Orlicz Norms of Sequences of Random Variables

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1 Orlicz Norms of Sequeces of Radom Variables Yehoram Gordo Techio Departmet of Mathematics Haifa 32000, Israel Carste Schütt Christia Albrechts Uiversität Mathematisches Semiar Kiel, Germay Alexader Litvak Techio Departmet of Mathematics Haifa 32000, Israel ad Dept. of Math. ad Stat. Sc. Uiversity of Alberta Edmoto, AB, T6G 2G, Caada Elisabeth Werer Departmet of Mathematics Case Wester Reserve Uiversity Clevelad, Ohio 4406, U. S. A. ad Uiversité de Lille UFR de Mathématique Villeeuve d Ascq, Frace emw2@po.cwru.edu Keywords: Orlicz orms, radom variables, 99 Mathematics Subject Classificatio 46B07, 46B09,46B45, 60B99, 60G50, 60G5 Partially supported by Nato Collaborative Likage Grat PST.CLG Partially supported by Frace-Israel Arc-e-Ciel exchage Partially supported by the Fud for the Promotio of Research at the Techio Partially supported by a Lady Davis Fellowship Partially supported by Natioal Sciece Foudatio Grat DMS ad by Nato Collaborative Likage Grat PST.CLG

2 Abstract Let f i, i =,...,, be copies of a radom variable f ad N be a Orlicz fuctio. We show that for every x R the expectatio E (x i f i N is maximal (up to a absolute costat if f i, i =,...,, are idepedet. I that case we show that the expectatio E (x i f i N is equivalet to x M, for some Orlicz fuctio M depedig o N ad o distributio of f oly. We provide applicatios of this result. Itroductio ad mai results Let f i, i =,...,, be idetically distributed radom variables. We ivestigate here expectatios E (x i f i (ω N, where N is a Orlicz orm. We fid out that these expressios are maximal (up to a absolute costat if the radom variables are i additio required to be idepedet. I case the radom variables are idepedet we get quite precise estimates for the above expectatios. I particular, let f,..., f be idepedet stadard Gauß variables ad let the orm o R be defied by z k, = k z i, where (zi i is the o-icreasig rearragemet of the sequece ( z i i. The we have for all x R c x M E (x i f i (ω k, c 2 x M, where the Orlicz fuctio is M(t = k e (kt 2, t < /(2k, M( =. This case is of particular iterest to us. I a forthcomig paper ([2] these estimates are applied to obtai estimates for various parameters associated to the local theory of covex bodies. Let us ote that i the case k = the orm k, is just the l -orm. Some of the methods that are used here have bee developed by Kwapień ad Schütt ([4], [5], [9], ad [0]. I this paper we cosider radom variables with fiite first momets oly. I the proofs of our results we assume that the radom variables have cotiuous distributios, i.e. P {ω f(ω = t} = 0 for every t R. The geeral case follows by approximatio. We defie the followig parameters of the distributio. Let f be a radom variable with a cotiuous distributio ad with E f <. Let t = t (f = 0, t 0 = t 0 (f =, ad for j =,..., ( t j = t j (f = sup { t P {ω f(ω > t} j }. Sice f has the cotiuous distributio we have for every j We defie the sets P {ω f(ω t j } = j. (2 Ω j = Ω j (f = {ω t j f(ω < t j }

