”f. .. \Jk E n L 3... rue ..i L L, k 9. I ,d :1 . .ufi ,Hn; nu _ H... “x’. .“SS‘ c 1 LIBRA p} ,1"a)’)“‘. ’: ._ "~ t 'v ‘. .I :“F 1' C“ .' .h . " i H" ‘,”.}""~7 E This is to certify that the thesis entitled THE REALIZATiON OF ORLICZ SEQUENCE SPACES AND HARMONIC ANALYSIS presented by James Milford Boyett has been accepted towards fulfillment of the requirements for Ph .D . degree in Stat ist ics and Probability cgkatloaa 2m“ Major profeSsor Date 7/1114 / / 0-7639 um suns BUCK BINDERY INC. . : hlfififmv emoms " ‘:'::::E‘9HAIICIIIGIII ‘ ABSTRACT THE REALIZATION OF ORLICZ SEQUENCE SPACES AND HARMONIC ANALYSIS By James Milford Boyett In this thesis we consider the inter-relation between the realization problem of L. Schwartz and harmonic analysis for Orlicz sequence spaces. A solution to the realization problem generalizing the work of Mustari on Lp-Spaces is presented in Chapter II. Harmonic analysis for such Orlicz sequence spaces is then carried out in Chapter III. The latter work generalizes some work of Kuelbs and 'Mandrekar. Finally, in Chapter IV an explicit form of the Fourier transform for Gaussian measures on an interesting subclass of these Orlicz sequence spaces is obtained and is exploited to study a Central Limit Theorem for this class. The results of the final chapter in- clude and extend some work.of M;N. Vakhania. THE REALIZAIION OF ORLICZ SEQUENCE SPACES AND HARMONIC ANALYSIS BY James Milford Boyett A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1974 TO Vera , T.J. , and Yvonne ii Acmowmncm'rs I wish to express my sincere appreciation to Professor V. Mandrekar for his guidance and patience during the preparation of this dissertation. I thank him also for advise and encouragement given during my study at Michigan State University. Also, I thank.Professors R.V. Erickson, D. Gilliland, and J. Shapiro for reading this thesis and for their helpful comments. I am also indebted to Noralee Barnes for her excellent and swift typing of the manuscript. I am grateful to the Department of Statistics and Probability, Michigan State University, NDEA, and the National Science Foundation for financial support. Finally, I must acknowledge the patience and assistance of my wife, Yvonne, in this endeavor. iii TABLE OF CONTENTS Chapter Page I INTRODUCTION 1 II REALIZATIONS 3 2.1 Introduction 3 2.2 Basic Facts and Properties of Orlicz Spaces 5 2.3 Realizations of Orlicz Sequence Spaces 14 III LEVY CONTINUITY THEOREM FOR ORLICZ SEQUENCE SPACES 21 3.1 Introduction 21 3.2 Preliminaries 23 3.3 Lévy and Bochner Theorems 29 IV GAUSSIAN MEASURES ON ORLICZ SEQUENCE SPACES 37 4.1 Introduction and Preliminaries 37 4.2 Representation of Gaussian Measures on Orlicz Sequence Spaces 40 4.3 A.Central.Limit Theorem for Orlicz Sequence Spaces 45 4.4 Concluding Remarks 48 BIBLIOGRAPHY 50 iv CHAPTER I INTRODUCTION We consider in this thesis two related problems. The first is the problem due to L. Schwartz of realizations for Orlicz sequence Spaces, which in the case of classical Lp-spaces was first studied by Mustari [22]. In a subsequent paper [23], Mustari also studied the question for separable Banach Spaces and obtained necessary conditions for realization. His work does not provide sufficient conditions in the case of separable Banach Spaces, nor does it apply if the space is an LP- Space (O< p< 1). Once the realization problem is settled we show that it pro- vides the extension of methods in [13] to the Orlicz sequence Spaces. The Bochner theorem and Lévy continuity theorem proved in [13] can then be generalized rather simply to the case of realizable Orlicz sequence Spaces. In essence, we essentially give our proofs as to adapt methods in [13]. Recently, using different methods other than those in [13], J. Kuelbs [12] has studied the generalization of the results of [13]. We note that although his method is general, it is also more complicated than the method of [13], and in case of Orlicz sequence Spaces, which is his main application, his results are included in ours. As a matter of fact our realization result shows that the most general Orlicz sequence spaces for which methods in [12] are applicable are precisely the ones studied in this thesis. Our results bring out both the potential as as well as the limitations of the methods in [13] while providing very simple proofs of the extensions of the work.in [13]. As an application of methods involved in the Lévy Continuity Theorem we establish a form of the central limit theorem for realizable Orlicz sequence Spaces generated by a convex function. This is done in the last chapter after obtaining the form of the Fourier transform of Gaussian random variables (vectors) taking values in these spaces. This last result generalizes some of the work of M;N. Vakhania [28]. Chapter II begins with some basic properties of Orlicz functions and Spaces and other well-known results that are used throughout this thesis. Our main result -- the realization theorem -- is given in section 2.3 and from it we obtain, as‘a corollary, Mbstari's results of [22]. Using this theorem we also obtain a partial solution to a problem formulated by Lindenstrauss and Tzafriri [18]. In chapter III we give a Lévy continuity theorem and a Bochner theorem for realizable Orlicz sequence spaces and Show that these theorems for certain Orlicz sequence Spaces considered in [11] are con- tained in [14]. CHAPTER II REALIZATICNS §2.l INTRODIIITION Let 0 be a o-algebra of subsets of a set a, and let P be a complete probability measure on (0,4) . We shall denote by ”(0.03) the vector Space of real valued a-measurable functions where equality of functions is understood to be almost everywhere (a.e.). 0n 71((n,d,P) we define the distance d(f,g) = (I m dP(w)- Then 7R(fl.d.P) with the topology induced by this distance (topology of convergence in probability) is a tapological vector Space (t.v.s.) . We Shall consider structural conditions on a t.v.S. B in order that there should exist a probability Space (0,43) and a vector Space isomorphism T mapping E into 7R(0.d.P) (T : E .. ’/l((fl,d,P)) such that both T and T-1 are continuous. Topological vector spaces E for which this happens are said to be realizable with the linear homeomorphism T being called the realization. We consider a Special case of this problem when E is a real F-space of real sequences with a Schauder basis, and the realization T is assumed to satisfy addi- tional conditions (cf. 52.3). Such realizations have proved useful in the study of harmonic analysis on certain vector spaces (e.g. [l3] and [11]). In particular, realizations for E - LP and E = Lp[0,l] (0 < p s 2) with the usual topologies have been established in [13] and [25]. D.H. Mushtari [22] examined this problem for the sequence Spaces LP and showed that only in the case o.