ(4N ) 2 4 q EΦ(N ), so ha ( β q { 1 EΦ(M ) α) δ q + 2 4q δ q 2 } EΦ(N ) Condiionally given G, ( k ) is a sequence of independen and symmeric random variables Therefore, by Levy s inequaliy, P (N G) 2P (N G) This implies (see eg [11, Proposiion 021]) ha for every increasing funcion φ : R + R + we have Eφ(N ) 2Eφ(N) This complees he proof of Theorem 11 As we menioned in he inroducion, inequaliy (23) exends a resul of Klass, who considered sequences (f k ) of special form On he oher hand, i follows from a resul of Kwapień [9] ha if f k = ( k 1 j=1 a j,kξ j )ξ k, where (ξ j ) is a sequence of independen zero mean random variables, hen (23) holds in he sronger form: EΦ( f k ) EΦ(c f k ), for some absolue consan c and for every convex funcion Φ Thus one may wonder wheher our resricions on Φ can be relaxed We wish o close his secion wih a negaive resul showing ha (23) does no hold for all convex funcions Φ (However, i is sill possible ha (23) holds under weaker assumpion han ours) Our example is an easy adapaion of an example due o Talagrand concerning comparison of ail behavior for sums of angen sequences This example was included in [5], and we refer he reader o he laer paper for deails ha are no included here 7

8 Proposiion 23 For every consan c > 0, here exiss a convex funcion Φ : R R and a sequence (f k ) such ha EΦ( f k ) ceφ(c f k ) Proof: We will show ha for every k N here exiss a convex funcion Φ and a sequence (f k ) for which EΦ( f k ) 22k+2 k 2 2 2k EΦ(k 4 f k ) Le (r n ) denoe he Rademacher random variables, ha is, a sequence of independen random variables such ha P (r n = ±1) = 1/2 Fix k N Given an ineger N 1 o be specified in a momen, define N 2, N k as follows: N i N i 1 = 2 (i 1) N 1, i = 2,, k Pu and hen Ω 1 = {r 1 = = r N1 = 1}, Ω i = Ω i 1 {r Ni 1 +1 = = r Ni }, i = 2,, k Define a sequence of random variables (v i ) by he formulas: v 1 = = v N1 = 1 v N1 +1 = = v N2 = 2I Ω1 v Nk 1 +1 = = v Nk = 2 k 1 I Ωk 1 We le f j = v j r j for j = 1, N k Then f j = v j r j, where (r j ) is an independen copy of (r j ) (cf [10, Example 431]) For 0 < δ < 1, le Φ δ be a convex funcion defined by Φ δ (x) = (x δkn 1 ) + Noe ha v j = kn 1, and herefore EΦ δ ( v j r j ) (1 δ)kn 1 P ( v j r j kn 1 ) On he oher hand, if v j r j < 4N 1, hen v j r j 4N 1 1, so ha wih δ = 1 1/(4N 1 ) we ge Φ δ ( k 4 v j r j ) (kn 1 k 4 δkn 1) + = 0 Since P ( v j r j 4N 1 ) k2 N 1/2 k 2 P ( v j r j kn 1 ), 8

9 (cf [5, op half of page 176]) we obain and i follows ha EΦ δ ( k 4 v j r j ) Φ δ ( k2 N 1 4 )P ( v j r j 4N 1 ) k k 2 kn 1 4 k2 N 1/2 P ( v j r j kn 1 ), EΦ δ ( v j r j ) EΦ δ ((k/4) v j r j ) 4(1 δ)kn 12 N1/2 k 3 = 2N1/2 N 1 k 2 22 N 1 k 2 2 2k, for N 1 2 k This complees he proof In he above example he sequence (f k ) may be consruced so ha f k is a randomly sopped sum of independen random variables (see Remark on p 176 of [5]) Thus, he conclusion of his Remark applies here as well 3 Rearrangemen invarian norm inequaliies Lemma 31 Suppose ha Φ and Ψ are wo Orlicz funcions Θ 1 (x) = 1 2Θ(x) Then k 2 k f Θ 1 inf{ f Φ + f Ψ : f + f = f # } 2 f Θ k+2 Le Θ = Φ Ψ, and Proof: To show he lef hand side, suppose ha f # = f + f, and f Φ + f Ψ 1 Then EΦ( f ) 1 and EΨ( f ) 1, and so EΘ 1 ( 1 2 f ) 1 2 EΘ(max{ f, f }) = 1 2 E max{θ( f ), Θ( f )} 1 2 (EΦ( f ) + EΨ( f )) 1 To show he righ hand side, suppose ha f Θ 1, ha is, EΘ( f ) 1 Le { f f () = # () if Φ( f() ) Ψ( f() ) 0 oherwise, and f = f # f Then we see ha EΦ( f ) EΘ( f ) 1 and ha EΨ( f ) EΘ( f ) 1, and he resul follows The nex lemma follows immediaely Lemma 32 Le Φ (x) = x p (x) q, where 0 < p < q < Then 2 1 1/p f Φ K (f, ) 2 f Φ Now we will prove Theorem 13, using he above Lemma From he hypohesis of Theorem 31, i follows ha f Φ g Φ Hence by Lemma 32, i follows ha K (f, ) 2 2+1/p K (g, ) Now he resul follows by he definiion of (p, q)-k-inerpolaion space Corollary 14 follows easily from Theorems 11 and 13 To show Corollary 15, we only need he following resul The mehods below are all fairly sandard in inerpolaion heory, and indeed if one is no concerned abou uniform esimaes, may be aken direcly from he lieraure 9