3 for j =,...,. Clearly, for all j =,..., Ideed Therefore we get P (Ω j =. Ω j = {ω t j f(ω < t j } = {ω t j f(ω } \ {ω t j f(ω }. P (Ω j = j j =. We put for j =,..., (3 y j = y j (f = f(ω dp (ω. Ω j We have y j = E f ad t j y j < t j for all j =,...,. j= We recall briefly the defiitios of a Orlicz fuctio ad a Orlicz orm (see e.g. [3, 6]. A covex fuctio M : R + R + with M(0 = 0 ad M(t > 0 for t 0 is called a Orlicz fuctio. The the Orlicz orm o R is defied by { } x M = if ρ > 0 : M ( x i /ρ. Clearly, if two Orlicz fuctios M, N satisfy M(t an(bt for every positive t the x M ab x N for every x R. Thus equivalet Orlicz fuctios geerate equivalet orms. I other words to prove equivalece of x M ad x N it is eough to prove equivalece of M ad N. Moreover, to defie a Orlicz orm M it is eough to defie a Orlicz fuctio M o [0, T ], where M(T =. Ay Orlicz fuctio M ca be represeted as M(t = t 0 p(sds, where p(t is o-decreasig fuctio cotiuous from the right. If p(t satisfies (4 p(0 = 0 ad p( = lim t p(t = the we defie the dual Orlicz fuctio M by M (t = t 2 0 q(sds,

4 where q(s = sup{t : p(t s}. Such a fuctio M is also a Orlicz fuctio ad x M x 2 x M, where is the dual orm to M (see e.g. [6]. Note that the coditio (4 i fact excludes oly the case M(t is equivalet to t. Note also that q satisfies coditio (4 as well ad that q = p if p is a ivertible fuctio. We shall eed the followig property of M ad M (see e.g. 2.0 of [3]: (5 s < M (sm (s 2s for every positive s. The aim of this paper is to prove the followig theorem. Theorem Let f,..., f be idepedet, idetically distributed radom variables with E f <. Let N be a Orlicz fuctio ad let s k, k =,..., 2, be the o-icreasig rearragemet of the umbers ( y i N ( j N ( j, i, j =,...,, where y i, i =,...,, is give by (3. Let M be a Orlicz fuctio such that for all l =,..., 2 The, for all x R M ( l k= s k = l 2. 8 x M E (x i f i (ω N 8 e e x M. Corollary 2 Let f,..., f be idepedet, idetically distributed radom variables with E f <. Let M be a Orlicz fuctio such that for all k =,..., M ( k j= y j = k. The, for all x R where c, c 2 are absolute positive costats. c x M E max i x if i (ω c 2 x M, Proof. We choose p big eough so that the l p -orm p approximates the supremum orm well eough (p = suffices. We cosider N(t = t p. This meas that for all t > 0 we have N (t = pt p ad N (t = ( p t p. 3

5 Therefore Thus With this we get N (t = t 0 N (sds = t 0 ( p s p ds = p p ( p t+ N (t = p p ( p p p p t p. N ( j N ( j = p p ( p By the Mea Value Theorem we get for j 2 p p ( p p + p j p p p p N ( j N ( j For sufficietly big p we have for all j with j Now we choose l = k ad get p. (( j p ( j p. p p ( p p + p (j p. N ( j N ( j 2. which implies the corollary. k y i l k s j 2 y i, j= Corollary 3 Let f,..., f be idepedet, idetically distributed radom variables with E f i =. Let k N, k, ad let the orm k, o R be give by x k, = k x i, where x i, i =,...,, is the decreasig rearragemet of the umbers x i, i =,...,. Let M be a Orlicz fuctio such that M ( = ad for all m =,..., The, for all x R ( m M y j = m. k j= where c, c 2 are absolute positive costats. c x M E (xifi(ω k, c 2 x M, 4