< p s 2 does such a linear homeomorphism exist. In this chapter we examine the problem for the Orlicz Spaces of sequences. We obtain necessary and sufficient conditions on the associated Orlicz function in order that a realization exists. This is achieved through careful analysis of the works of [3] and [26] where the Sufficiency of these conditions in the context of function spaces was studied. After giving necessary notation and terminology in the next section, we present our main result in the last section. §2.2 BASIC FACTS AND PROPERTIES OF ORLEZ SPACES 2.2.1 DEFINITION. An Orlicz function :9 is a continuous, even, non-negative function non-decreasing for positive x such that (p(O) - 0, (p(x) >0 for x i 0. For a sequence of real scalars a a {an} we write pcp“) = 2:'l¢(an) and let ch = {a - {an} : 3 ). €E+ - [0,») 9 pcp().-1a) < a}. We also write p(a) when no misunderstanding is likely from the omission of the subscript. If for a sequence ak = {a:} C2 I’cp and a = {8“} 64¢ p(a - 8k) ‘ E:=1¢(an - 8:) converges to zero as it tends to infinity, then we say "{ak} converges in the mean to a". If (p is a convex Orlicz function, then ch with the norm “a“cp = inf“, > 0 : p(x-la) s l} (Luxemburg norm) is a Banach space. In the Banach Space (Lq), ““q’) convergence to zero in the norm is equivalent to convergence to zero in the mean. The Space ‘29 with the norm ““cp is called an Orlicz sequence Space. For the most part properties relating to convex Orlicz functions are taken from [10], [17] and [29]. 2.2.2 EXAMPLES. 1) Let (p(x) -- m" (1 s p O generate the same Orlicz sequence space. 2.2.3 PROPOSITION ([17], p. 127). Let (p be a convex Orlicz function with tp(x) =- J‘gx‘p(t)dt. Then p(O) > 0 if and only if (iff) Lap is isomorphic to (.1. As our problem has been studied for the above case, we consider Orlicz functions which include convex functions with p(O) = 0. The function g(s) = sup t is then a right-continuous, non-decreasing function defined 3512151: non-negative reals such that q(O) - O and q(s) > 0 for s > 0. The function ¢(x) = ng‘q(s)ds is a convex Orlicz function, and following [10] we call it the complementary func- tion of (p. It is easy to see that the relation of being complementary is symmetric. 2.2.4 m. If (90!) = m" (1 < p are isomorphic as Banach Spaces). As stated in remark 2.2.4, for (p(x) - \x‘p (p > 1), mt) - M9 where 1/p + l/q - 1. The following proposition is an analogue of the classical result on inequalities involving complementary functions. 2.2.6 PROPOSITION. Suppose that q) is a convex Orlicz func- tion with the complementary Orlicz function t. Then, i) For all x,y 2 0, xy s (p(x) + y(y) (Young's inequality) ii) For all x ech’ z xnyn s \Hxlllcppwo) if pwm s 1 . For a convex Orlicz function (p we can define another vector Space of real sequences by hep a {a - {an} : V l > O, p(x-la) < co}, and since hq) is a subset of LCP we can consider it with the norm ““e' This new Space htp with norm “HP has played a significant role in the study of the topological duals of Orlicz sequence Spaces, and was introduced by Gribanov [7] who established the following result. 2.2.7 PROPOSITION. Let cp be a convex Orlicz function. Then ht? is a closed subspace of Lg). The dual Space of a t.v.S. E is the vector space 8' whose elements are the continuous linear functionals on' E. 3' will always be considered as having the weak-star topology ([24], p. 66). That is the topology induced by pointwise convergence. For y 68' and x €13 we denote as (in?) (Ct,y>) the evaluation of y at x (i.e., y(x) - (x,y)). 2.2.8 PROPOSITION. If :9 is a convex Orlicz function having complementary Orlicz function y, then h; is isomorphic to Lq). Given a convex Orlicz function the sequence Space 4,“) is linear. In general, however, the space ch associated with an Orlicz function tp need not be linear. An important class of Orlicz functions for which 4‘? is a vector space are those which satisfy the so called Az-condition in a neighborhood of the origin. In addition the Az-condition ensures us that the unit vectors (ek = (0,...,O,l,0,...), l in kth coordinate)" form a Schauder basis for the Space ([17], [18]). 2.2.9 DEFINITION. .An Orlicz function m is said to satisfy the AZ-condition for small x if there exists x0 >40, h > 0 such that (p(Zx) S hcp(x) for O s x S x0. Since m is assumed to be non-decreasing it is obvious that h 2 l; furthermore, for every ). > 0 there exists h(),) > 1 such that (p(xx) s h(),)q>(x) for O s x 5 x0 (cf. section 3.3). 2.2.10 PROPOSITION. Let (p be a convex Orlicz function with p(0) = 0. Then the following are equivalent: 1) (p satisfies the Az-condition for small x. ii) L = h P T iii) Lap is separable. 2.2.11 EXAIPLES. l) (90:) '- e‘x‘ - \x‘ - l is an example of a convex Orlicz function which does not satisfy the Az-condition. Incidently, the complementary function to (9 does satisfy the Az-con- dition ([10], p. 27). 2) cp(x) - \x‘p (O < p < l) is an example of a non-convex Orlicz function which satisfies the Az-condition for all x. For the Orlicz functions satisfying the A -condition, the associated 2 space Lt? need not be a Banach space. For example, consider the case :90!) - \xlp (0 < p < 1). However it is a t.v.S. as the following proposition [21] Shows. 2.2.12 PROPOSITION. Let cp be an Orlicz function satisfying the AZ-condition for small x. Then 4"? with the quasi-norm “a“q’ ' inf“, > 0 : p(fla) s l} is an F-Space in which convergence to zero in this quasidnorm is equivalent to convergence to zero in the mean. When the Orlicz function q) satisfies the Az-condition for small x, LT ‘with the quasi-norm given above will also be called an Orlicz sequence Space. EXAMPLE: Let cp(x) = m" (o < p< 1). Then “WWW is the classical Lp-space with the usual tOpology. Note however that H.“¢ 2 “.HP (“a“p = 2:;1\an\p : the usual quasi-norm on LP). Since we are interested in the Orlicz sequence spaces as topological vector Spaces, we need to know'when if ever, distinct Orlicz functions generate the same topological vector space of sequences. This depends on the behavior of the function in a neighborhood of the origin as the next definition and subsequence proposition show. 2.2.13 DEFINITION. Two Orlicz functions q) and y are said to be equivalent for small x (for large x) denoted q) I. r (q,9 q) if there exist an xo >20 and strictly positive constants b1,b2,kl,k2 such that blcp(k1x) S $(x) S b2.o such that quKx) s O y(x) s kch(x) for 0 < x s x (x z x The next proposition is o 0" basic to the theory of Orlicz Spaces and is taken from [21]. 