10 Lemma 33 Given p 0 > 0, here is a consan c p0 > 0 such ha if p, q p 0, hen L is a (p/2, 2p)-K-inerpolaion space wih consan bounded by c p0 Proof: Firs le us define some norms For p q, le f a() = inf{ 2/p f p/2 + 1/2p f 2p : f + f = f # }, f b() = ( 1 2/p ( 1 1/2p f # (s) ds) p/2 + f # (s) ds) 2p 0 Clearly f a() f b() Also f a() min{1, 2 1 2/p }f # () This is because if f # = f + f, hen ( 1 2/p f p/2 + 1/2p f 2p ( min{1, 2 1 2/p } 2/p ( f (s) ds) p/ /p ( f (s) ds) p/2 1 + min{1, 2 1 2/p }f # () 0 ( 1 f # (s) p/2 ds 0 Nex, given a funcion f, le us define he funcion Hf() = f b() 0 ) 1/2p f (s) 2p ds ) 2/p f (s) p/2 ds ) 2/p Then i follows ha Hf 32 1/ min{} f To see his, firs noe ha Hf ( H 1 f q + H 2 f q )1/q, where ( 1 H 1 f() = ( 1 H 2 f() = 0 ) 2/p f # (s) p/2 ds, ) 1/2p f # (s) 2p ds We will use wo properies of L Firs, if v q, and if f 1, f 2,, f n are funcions, hen ( n ( (f # n 1/v i )v f i ) v i=1 Second, if we define he operaors D a f() = f # (a) for 0 < a <, hen D a f = a 1/p f (The firs propery follows from Minkowski s inequaliy for L q/v, he second is a simple change of variables argumen) Le u = max{p/q, 2}, v = min{q, p/2} Then i=1 ( 1 2/p H 1 f (D a f # ) da) p/2 0 10

11 ( ) 2 un 2/p (D a f # ) p/2 da 2 u(n+1) ( ) 2/p 2 un (D 2 u(n+1)f # ) p/2 ( 4 n (D 2 u(n+1)f # ) v ( 4 n D 2 u(n+1)f # v ( 1/v 4 n 2 )) u(n+1)v/p f ( 2 1 n f 4 1/v f Now le u = max{2p/q, 1}, v = min{q, 2p} ( 1/2p H 2 f (D a f # ) da) 2p 1 ( ) 2 u(n+1) 1/2p (D a f # ) 2p da 2 un ( ) 1/2p 2 u(n+1) (D 2 unf # ) 2p ( 2 n+1 (D 2 unf # ) v ( 2 n+1 D 2 unf # v ( 1/v 2 n+1 2 )) unv/p f ( 2 1 n f 11

12 4 1/v f Finally, o finish, suppose ha K p/2,2p (f, ) K p/2,2p (g, ) for all > 0 Then i follows ha f # () max{1, 2 2/p 1 } f a() max{1, 2 2/p 1 } g a() max{1, 2 2/p 1 } g b() =2 2/p Hg() Hence f 2 2/p Hg 128 1/ min{} g Acknowledgmens The second named auhor would like o express warm graiude o Vicor de la Peña for inroducing him o his problem Research of boh auhors was parially suppored by separae NSF grans The second named auhor was also suppored by he Research Board of he Universiy of Missouri References [1] J ARAZY and M CWIKIEL A new characerizaion of he inerpolaion spaces beween L p and L q, Mah Scand 55 (1984), [2] D L BURKHOLDER Disribuion funcion inequaliies for maringales, Ann Probab 1 (1973), [3] D J H GARLING Random maringale ransform inequaliies, Probabiliy in Banach Spaces, 6 (Sandbjerg, Denmark, 1986), , Progr Probab 20, Birkhäuser, Boson (1990) [4] P HITCZENKO Comparison of momens for angen sequences of random variables Probab Theory Relaed Fields 78 (1988), [5] P HITCZENKO Dominaion inequaliy for maringale ransforms of a Rademacher sequence Israel J Mah 84 (1993), [6] P HITCZENKO On a dominaion of sums of random variables by sums of condiionally independen ones Ann Probab 22 (1994), [7] W B JOHNSON and G SCHECHTMAN Maringale inequaliies in rearrangemen invarian funcion spaces, Israel J Mah 64 (1988), [8] M J KLASS A bes possible improvemen of Wald s equaion, Ann Probab 16 (1988), [9] S KWAPIEŃ Decoupling inequaliies for polynomial chaos, Ann Probab 15 (1987), [10] S KWAPIEŃ and W A WOYCZYŃSKI Semimaringale inegrals via decoupling inequaliies and angen processes, Probab Mah Sais 12 (1991), [11] S KWAPIEŃ and W A WOYCZYŃSKI Random Series and Sochasic Inegrals Single and Muliple, Birkhäuser, Boson (1992) [12] M LEDOUX and M TALAGRAND Probabiliy in Banach Spaces Springer, Berlin, Heidelberg (1991) 12

13 [13] J LINDENSTRAUSS and L TZAFRIRI Classical Banach Spaces Funcion Spaces, Springer, Berlin, Heidelberg, New York (1977) [14] S J MONTGOMERY SMITH Comparison of Orlicz Lorenz spaces, Sudia Mah 103 (1992),