6 Clearly, Corollary 3 implies Corollary 2. We state them separately here, sice the proof of Corollary 3 is more ivolved. We could argue i the proof of this corollary i the same way as i the proof of Corollary 2. But it is less cumbersome to use the lemmas o which Theorem is based. Proof. Let ɛ > 0 will be specified later. Cosider the vector z = (,...,, ɛ,..., ɛ [ k ] + ( [ k ]ɛ, where the vector cotais [ ] coordiates that are equal to. (For techical reasos we k require that all the coordiates of z are ozero, otherwise the fuctio M might ot be well defied. First we show that if ɛ is small eough the for every x R (6 c x k, + max x iz ji c 2 x k,. i j,...,j To obtai this we observe first that we ca choose ɛ so small that we ca actually cosider the vector z = (,...,, 0,..., 0/[ ] istead. By Lemma 7 we have k c s i (x, z + max x i z ji s i (x, z, i j,...,j where s l (x, z is the decreasig rearragemet of the umbers x i z j, i, j =,...,. O the other had, k x i s i (x, z /k k k x i 2 x i. [/k] Let N be a Orlicz fuctio that satisfies N ( k z i = k/. Lemma 5, Lemma 9, ad iequalities (6 imply c 3 x N x k, c 4 x N for some absolute costats c 3, c 4. Clearly, N ( j N ( j = z j. Now we apply Theorem to the Orlicz fuctio N ad obtai the umbers s k ad the fuctio M as i the statemet of Theorem. Choosig ɛ small eough we obtai s = = s = ([ ] + ( [ [ k ] k k ]ɛ y s = = s [ k ]+ 2[ = ([ ] + ( [ k ] k k ]ɛ y 2. s ( [ k ]+ = = s [ k ] = ([ k ] + ( [ k ]ɛ y. 5

7 The followig umbers s k, k = [ k ] +,..., 2, are all smaller tha ɛy. Sice j= y j = Ef i =, we get 2 k= s k =, which meas M ( = ad j[ k ] s i = [ ] j k [ ] + ( [ ]ɛ y i. k k This meas that for j =,..., ( M [ ] j k [ ] + ( [ ]ɛ y i k k = j[ k ] 2. Therefore there are absolute costats c ad C such that ( j c j k M y i C j k. Theorem implies the result. Remark. I particular i the proof we get that for every x R (7 c x k, + max x i z ji c,k x k,, i j,...,j where c = ( / ad c,k = k /[ k ] 2. Theorem 4 Let f,..., f, g,..., g be idetically distributed radom variables. Suppose that g,..., g are idepedet. Let M be a Orlicz fuctio. The we have for all x R E (x i f i (ω M 6e e E (x ig i (ω M. Remark The subspaces of L with a symmetric basis or symmetric structure ca be writte as a average of Orlicz-spaces, more precisely: the orm i such a space is equivalet to a average of Orlicz-orms. Thus our theorems ad corollaries exted aturally (for subspaces of L with a symmetric basis see [] ad for the case of symmetric lattices see [7]. 2 Proofs of the theorems To approximate Orlicz orms o R we will use the followig orm. z R m with z z 2 z m > 0 deote ( ki x z = z j x i. max k i=m j= Give a vector 6

8 I this defiitio we allow some of the k i to be 0 (settig 0 z j = 0. The followig lemma was proved by S. Kwapień ad C. Schütt (Lemma 2. of [5]. Lemma 5 Let, m N with m, ad let y R m with y y 2 y m > 0, ad let M be a Orlicz fuctio that satisfies for all k =,..., m The we have for every x R M ( k y i = k m. x 2 y x M 2 x y. Remark. Note that for every Orlicz fuctio M there exists a sequece y y 2 y m > 0 such that ( k M y i = k m for every k m. Because of Lemma 5, to prove both our theorems it is eough to prove the followig propositio. Propositio 6 Let f,..., f be idetically distributed radom variables (ot ecessarily idepedet. Let N be a Orlicz fuctio ad deote z j = N ( j N ( j, j =,...,. Let s = (s k k R 2 be the o-icreasig rearragemet of the umbers y i z j, i, j =,...,, where the umbers y i, i =,...,, are give by (3. The, for all x R E (x i f i (ω z 2 c x s, where c = ( / > /e. Moreover, if the radom variables f,..., f are idepedet the for all x R x 2 s E (x i f i (ω z. To prove this propositio we eed lemmas 7. Lemma 7 Let a i,j, i, j =,...,, be a matrix of real umbers. Let s(k, k =,..., 2, be the decreasig rearragemet of the umbers a i,j, i, j =,...,. The c s(k k= j,...,j = max a i,j i i where c = (. Both iequalities are optimal. 7 s(k, k=