10 2.2.14 PROPOSITION. Let <9 and y be Orlicz functions. Then, 1) Lap") iff (92¢ ii) If cpg y, then pcp(al3 40 as k-oco iff p'(ak) do as k .. co. iii) If (p satisfies the Az-condition for small x and (pi y, than V satisfies the AZ-condition for small x. 2.2.15 R_E_M_AR_R_. AS an example of the above, observe that, in view of propositions 2.2.12 and 2.2.14, if :9 and] V are two Orlicz functions such that cp E t and :9 satisfies the Az-condition for small x, then ch and L] are isomorphic as topological vector Spaces. Since the topology and vector Space structure of an Orlicz sequence Space is dependent only on the behavior of the Orlicz function in a neighborhood of the origin, any given Orlicz function satisfying the Az-condition for small x can be replaced by an equivalent Orlicz function satisfying the Az-condition for all x ([10], p. 24). Similarly, an Orlicz func- tion which satisfies the Az-condition for small x can be replaced by an equivalent Orlicz function which is strictly increasing ([21], p. 104). In particular Orlicz functions in chapter III will be assumed to be strictly increasing as well as satisfying the AZ-condition for all x. Two real valued functions f and g are said to be asymptotically equivalent at x - 0 if lim f(x) /g(x) - c > 0, denoted by f ~ g as x -+ 0. It is easy to see thatxfgr two Orlicz functions (9 and V: q) .. y as x -+ 0 is sufficient for cp E y, but the converse is not true. Facts concerning Orlicz function spaces are very similar to the above for Orlicz sequence Spaces. Aside from obvious differences some distinction is made necessary because Orlicz function Spaces over an ll interval of finite measure are determined by the behavior of the Orlicz function in a neighborhood of infinity and not the origin. We also need to assume that an Orlicz function is increasing with lim cp(x) -= as. For 771 = 77(([0,1], B[0,l], Lebesgue measure) let-”p(pfi) Jpgtp(f(x))dx where dx indicates the integration with respect to Lebesgue measure on [0,1]. We let LC? - [f 67R : 3 x 6 2+ 3 pCPQ-lf) < co}. Analogous to the situation existing in the sequence Space case a further assumption is needed to ensure that ch will be a linear space. 2.2.16 DEFINITIQW. An Orlicz function tp is said to satisfy the AZ-condition for large x with Az-constant h 2 0 if there exists an x0 > 0 such that (p(Zx) s hq)(x) for x 2 x0 . Suppose that tp is an Orlicz function satisfying the AZ-con- dition for large x. Then Lq, with the usual vector addition and multiplication by a scalar is a linear Space and becomes an P-Space under the quasi-norm “f“cp - inf“, > 0 : p(flf) s 1}. I'm with this quasi-norm will be called an Orlicz function Space. A sequence )ao. {£13 c: ch is said to "converge to zero in the mean" if lim pcp(fk It is well known ([10], p. 76) that convergence in the qufST-norm “'th is equivalent to convergence in the mean. Since our study deals with realizations of Orlicz function spaces as t.v. Spaces we need to know when distinct Orlicz functions generate the same function Spaces. 2.2.12 PROPOSITICN. Let (p and t be Orlicz functions. Then, L i L L iff cp ~ ) tp v I 12 11) 1t n9 1: then pcp(fk) -. o as k .. co iff flak) —. o as k _. no. iii) If (p satisfies the Az-condition for large x and (9 £3 vb, then y satisfies the AZ-condition for large x. Two real valued functions f and g are said to be asymptotically equivalent at infinity if lim f(x) /g(x) = c > 0 de- noted f ~ g as x -+ on. Then for two OrI-i'zz functions 1;) and y, tp ~ 1) as x -+ co implies (p E y, but the converse need not be true. The study of Orlicz Spaces seemsto naturally divide into two classes depending on how the rate of growth of the associated Orlicz function compares to the function f(x) - x2, and with this motivation we nuke the following definition. 2.2.18 DEFINITION. Let K(2,0) (K(2,oo)) be the collection of Orlicz functions m such that there exists Orlicz function y with qag y and y(x)/x2 is a.non-increasing (non-decreasing) function of x. The usefulness of this definition evolves from the next proposi- tion which can be found in [20]. 2.2.19 PROPOSITIm. Let tp be an Orlicz function such that cp E 1((2,0) (1p 6 K(2,an)) . Then there exists Orlicz function t such that q>3 y and ng/r) is a concave (convex) function of x. 2.2.20 RM. Due to this proposition and the fact that equi- valent Orlicz functions generate the same Orlicz Spaces, when m E K(2,0) (m €‘K(2,a9) we will assume without any loss of generality that m o,f is a concave (convex) function. We conclude this section by giving notation and well known de- finitions that will be used throughout this thesis. 13 For two real valued functions f and 3 we will write f(x) A g(x) for min{f(x),g(x)} and f(x) V g(x) for max{f(x),g(x)}. x x The indicator function of a set A will be denoted by [A], i.e. [A] - [A](m) ={1 if u’ EA 0 if m £EA If X is a symmetric (about zero) infinitely divisible random variable, then the characteristic function of X is given by xx(t) = exp[-ozt2/2 - [3(1 - cos ut)dM(u)} where a 2 0 and M is a Lévy measure on ‘2_ That is, M is a measure on g? such that I: dM(u) < m and I; u2dM(u) < m. A complete discussion of this representation for infinitely divisible distributions appears in ([6], p. 70). 14 §2.3 REALIEATIONS 0F ORLICZ SEQUENCE SPACES Let E be a vector Space on which an invariant metric d is defined. Furthermore,suppose that the vector space operations are continuous with reSpect to the topology on E induced by d, and that the metric Space (E,d) is complete. 0n E define the quasi-norm H'H by urn = d(x,0). Then (E,“o“) is an F-space ([24], p. 8). Now suppose that (E,“-“) is a real F-Space of sequences of real numbers with a Schauder basis {ek}. We say that the sequence space (E,\|-\\) is realizable if there exist a probability space (e.g.?) and a linear homeomorphism T : E ~7R(fl,d,P) such that the random variables in T(E) are symmetric about the origin and {T(ek)} is a sequence of independent identically distributed (iid) random vari- ables. (7ll(fl,d,P) is assumed to have the topology of convergence in probability.) In order to study the realizations of sequence Spaces a dif- ferent type of Space is needed. Suppose that X is a random variable symnetric about the origin, and let {Xn} 'be a sequence of independent random variables distributed as X. Define by {X the Space of all real sequences such that (2:;1 anxn} is Cauchy in probability. We say that the Space Lx is generated by the random variable X. ‘Note that since we are dealing with independent random variables, 9X e {{an} : 2:81 an)!n converges a.s.]. Our main effort is to eStablish that q) 6 K(2,0) is necessary and sufficient for the sequence Space LT (section 2.2) to be realizable. In particular we exhibit a symmetric random variable X such that L 3!. . (p X 15 The key to the development is the following lemma which can be found in [3] and [26]. 2.3.1 LEMMA. Suppose that m is an Orlicz function satisfy- ing the A -condition for small x and m 6 K(2,0). Then there exists 2 o 2 0 and a Lévy measure M such that 2 2 2 cp 3 o x + fish A 1)dM(u) 2.3.2 Emu. 1) If cp(x) - MP (0 < p< 2) then o = o and dM(u) - u-l-pdu. 2) If qu) - x2, then a = 1 and M. is the zero measure on Er. The next theorem characterizes those Orlicz sequence Spaces which can be generated by a symmetric random variable. In fact it shows that the Space LX for X a symmetric random variable is always an Orlicz sequence Space. 2.3.3 THEOREM. If X is a random variable symmetric about zero, then there exists an Orlicz function 1;) such that {‘X - Leg and m e K(2,0). Conversely, if m is an Orlicz function from the class K(2,0), then there exists a random variable symmetric about zero such that {'cp -—- LX' The equalities that appear above are equalities between vector spaces. In fact the mapping I(x) - x is a vector space isomorphism. PRQQE, First Suppose that X is a random variable symmetric about zero and consider E - LX' Then [an] 6E iff zit-1 aan con- verges a.s. The Kolmogorov three series theorem ([19], p. 237) implies the existence of A >10 such that 1) z:_1 P{‘anxn\ > A} < e 11) £31 E(anxn[\anxn\ s A]) < a 16 111) 3:..1 Var(aan[\aan| s A]) < e . Since the random variables {Xn] are symmetric, the series ii) is zero. Notice that using the symmetry of X i) and iii) can be written as .. A/laIn l2 220‘15/‘anidF(x) < co, and 22215130 x 2dF(x) < as where F(x) is the cumulative distribution function of’ X. Now define the function qA by qA(A) = I:(x2x2 A A2)dF(x). Then the above can be stated as {an} 611 iff 2:31 qA(an) < e. Clearly we can see the function qA is an even function of A non-decreasing for A 2 0 with qA(O) = 0 and q(x) > O for A i 0, and the Lebesgue dominated convergence theorem shows that q is con- A tinuous. Hence is an Orlicz function and since for x 2 0, (1A (4x2x2 A A2 ) s 4(x2x 2 A A2) for all A > 0, qA also satisfies the AZ—condition for all A > 0. ‘Now let q(x) - q1(X) and observe that qA (1,) - AZqOJA); thus, qA fl q and we conclude that {an} 61?. iff 2:=1q(an ) < 0" by proposition 2.2.14. The above Shows that E = Lq and so there remains only to Show that q E K(2,0), but this follows from q(),)/),2 - g(xz A l/x2)dF(x). Conversely suppose that m is an Orlicz function satisfying the Az-condition for small x and m e K(2,0). Then we exhibit a symmetric random variable X so that [S’. Ax. Lemma 2.3.1 implies there exist 0 2 0 and a Lévy measure M, such that mCx) g 02x2 + 53(x2u2 A l)dM(u). To simplify the details we consider three cases; the first case is to assume that the Lévy measure M is identically zero. Then ch '3 L2 and Chebyshev's inequality implies that if {an} 6 {,2 then 17 {a } 6.9x where X1 is a Gaussian distributed random variable with n 1 mean zero and variance 1. Using ([2], proposition 8.37, p. 177) it is easy to see that {an} ELXI implies {an} 61,2 ; thus, we have :9 X1 For the second case we will assume that a = 0. ‘Let XM, be a random variable with the characteristic function, xxM(t) = expf-f3(l - cos xt)dM(x)] and so X is a symmetric infinitely M o o . Q diViSible random variable. Recall that LXM {{an] . zn=l aan con- verges a.s.] where {Xn} is a sequence of iid random variables dis- tributed as XM' ‘We Show that cw =‘QXM. Suppose that {an} t L¢. Th m jm(a2u2 A l dM( d 1 ti 1a m dM(u) < m' en £n=l O n ) u) < m. an n par cu r 2h=1 I:/\an\ . hence on (2.3.4) zn___1 finan‘u - cos u ant)dM(u) < so for all t. 2 It can be easily shown that (l - cos x/2) z x /2 as x ~10 and so there exists a > 0 such that for \x\ < e. (1 - cos x) s x2/2. Thus, for O s u s l/[a ], it] < e we note n l/la \ l/|a l tzazuz f ‘ n (l - cos a u t)dM(u) s I n --2-—-M(du) O n 0 2 2 l/‘a i S. n 2 2 s 2 I0 anu M(du) a. .l/lanl 2 2 and because En=l J0 anu F(du) < m we must have for \t\ < e, m l/la \ anl $0 n (l - cos anu t)dM(u) < m. Together with (2.3.4) this implies that as Q En=1 A0(1 - cos anu t)dM(u) < m for \t‘ < e . 18 Hence ([2], theorem 8.38, p. 177) implies that 2:121 aan converges a.s. That is L c . <9 ‘XM Q Now suppose {an} ELXH. Then En=l aan converges a.s. and E X _. 0 in probability as k _. 0°. Since x m (t) = n-k an n >2"n=kanxn ex {. 0° ”(1 - cos u a t)dM(u)} we note 200 [ma - cos u a t)dM(u) —» O 9. rn=k do n ’ n=k -10 n uniformly on compact subsets of R as n -+ as. In particular this implies lim I?) E:=k [3(1 - cos u ant)dM(u)dt = 0. Using Tonellivs “no ! ‘2‘ k theorem and integrating out the variable t we get I a no sin anu : 11m nn=k J0(1 - —aT-)dM(u) = 0, but since there ex1sts a constant 15‘, u. k—«ao n b c>0 such that l-—-—-81:x2c(le2) forall x>0, on on 2 2 co 2 2 11:: zn___k loan" A l)dM(u) - 0. Thus, 2n=l g(anu A l)dM(u) < e implying that {an} Ech' This concludes the proof of this case since i b . [an] was an ar itrary vector from {,xM For the final case, suppose that o > 0 and M is not the + nw 2 2 zero measure on R . Then let ¢1(x) = ]0(x u A l)dM(u) where 2 ¢(x) E 02x 4- [g(xzuz A l)dM(u). Observe that tp 61((2 ,0) implies that tp €K(2,0) and furthermore that L C152. Thus, 2!, =4, and 1 (P1 ‘9 (P]. the question reduces to the second case. AS a corollary to the above proof we have the following. 2.3.5 COROLLARY. Suppose that q) is an Orlicz function satisfying the AZ-condition for small x and tp é 1((2 ,0). Then I, 8 {X where the random variable X is as given in the above theorem CP k k for each case, and for {a }c ch’ a -o O as k _. co in {'cp iff co . . {final 8:31“) -+ 0 as k _. as in probability. PROOF. For \t] < S with e as chosen in the above proof, it 2m (“8‘3 2 ~1og x (t); hence, if a _. 0 in l, as n=l n zoo a‘3( (p n=1nn 19 as k «'m then fi:=1 aEXn a 0 in probability. Conversely, we saw that if -1og Xfi kx (t) ~ 0 as k a m uniformly on compact sub- a =1 n n sets of Rh then a 22:=1(a:)2 + 2:;1 j3((a:)2u2 A l)dM(u) a O as k a m; therefore, ak a 0 in L as k a m. Now we prove our main theorem. 2.3.6 THEOREM. Suppose that m is an Orlicz function satisfy- the Az-condition for small x. The orlicz sequence space L is realizable as a space of random variables iff m E K(2,0). 