9 Proof. Both expressios j,...,j = max a i,j i i ad s(k k= are orms o the space of -matrices. We show first the right had iequality. The extreme poits of the uit ball of the orm k= s(k are up to a permutatio of the coordiates of the form (ɛ a, ɛ 2 b, ɛ 3 b,..., ɛ 2b with a b 0, a + ( b =, ad ɛ i = ±, i 2. This meas that such a matrix has the property: The absolute values of the coordiates are b except for oe coordiate which is a. We get max a i,j i = a + b =. i j,...,j = Now we show the left had iequality. Clearly, we may assume that at most coordiates of the matrix are differet from 0. Next we observe that we may assume that for each row i the matrix there is at most oe etry that is differet from 0. I fact we may assume that this is the first coordiate i the row. Now we average the ozero etries, leavig us with the case that all ozero coordiates are equal. I fact, we may assume that these coordiates equal. Thus max i a i,ji takes the value 0 or. I fact, it takes the value 0 exactly out of times. It follows j,...,j = ( max a i,j i = ( i, which proves the lemma. Lemma 8 Let a i,j,k, i, j, k =,...,, be oegative real umbers. Let s l, l =,..., 3, be the decreasig rearragemet of the umbers a i,j,k, i, j, k =,...,. The 2 2 l= s l 2 j,...,j k,...,k max i a i,j i,k i 2 l= s l. Proof. The right had iequality is show as i Lemma 7. For the left had iequality we use here a coutig argumet. 8

10 Note that without loss of geerality we may assume that the sequece {s k } is strogly decreasig. There are exactly 2 2 out of 2 multiidices (j,..., j, k,..., k such that max i a i,j i,k i = s. Now we estimate for k 2 how may multiidices there are such that max a i,j i,k i = s k. i Clearly, oe of the coordiates a i,ji,k i has to equal s k, but oe of these coordiates may equal s j for j =,..., k. The secod coditio meas that for every i (except for the row with the coordiate equal to s k there are ji k coordiates that have to be avoided ad jk i = k. Let us assume that the coordiate that equals s k is a elemet of the first row. This leaves us with ( 2 ji k multiidices. Therefore we get 2 j,...,j k,...,k i=2 max i a i,j i,k i 2 2 s k k= 2 k= i=2 ( jk i 2 ( s k k 2 k= s k, sice k= j= That completes the proof. ks k = s k = ( 2 k= j= k=j s k j 2 j= k= ( 2 s k j s k s k k= k= = k= s k. Lemma 9 Let N, ad let y R with y y 2 y > 0. The we have for x R c x y + max x iy ji x y, i j,...,j where c = (. 9

11 Proof. We show the right had iequality. By Lemma 7 + j,...,j max x iy ji i s k (x, y, k= where {s k (x, y} k 2 is the o-icreasig rearragemet of { x i y j } i,j. Therefore there are umbers k i, i =,...,, with k i = such that + j,...,j max x iy ji i k i x i y k x y. k= Now we show the left had iequality. By Lemma 7 c s k (x, y + k= j,...,j = max x iy ji. i Therefore, we have for all umbers k i, i =,...,, with k i = c k i x i y k + k= j,...,j = max x iy ji. i The result follows by defiitio of y. Lemma 0 Let f,..., f be idepedet, idetically distributed radom variables with E f <. Let y j, j =,...,, be defied as i (3. Let be a -ucoditioal orm o R. The we have for all x R + j,...,j = (x i y ji E (x i f i (ω. Proof. Let t j (f i ad Ω i j := Ω j (f i, i, j, be defied by ( ad (2. Sice the fuctios f i, i =,...,, are idetically distributed, the umbers t i (f j do ot deped o the fuctios f j. Below we will write just t j. For j,..., j with j,..., j we put Ω j,...,j = Sice f,..., f are idepedet we have Ω i j i. P (Ω j,...,j =. 0