2599:, If m 6 K(2,0), then by theorem 2.3.3 there exists a symmetric infinitely divisible random variable X such that LT = LX' Let E = {z:_ _1 anX n :{an ] E with {Xn } iid as X. Then E with the topology of convergence in probability is a t.v.s. Define T mapping ALT into E (T :.{,(P a‘E) by T([an}) = fi:;1 aan. This linear mapping is well-defined and injective Since {an} E‘Lm iff {an} 6 AX, and corollary 2.3.5 shows that T is bi-continuous. Thus L¢ has the realization given by T. Now suppose that LC? is realizable,i.e. there exist 77:01.43) and T : {’cp ~Wl(fl,d,P) such that T is a linear homeomorphism and {T(ek)} is a sequence of iid symmetric random variables where ek is the kth unit vector of qu Then for [an ] €.L¢ , T({an ]) = E: a T(en ). =ln Let X = T(el). Then ALT = Ax and since T is a linear isomorphism this is equality between vector Spaces. Theorem 2.3.3 Shows that AX = Ly where t is an Orlicz function from the class K(2,0). Then using the bi-continuity of T and corollary 2.3.5 we conclude .L¢ = Li (equality as t.v. Spaces) and hence, m E K(2,0). 2.3.7 EXAMPLE. 1) Let (p(x) :- m" (o < p < 1). Then from example 2.3.2 we get a = O and dM(u) = u-1-pdu and recognize that 20 exp{-I:(1 - cos xu)u_1-pdu] is the characteristic function of a symmetric Stable random variable with index p. Thus {an} €.L¢ iff E X converges a.s. with {Kn} iid symmetric stable of index p. n=l an n As a corollary to theorem 2.3.6 we obtain the results of Mushtari [22]. 2.3.8 COROLLARY. The sequence Space LP for p > 2 is not realizable as a Space of random variables. This result can also give some information on a problem suggested by J. Lindenstrauss and L. Tzafriri [18]. There they prove the following: 2.3.9 PROPOSITION. Every Orlicz sequence Space generated by a convex Orlicz function m contains a subspace isomorphic to LP for some p 2 1. The question raised is can more be said about the possible values of p. Our results Show the following: 2.3.10 THEOREM. If m is a convex Orlicz function and m €_K(2,0), then 1L¢ contains a subSpace isomorphic to LP for some p, 1 S p s 2. 23992, The proof is immediate from theorem 2.3.6, corollary 2.3.8, and proposition 2.3.9. "IL. 111‘! < r z-. ,. 1 . CHAPTER III LEVY CONTINUITY THEOREM FOR ORLICZ SEQUENCE SPACES §3 .1 INTRODIIITLON_ In this chapter we generalize the Lévy continuity theorem to the case of Orlicz sequence spaces when the associated Orlicz function is in the class K(2,0). This extends the results of [13] which handles the Situation for the classical LP (0 <'p < a) Spaces. For completeness and ease of reference a Statement of the Bochner theorem proved in [11] is included. We also show that the reSults of Kuelbs and Mandrekar can be applied to Lq: when the Orlicz function m is in the class K(2,m); thus, [14] includes the reSults given in [11] for this class of Orlicz Spaces. Our proof of the Lévy continuity theorem is based on the realization theorem and adaptation of techniques from [13] to this case. Recently, by using an extension of characteristic functions due to L. LeCam [16], J. Kuelbs has obtained some results which constitute generalization of the work in [13]. However, his proofs are complicated due to the generality of his approach and the precise sequence Spaces to which Such results are applicable are not known. Extensions of techniques of [8], [13] are of interest due to the simplicity of the approach as brought out in the recent work of [11]. Our approach shows that the problem of Levy continuity theorem is intimately connected with the existence of a realization of the Space. In turn the realization 21 22 problem is intimately related to the structural problems of Banach Spaces as our theorem 2.3.10 indicates. First we establish some terminology and preliminaries in section 3.2 and give our results in the last section. 23 §3 .2 PRELIMINARIES Let .L denote the vector Space of all sequences of real numbers with the topology of coordinate-wise convergence (Tychonoff topology). We will frequently think of the Lap Spaces as being subsets of 1!, (cf. section 4.2) and if x 64, we define "U X II N (x1,...,xN,O,...) (O,...,O 0 >4 II N ’XN+1 ”‘N+2 ’ ‘ ° ') We denote the coordinate functionals on L by Bj’ j = 1,2,... . That is, Bj :1, —~ ,9 is given by ej(x) = xj for all x EL and for all j. For m 6 K(2,0) we denote by km the probability measure on the Borel subsets, B, of L by taking the product measure on 4, such that the coordinate functionals have independent infinitely divisible laws with Fourier transforms exp[ -ozt2/2-j':(l - cos ut)dM(u)} where 2 e o and the Levy measure M are given by lemma 2.3.1. 3.2.1 DEFINITION. The Fourier transform (or characteristic functional) of a probability measure p, on the Borel subsets of a t0pologica1 vector Space E is the function x defined on E' (the t0pologica1 dual of E) by 7100 = is exp{i(y.x)}du(y) for x 68' 3.2.2 REMARK. If E' contains enough linear functionals to separate points of E and if p. is a tight Borel probability measure (cf. definition 3.2.7) , then x determines u uniquely on the Borel subsets of E. In particular we study the case where E = L and 1p 6 K(2,0) . While ch may not be locally convex, the coordinate functionals are contained in {’1}; and so L' does separate points of cP L . More will be said later concerning the tightness of Borel proba- T bilities on L . m 3.2.3 LEM. If p. is a probability measure on the Borel subsets c. of L¢, m €.K(2,0), then the function ~ N (X1Y) = [gm Ek=1 Bk(x)5k(Y) is a B A Omeasurable function on L X L , and (Ac? X u)({\(x.y)"! < col) = 1. The proof of this lemma is very similar to the proof for the Lp (O < p s 2) case as given in ([13], lemma 3.1, p. 222) and will be omitted. 3.2.4 DEFINITION. If u is a probability measure on the Borel subsets of LT, m E K(2,0), then we define the extended Fourier trans- form of u on L by 32(X) = II. exv{i(y.x)"}du(y) for x 6 L - W 3.2.5 REMARK. The extended Fourier transform of a probability u on the Borel subsets of LCp is a Borel measurable function on L which is finite almost everywhere with reSpect to the measure A and which is equal to x(x) ='IL exp{i(y,x)]du(y) for all x €.Lé. Thus i is truly an extension of T; from .Lé to L. Let S be a metric space with I denoting the Borel sets in S. We need the following concepts from the theory of weak convergence of measures [1]. 