12 For (j,..., j (i,..., i we have Ω j,...,j Ω i,...,i =. Usig this ad the ucoditioality of the orm we obtai E (x i f i (ω = (x i f i (ω dp (ω j,...,j = j,...,j = Ω j,...,j (x i = + j,...,j = Ω j,...,j (xi y ji. For the last equality we have to show f i (ω dp (ω = + y ji. We check this. The fuctios Ω j,...,j are idepedet. Therefore we get f i (ω dp (ω = Ω j,...,j f i (ω dp (ω f i χ Ω i ji, χ Ω j,..., χ Ω i j i, χ Ω i+ j i+,..., χ Ω j Ω f i (ω χ Ω j χ Ω j dp (ω = + Ω iji f i (ω dp (ω. Lemma Let f,..., f be idetically distributed radom variables (ot ecessarily idepedet with E f <. Let y j, j =,...,, be defied as i (3. Let z z 2 z 0. Let s k (x, y, z, k =,..., 3, be the decreasig rearragemet of the umbers x i y j z k, i, j, k =,...,. The we have for all x R k,...,k E max i x iz ki f i (ω 2 k= s k (x, y, z. Proof. Let µ be the ormalized coutig measure o {k = (k,..., k k,..., k }. For i =,..., defie the fuctios ζ i : {k = (k,..., k k,..., k } R, i =,...,, by ζ i (k = z ki ad we put { } Λ i = (ω, k x iζ i (kf i (ω = max x lζ l (kf l (ω. l

13 We may assume that the sets Λ i, i =,...,, are disjoit. I case they are ot disjoit, we make them disjoit. Therefore P µ(λ i =. We defie umbers λ i ad sets Λ i, i =,...,, by P µ{(ω, k ζ i (kf i (ω λ i } = P µ(λ i ad Λi = {(ω, k ζ i (kf i (ω λ i }. The existece of these umbers λ i follows from the cotiuity of distributio of the fuctios f i (cf. defiitio of t j (f. We have ad Λ i = P µ( Λ i = {k ζ i (k = z l } {ω f i (ω λ i z l }. l= Sice µ{k k i = l} = µ{k ζ i (k = z l } = P µ( Λ i = As i the previous lemma we deote For (i, l we choose j i,l = if t λ i z l otherwise. The we have ad we get l= P {ω f i (ω λ i z l }. Ω i j = Ω j (f i = {ω t j f i (ω < t j }. ad j i,l with t ji,l λ i z l < t ji,l {ω f i (ω λ i z l } {ω f i (ω t ji,l } = {ω f i (ω λ i z l } {ω f i (ω t ji,l } = settig 0 j=ω i j =. Therefore we have = P µ( Λ i = P l= { ω f i (ω λ i z l } 2 j i,l Ω i j j i,l l= Ω i j, P j i,l j= Ω i j.

14 Thus we get which gives us 2 (j i,l, i,l= 2 2 j i,l. By the defiitios of the sets Λ i ad Λ i we obtai E max x iζ i (kf i (ω = x i ζ i (kf i (ω dp (ωdµ(k i k Λ i x i ζ i (kf i (ω dp (ωdµ(k. Λ i Sice Λ i ( l= {k ζ i (k = z l } j i,l j= Ωi j, k Sice 2 2 i,l= j i,l, we get E max i x iζ i (kf i (ω i,l= x i z l f j i (ω dp (ω i,l j= Ωi j j i,l x i z l y j. l= l= j= k E max i x iζ i (kf i (ω 2 2 s i (x, y, z 2 s i (x, y, z. Proof of Propositio 6. Let t l, l =,..., 3, deote the decreasig rearragemet of the umbers ( xi y j N ( k N ( k, i, j, k =,...,. The, by defiitios of the umbers s l, there are umbers k i with k i = 2 such that l= t l = k i x i s l, settig 0 l= s l = 0. Moreover, for every umbers m i with m i = 2 we have l= t l l= m i x i s l, 3 l=