3.2.6 DEFINITION. A sequence of probability measures {Pu} on I is said to converge weakly to the probability measure P on 25 I if ijdPn a IsfdP for every bounded continuous real valued func- tion defined on S. If {Pn} converges weakly to P, we write Pn =rP. 3.2.7 DEFINITION. A probability measure P on (8,3) is said to be tight if for every positive a there exists a compact set K CZS such that P(K) > 1 - e- We shall deal only with Orlicz spaces for which m satisfies the AZ-condition, and by proposition 2.2.10 these .Lw Spaces are separable. The L Spaces are a priori complete being F-Spaces; hence, every probability measure on LCp is tight ([1], p. 10). 3.2.8 DEFINITICN. A family {paz oz EA} of probability measures on S are said to be tight if for every positive e there exists a compact set X C S such that ”d(K) > 1 - e for all a 6A. 3.2.9 DEFINITIW. A family [110: a 6A} of probability measures on (8,1) is said to be relatively (conditionally) compact if every sequence of elements {pork} (ark 6A, k = 1,2,...) contains a weakly convergent subsequence. The next proposition is due to Prohorov and can be found in [1]. 3.2.10 PROPOSITION. Let S be a separable and complete metric space. Then a family {pa: 0 EEA} of probability measures on (8,3) is tight iff the family is relatively compact. We shall be dealing with tight sequences of measures on L for m E K(2,0) and satisfying the AZ-condition. In order to do this we need a description of the compact subsets of Such spaces. For the LP case such a description is readily available [4]. We give the following generalizations which do not seem to be available in the literature. 26 Suppose that m is an Orlicz function satisfying the Az-con- dition for 0 s x s x with constant h 2 l (i.e. (p(Zx) s h(p(x) 0 for 0 s x s x0). We now assume without any loss of generality, that .9 is strictly increasing (remark 2.2.15). If mathl, \s‘] 5 x0 we have the following (p(t‘s) = 0, 3 J 3 for all i,j 2 J, p(f - f )< a. By assumption ii) 3 N 3 Sup pN(f) < min{rp(x0), e/3h]. Then, e V - N _ p(fk' - fk') finalcp(en(fk.) Sn(fko)) + p (fk. fk) 1 J 1 J 1 j Since iljim iBn(fk.) ‘ Bn(fk')l = 0 and (p is continuous at the S “m 1 origin, a J a v i,j 2 J, £=ltp(fin(fk) - enak )) < 6/3. Then by the 1'. assumption n1ax{\5n(fk )1, \Bn(fk )\} < x0 for all i,j and n 2N + 1, i J using the fact that (p is increasing, and equation 3.2.11, we find N p (fk - fk) S hpN(fk) + hpN(fk) S 23/3 . i J 1 J Hence, p(f - fk ) < e for all i,j 2 J. i J Conversely,suppose that K c: [”29 is compact. Then clearly i) k holds since p is a continuous function on ch. Now take 1 > e > O. L where or = min[tp(xo) 1 e/Zh], but if “f“cp S 1 then p(f) a “fum’ so Then 3131,...,£ EKsuchthat for feKai,lsisL9“f-f1\\ <0, P p(f - £1) < a. Now choose N so that pN(fi) < o for all 1, l s i 5L. Take f 6K and choose i so that p(f - f1) S 01. Then, pN(f) == pN(f - fi + fi) s hpN(f -fi) + hpN(fi) implying that pN(f) < 3. Since 1 > e > O was arbitrary and f was an arbitrary vector from K, condition ii) is verified. 3.2.14 THEOREM. Suppose (p is an Orlicz function satisfying the Az-condition for all x with Az-constant h. A set R C (1‘? is compact if for every 6 > 0 3 x1,...,xr E L:P such that r KCU S(x j=1 ) s 5}. 16) where S(x-1,6) = {y GLCP : p(y _ X J J 28 PRQQE, For the proof it will suffice to Show that under the stated hypothesis, conditions i) and ii) of theorem 3.2.13 are satisfied. Take e > 0. Then by assumption 3 x1,...,xr ELCP such that K': JE=1 S(xj,e/2h). For f E'K choose i, 1 s i S r, such that f E S(xi,e/2h). p(f) = p(f - Xi + xi) S hp(f - xi) + hp(xi) S e/Z + h SUP p(xi) < m . lsisr Hence sup p(f) < m and the first condition is shown. Now choose fEK N0 such that N 2 N0 implies pN(x.) < e/Zh for all 1, 1 s 1 s r. 1 Then again we see pN(f) 5 hpN(f - xi) +'hpN(xi) yielding pN(f) a 3/2 + hpN(xi) < 6- Therefore, lim Sup pN(f) = 0, and the Nflm f6( theorem is proved. 29 53.3 LEVY CONTINUITY THEOREM AND BOCHNER'S THEOREM We now prove Lévy Continuity Theorem for Orlicz sequence Spaces with m E_K(2,0). The section also includes a statement of Bochner's Theorem and concludes by showing that the case m E K(2,m) is derivable from the work in [14]. This is done to Show that the work in [11] which uses techniques similar to [13] is included in [14]. In this section until further notice, m will denote an Orlicz function satisfying the AZ-condition for all x with Az-constant h, and we define ¢(t) by 2 2 m (3.3.1) S(t) = o t /2 + [0(1 - cos xt)dM(x) where o > 0 is the constant and M is the Lévy measure of lemma 2.3.1. Recall that corollary 2.3.5 implies q): y; hence, i will generate the same topology on L as does ¢° In particular for {xk}Cch, pcp(x|3 -»0 as k—Om iff p'(xk) —.0 as k-o0;hence, for every 6 > 0 there exists 6(a) = 6 > 0 such that (3.3.2) if p¢(X) > c then p¢(x) > 5 We begin the proof of Lévy's Continuity Theorem with the following lemmas. 3.3.3 LEWIA. If {a : 0 EA} is a family of probability "‘—" o measures on .Lcp such that lim Sup J (u. ) = 0 N—m; wo as. "'Y a where J < ) =1‘ 1 - «mm-[13" vwx ) + pN(x)]}du (x) NsY “a LT n=1 n V o Then {110: o EA] is conditionally compact. 30 PROOF. Take e20,0<81-e for ySy0,N2N0 and all a E A. With no loss of generality we take Y0 < 1. For x 6 EC, :1: MW“) < S/Zh < l < $2 therefore, “131‘!me <3: . Since {x E PNch : “PNxHq;< l/y} is a bounded subset of a finite dimensional - l. Space, we can find x1,...,xr 1n PNLCp Such that “Xi‘\¢< l/y and for all x EL with lPx <1/ we get min Px -x. }duk(y> ltwex"{‘("’z”d”k‘z’d"o 1' ° , - d d d 1:"! L’Jtcpjéexfiflx Y 2)} ACP(X) uk(y) 111(2) lim hwjflpexvi -pw(y-2) lduk(y)duk(2) I j exPi‘P (Y‘z)}du(y)dp(z) = A . Lt? Le? * Similarly ‘1‘;ka (um converges to A which is Iblxl dlcpo Thus {Xk} converges in mean-square to i, and so also in A -measure. 32 Conversely, using the preceding two lemmas we find {pk} is a conditionally compact sequence. Thus there exists a Subsequence {pk ] J converging weakly to a probability measure v with Fourier transform g. Then g = lim Xk = x, and the uniqueness of the Fourier transforms for measures implies {”k } must converge weakly to a (i.e. u = v J on L ). Furthermore this shows that any convergent Subsequence of (p I {pk} must converge to u. Hence, every Subsequence of {pk} in turn [1 has a Subsequence which converges to u and so {uk} converges weakly to H1 since weak convergence in this case is metric convergence. The Bochner theorem for these Spaces was proved in [11] using 5 techniques developed in [13]. In the following statement of the theorem let C denote the complex numbers and a positive definite function is a function satisfying £:,j=lzi£jf(ri - rj) 2_0 for any finite collection of real numbers r1,...,r and complex scalars N 21,...,zN. 3.3.6 BOCHNER THEOREM. Suppose that m isian Orlicz function satisfying the AZ-condition for small x and T E'K(2,0). Let f : L} —oC. Then T f(y) = j, expiuxsndmx) (y esp m for some Borel probability measure on LQp iff f is positive definite, sequentially weak-star continuous and H II f(0) = 11mN f,.f(y)dxcp(eN.y) T where r = - ,Lexmuxsndxcpuww II;=1€XP( emwejoo». 