15 which meas By Lemma 9 l= E (x i f i (ω z c + t l = x s. k,...,k E max i x iz ki f i (ω By Lemma E (x i f i (ω z 2 c l= t l = 2 c x s. Now we show the moreover part of the Propositio. By Lemma 0 E (x i f i (ω z + j,...,j = (x i y ji z. By Lemma 9 E (x i f i (ω z 2+2 j,...,j k,...,k max (x iy ji z ki. i By Lemma 8 E (x i f i (ω z t 2 l = x 2 s, which proves the propositio. l= Remark. Usig (7 ad repeatig the proof of Propositio 6 we ca obtai estimates for the costats i Corollary 3. Namely, for every f,..., f satisfyig the coditio of the propositio we have E (x i f i (ω k, 2 c x s, where s = (s l 2 l= is the o-icreasig rearragemet of the umbers y iz j, i, j, z = (,...,, 0,..., 0/[/k]. Moreover, if f,..., f are idepedet the x s 2c,k E (x i f i (ω k,. I particular, we have the variat of Theorem 4 for k, : (8 E (x i f i (ω k, 4c,k c E (x i g i (ω k,, where f,..., f satisfy the coditio of Propositio 6, g,..., g are idepedet copies of f, ad c,k = k /[ k ] < 2, c = ( / > /e. Let us ote that takig 4

16 m = k([/k] + ad applyig the (8 for the sequeces ( x i f i i m, ( x i g i i m, where x = (x, x 2,..., x, 0,..., 0 we obtai (9 E (x i f i (ω k, 4e e E (x ig i (ω k,, sice c m,k =. 3 Examples I this sectios we provide a few examples. We eed the followig two lemmas about the ormal distributio. Lemma 2 For all x with x > 0 2π (π x + x 2 + 2π e 2 x2 2 π x e 2 s2 ds 2 π x e 2 x2. The left had iequality ca be foud i [8]. The right had iequality is trivial. Lemma 3 Let f be a Gauß variable with distributio N(0,. Let the umbers t j, y j be defied by ( ad (3. The there are absolute positive costats c, c 2, c 3 such that (i for all j /e we have l t 2 j j 2 l ad j c l exp ( t 2 /2 c 2 l, (ii for all 2 j /e we have l y 2 j j 2 l j ad l y c 3 l. Proof. The iequalities for t ad y follow by direct computatio. The iequalities for the y j s follow from the iequalities for t j s, sice t j / y j t j / for every 2 j. Let us prove the iequalities for t j s. By defiitio P {ω f(ω t j } = j. This meas 2 π t j e 2 s2 ds = j. 5

17 By Lemma 2 we get 2π (0 e j 2 t2 j (π t j + t 2 j + 2π 2 π t j e 2 t2 j. First we show t j 2 l. For this we observe that j s e 2 s2 is decreasig o (0,. Suppose ow that for some j we have t j > 2 l. Therefore, usig (0, we get j j 2 π t j e 2 t2 j 2 π Thus we have 2 l j 2 π, which is ot true if e j. We show ow that l t 2 j j. The fuctio is decreasig o (0,. Suppose ow 2 l j 2π (π x + x 2 + 2π e t j < 2 l j. The we have by (0 j 2π e 2 t2 j 2π (π t j + t 2 j + 2π (π l + l + 2π 2 j 2 j 2 x2 j. ( j 4, which is false for j /e. That proves the lemma. Example 4 Let f,..., f be idepedet Gauß variables with distributio N(0,. Let M be the Orlicz fuctio give by 0 t = 0 M(t = e 3/(2t2 t (0, e 3/2 (3t 2 t. The we have for all x R c x M E max i x if i (ω C x M, where c ad C are absolute positive costats. 6