33 = ,..., 11 th I: Z 0 for any sequence of EN (eN,l €N,N) sue a 6N,j (for all N and j) and lim max gN = O, and when w is given N—m lSjQ‘l ’ by equation 3.3.1. Now we consider the case where the Orlicz function is from the class X(2,a9: still satisfying the AZ-condition for all x 2 0. We derive the Lévy continuity theorem and Bochner theorem for the sequence space LT ‘by Showing that Lq> satisfies conditions given in the work of Kuelbs and Mandrekar. First we give necessary notation and terminology from [14], state the theorems, and then show that their hypotheses are satisfied by this class of Orlicz sequence Spaces. Let Lco denote the Banach Space of all bounded sequences of real numbers with the usual supremum norm, and L: the positive cone of Lm. E will be a real F-space with a basis {bu}. 3.3.7 DEFINITION. If A EL: and (no: (1! EA] is a family of probability measures on E such that x GEE ° m A 2 < m] ' 1 ”J ' zk=l kxk for each a EA, we say A is sufficient for the family [11 : 0 EA]. ‘ a 3.3.8 DEFINITION. A family of probability measures 4.. {”02 a 63A] on E is a x-family for some 1 ELco if A is sufficient for {u : 0 63A] and for every 3,6 >10 there is a sequence {3N} a such that ( EE ° °° x2 6 >1 - ”Cy-x ' zk=N+lxk k < 1 6 implies a: nap: EE . “2k=N+1xkka< b(6)} > 1 - (a + a") where lim a“ = 0 and b is a strictly increasing function on N [0,e) with b(0) =0. 34 Now let o(°) be a convex function on [0,m) such that q(O) = 0 and q(s) > 0 if s > 0. Further, assume that for every compact K of E there exists an r > 0 such that y E;K implies 2 A(y) = 2T=1 q(yi) < r, and for every r > 0 there exists M > 0 Such m 2 that A(y) < r implies 21=1a(yi) S My(“y]) where y(-) is another continuous function on [0,m) such that y(0) = 0. 3.3.9 DEFINITION. If the quasi-norm, H-“, on E admits the .1 g existence of functions a(*) and y(-) having the above properties we will say that it is accessible. If the quasi-norm on E is accessible then by the Try-topology 9 we will mean the tOpOlogy on E' generated by taking as a subbase all A translates of sets of the form {x E;E' : T(x,x) < 1] as T(°,-) varies over the symmetric, positive definite, bilinear forms on E' which are jointly weak-Star sequentially continuous on E' and satisfying ::=la(tkk) < m where tkk = T(Bk’ek) with Bk the kth coordinate functional on E (i.e. ak(x) = xk). 3.3.10 BOCHNER THEOREM [14]. If E has an accessible quasi- norm then a function x on E' is the Fourier transform of a proba~ bility measure iff i) ((0) = l, A is positive definite, ii) X is continuous in the T -tOpology, 0 iii) the family of measures {on} correSponding to x(PN(-)) 4. has a Subsequence which is a A—family for some A EL0° satisf in l' m t r 0 whenever fim ‘ y g i” z1=k"1 11 1:1“ Here a(-), of course, is the function aSsociated with the (‘11) < w. accessibility of the quasi-norm. 35 For ¢ E K(2,¢0 we now let AcP denote the product probability on L such that the ith coordinate is Gaussian with mean zero and variance Ai > 0, and A E.é: is chosen so that it is sufficient for the probability measure u on E. 3.3.11 LEVY CONTINUITY THEOREM [14]. Let ink} be a sequence of probability measures on E with Fourier transforms {xk]. Then Yuk} converges weakly to a measure p with Fourier transform x iff {”k} is a A-family for some A E L: which is also sufficient for u, {Xk} converges to x on a subset of E' which is dense in E' with reSpect to weak-star sequential convergence, and {RR} converges in AT measure to i, To show these results apply to Lm when m E K(2,m) we need only Show H-“w is accessible since we already know .LQp is an F-Space with a basis (T satisfies Az-condition). Let q(s) = mQ/S). Then by proposition 2.2.19 we know a is a convex function on [0,m) such that q(O) = 0 and q(s) > 0 for s > O. For K a compact subset of L , theorem 3.2.13 implies T __ Q co 2 . jug}: pep”) < co and pch’) - Ei=1 0 a constant M and continuous function y(°) such that y(0) = 0 such that A(y) < r implies A(y) S.My(“yflq). Since a is convex so is T (composition of convex functions) and then using the Az-condition we note m(Ax) S AqKX) for 0 < A S l 2n-l and (p(Ax) S hnq;(x) for < A S 2“, n = 1,2,... . Now define h), OsAsl Y0.) = l (hn+l - hn)A + 2hn _ hn+1 2n-l g A n IA N :1 ll H 'N 2n-l 36 Then clearly y is a continuous piece-wise linear function and «((0) = 0. Recall that since (p is convex “y‘lcp = inf{¢ > 0 : p is Gaussian with mean for each linear functional y E_E' (the topological dual of E). If L (the Space of all real sequences x = {xn}) is given the Tychonoff topology, then it becomes a separable reflexive Fréchet ([24], p. 8) Space. Let Lo denote the linear subSpace of L which consistscu those elements of L containing only a finite number of nonzero coordinates. Then the topological dual of L is L , i.e. 0 1 ”.0. 37 38 4.1.2 PROPOSITION. ‘Let T be an Orlicz function, then LT is a Borel subset of L. PRQQE, The coordinate functionals defined on L are Borel measurable maps (i.e. Bj : L a E’ defined by Bj(x) = xj for all x E L is a Borel measurable map for all j). Since m is a con- tinuous function, m o Bj is Borel measurable for all j; thus, i§=1T o Sj defines a Borel measurable map on L for all N. That Lm is a Borel subset of I now follows from these remarks and that J to co m L - U 1 flm=1[x €‘L . Ej=1¢(8 T n= (x)) S n} . J The above proposition implies that the Borel subsets of Lm are Borel subsets of L; hence, a measure a defined on the Borel subsets of LT) can be considered as a Borel measure on L. 4.1.3 DEFINITION. A matrix S = (31 ) is said to be positive J definite if eSe' 2 0 for all e E.L (e' denotes transpose of the vector e). If S = (Sij) is a symmetric positive definite matrix then (4.1.4) sij S/sii jsjj for all 1 and J This is easy to see by considering in definition 4.1.3, the vector e having - /§;;- in the ith coordinate, ngzf in the jth coordinate and zero's elsewhere. The next proposition is well known and can be found in [28]. 