18 Proof. It is easy to see that there are absolute costats c, c 2 such that c k l(e/k k l(/j c2 k l(e/k j= for every k. Sice j= y j = E f = 2/π, Lemma 3 implies for every k ( c 3 k l(e/k k j= k l(e/k y j c 4, where c 3, c 4 are absolute costats. By the coditio of the example, M (t = 3/(2 l t o (0, e 3/2. Thus M (t 3/(2 l(e/t o (0,. By (5 we observe t 2 l(e/t/ 3 M (t 2t 2 l(e/t/ 3. Takig t = k/ ad usig ( we get for every k c 5 j= k y j M (k/ c 6 k y j, j= where c 5, c 6 are absolute costats. Applyig Corollary 2 we obtai the result. The ext example is proved i the same way as the previous oe, we just use Corollary 3 istead of Corollary 2 at the ed. Example 5 Let g i, i =,...,, be idepedet Gauß variables with distributio N(0,, k, ad x = k x i. Let 0 t = 0 t M(t = 2 k e 3/(2k2 t (0, /k e 3/2 (3t 2/k t /k. The for all λ R we have c λ M E (λ i g i (ω c 2 λ M, where c ad c 2 are positive absolute costats. The followig example deals with the momets of Gauß variables. 7

19 Example 6 Let 0 < q l, a q = max{, q}, g i, i =,...,, be idepedet Gauß variables with distributio N(0,, ad f i = g i q, i =,...,. Let 0 t = 0 M(t = ( a k exp q / (kt 2/q t (0, t 0 at b t t 0, where t 0 = ( q/2 2aq, k q + 2 a = q + 2 e q/2, eqkt 0 b = 2 eqk e q/2. The for all λ R we have cq (ca q q/2 λ M E (λ i f i (ω C (Ca q q/2 λ M, where 0 < c < < C are absolute costats ad x = k x i. This example is proved i the same way as the previous two examples. We use that ( q/2 k ( q/2 ( q/2 k l(/k l(/j 2k l(/k for every k /e q ad that j= for some absolute positive costats c, C. ca q (E g(ω q 2/q Ca q Fially we apply our theorem to the p-stable radom variables. Let us recall that a radom variable f is called p-stable, p (0, 2], if the Fourier trasform of f satisfies E exp ( itf = exp ( c t p for some positive costat c (i the case p = 2 we obtai the Gauß variable. Example 7 Let p (, 2. Let f,..., f be p-stable, idepedet, radom variables with E f i =. Let k ad x = k x i. Let { k M(t = (ktp t [0, /k] pt + (p /k t > /k. The for all x R c p x M E (λ i f i (ω C p x M, where c p, C p are positive costats depedig o p oly. I particular, c p x p E max x if i (ω C p x p, i where p deotes the stadard l p -orm. 8

20 Proof. There are positive costats c ad c 2 depedig o p oly such that for all t > c t p P {ω f(ω t} c 2 t p. Thus ( /p ( /p c j tj c 2 j. Repeatig the proof of Example 4 we obtai the desired result. Refereces [] D. Dacuha-Castelle, Variables aleatoires exchageables et espaces d Orlicz, Semiaire Maurey-Schwartz, Ecole Polytechique, expose o. 0 et. [2] Y. Gordo, A. E. Litvak, C. Schütt, E. Werer, Geometry of spaces betwee zooids ad polytopes, preprit. [3] M. A. Krasosel skii, J. B. Rutickii, Covex fuctios ad Orlicz spaces. P. Noordhoff Ltd., Groige 96. [4] S. Kwapień ad C. Schütt, Some combiatorial ad probabilistic iequalities ad their applicatios to Baach space theory, Studia Math. 82 (985, [5] S. Kwapień ad C. Schütt, Some combiatorial ad probabilistic iequalities ad their applicatio to Baach space theory II, Studia Math. 95 (989, [6] J. Lidestrauss, L. Tzafriri, Classical Baach spaces. I. Sequece spaces. Ergebisse der Mathematik ud ihrer Grezgebiete, Vol. 92. Spriger-Verlag, Berli- New York, 977. [7] Y. Rayaud, C. Schütt, Some results o symmetric subspaces of L, Studia Math. 89 (988, [8] M. B. Ruskai ad E. Werer, Study of a Class of Regularizatios of / x usig Gaussia Itegrals, SIAM J. of Math. Aalysis vol. 32, o. 2, [9] C. Schütt, O the positive projectio costat, Studia Math. 78 (984, [0] C. Schütt, O the embeddig of 2-cocave Orlicz spaces ito L, Studia Math. 3 (995,