4.1.5 PROPOSITION. If u is a Gaussian measure on L then the Fourier transform of u is given by (4.1.6) x(f) = expii ] f 6 {.0 9 39 where = 2:31:39“, 111 = {on} e L and = gikflsjkfjfk with the matrix (sjk) being symmetric and positive definite. Con- versely, all functionals of the form 4.1.6 defined on L0 are the Fourier transforms of Gaussian measures on L. A relation between moments of a Gaussian random variable and its standard deviation is given by the next proposition [27]; the proof of which is straightforward and can be found through a change of vari- ables and the definition of the gamma function, F. 4.1.7 PROPOSITION. Let X be a Guassian random variable with , 2 mean zero and variance 0 . Then, (4.1.8) Elxlp = c(p,2)(EX2)p/2 9 where c(p,2) = F(p +.1/2)/(2“)1/2. We also have need for the following Special case of a result of Fernique [5] pertaining to the integrability of Gaussian vectors. 4.1.9 PROPOSITION. If A is a Gaussian measure on an F- space (E,u-“), then [Ellxndmm ., . 4O §4.2 REPRESENTATION OF GAUSSIAN MEASQRES ON ORLICZ SEQUENCE SPACES In this section we will characterize Gaussian measures on Orlicz sequence Spaces which are generated by a convex Orlicz function m where m E K(2,0) by giving a representation for their characteristic functionals. For convex Orlicz functions m with a right derivative p(t) that is strictly positive at the origin (p(O) > 0) prOposition 2.2.3 says that LT is isomorphic (as a topological vector Space) to L1. Since the form of Fourier transforms of Gaussian measures on L1 are well known (e.g. [28]), we assume that p(O) = 0. 4.2.1 THEOREM. Let m be a convex Orlicz function satisfying the Az-condition for all x with mCx) = ng‘p(t)dt. Furthermore, suppose that m 6 K(2,0) with p(O) - 0. Then u is a Gaussian measure on .Lw with mean a epr iff the Fourier transform x of p defined on L; has the form (4.2.2) x(f) = exp{i - 1/2 } for all f €.Lé , where = 2m _ 8. f.f with (s ) being a symmetric positive definite matrix such that 2:;1qKsEk) < a. In addition, 8 can be taken as (x-a)Bk(x-a)du(x) for j = 1,2,...; k = 1,2,... 81k 8 JP; SJ {5992, Let q; be a convex Orlicz function satisfying the hypothesis of the theorem, and let V denote the Orlicz function complementary to m (2.2.4). Now suppose that u is a Gaussian measure on L¢"with mean a E LT. The Bochner theorem 3.3.6 for AL? implies that the Fourier transform x of u on L; is weakrstar sequentially continuous; thus, x(f) = lime(PNf) for f €12"). Now (,ng L and PNf 6 L0 41 for all N, so thinking of p, as a Gaussian measure on I, and applying proposition 4.1.5 we get the existence of a symmetric positive definite matrix S = (Sjk) with sjk 3 ILBj(X-a)6k(x-a)du(X) = J"; Bj(X-a)_6k(x-a)dp,(x) for j = 13,-”; k = 1,2,...; and such that . _ . . N x(f) — lime(PNf) — limNexph - 1/2 2j,k=1sjkfjfk} for f 6 Lc'p. Note that limN for f 6L. To complete this part of the proof we need to show IImNZ‘N:=18jkfjfk exists for all f E {,c'p and that {BER} 6 LCP. We do this by first proving the following lemna. 4.2.3 LEMMA. With the notation as given above {SER} 6 ch. PROOF. Since p. is Gaussian with mean vector a 6%, Bj(-) is Gaussian with mean = 3j(a) for j = 1,2,... . By proposition 4.1.7, IL ‘Bj(x'a)‘du(x) = E‘BJXX-a); = C(1,2)[E(Bj(x_a))2]%’ (p i.e., (4.2.4) = (1/c(l,2))IL \Bj(x-a)\du(x) ‘P Proposition 2.2.8 implies that h; is isomorphic to {,q), and so to complete the proof of the lemma it will suffice to show on k stlskk r > 0 such that O¢(Bk(Y)/r) S 1. Now using 4.2.4, Bk”) < on for all y 6 hV. Take y 6 11¢. Then there exists \zflls’é skkakon s <1/c(1 2)) hfmm‘“ -a)HBk(y)|du(X) Multiplying the right hand side by l and applying proposition 2.2.6, 42 *2 zislskkekm s p (y/sj, uxx-amwd.., * <9 but the right hand side is finite since p*(y/r) s l and proposition 4.1.9 yields I; “\x-a“\q§u(x) < m. Now define SN : hW » 5’ by A: ‘P . . SN(y) ‘£:=lskkek(y)' The above shows that the limit on N of SN(y) exists for every y E hV, and applying the uniform boundness principle [24], S(y) = limNSN(y) is a bounded linear functional on by. By the earlier remark that h; can be identified with qu there must exist a unique x E.L¢ such that S(y) = - zk=16k(x)ak(y) for all .w% _& *, but S(y) zk=lskkak(y) implies that x - {skk} and so 2:=1¢KSER) < m completing the proof of the lemma. Y €.h Upon proving the lemma we can finish the "if" portion of the proof of the theorem by showing that limN£:,k=lsjkfjfk exists for all f €.L$r Since (sjk) is a symmetric positive definite matrix, in- eQuality 4.1.4 can be used to derive, . 5 ‘£:,k=18jkfjfk\ s fi§,k=lsjk‘fj“fk‘ s (i:=ls?j\fj\)i:=lskk‘fkl . Then by the above lemma {83k} €.L¢_ implying that we have a bound independent of N by proposition 2.2.6(i) completing the proof. Conversely suppose that g(f) = exp{i - 1/2 } where (Sjk) is a symmetric positive definite matrix such that a: 5 _ co zkequskk) < m and - 2j,k=lsjkfj fk. Let {nN} denote the sequence of Gaussian measures on Lq> with Fourier transform xN(f) = xflPNf) for N = 1,2,... . Since for each N we are dealing with a measure on a finite dimensional Space, the form of xN(f) implies that ”N is a Gaussian measure. Now to Show that the sequence of measures {“N} is conditional compact. Take 3,5 > O. 43 L m “NW Ech : p(pm > a} 5 1/5 §i¢2i=n+1¢wfimdwfl = 1/5 filmlhjmdflwwm Since (p G. K(2,0) proposition 2.2.19 implies that (p o ./ is con- cave; thus, . m 2 My e LC? = 9:)(3’) > a, s 1/5 zi=L+1 6} < e for N 8 1,2,... . From this fact and utilizing the techniques in the proof of lemma 3.3.3,'we see that the sequence of measures {nu} is conditionally compact and must converge weakly to some measure a on .Lw. Since x(f) = limhxiPNf) for each f €.L$, x is the Fourier trans- form of the measure p. We are done if u can be concluded to be Gaussian. Choose y E (,4). Then the sequence of Gaussian random vari- ables {} converges to the random variable <-,y>; hence, the random variable <-,y> must be Gaussian for each y 6 Lc'P. Therefore 9 is a Gaussian measure on cw. As a corollary to this theorem we get the result of MJ‘Nicolas Vakhania [28] (see also [15]). 4.2.5 COROLLARY. x(f) == exp{i - 1/2 } for f 6 LI; is the Fourier transform of a Gaussian measure p on LP 44 with mean a Ein (1'< p s 2) iff (sjk) is a symmetric positive as p/Z m = f o definite matrix Such that zj,k=lsjkfj k and Ek=18kk < a: 45 §4.3 A CENTRAL LIMIT THEOREM.FOR QRLICZ SEQUENCE SPACES We now study a central limit theorem for sequences of independent identically distributed random variables (vectors) taking values in an Orlicz sequence Space Lw. 4.3.1 THEOREM. Suppose that m is a convex Orlicz function satisfying the AZ-condition for all x with m G K(2,0). Let 21,22.... be independent identically distributed random vectors with values in Lq> and having zero expected values. Let “N denote the ’5 meaSure induced on ch by YN 8 N- (21 +...+ ZN) and LL denote a measure equivalent to al. Then if (a 3 2) ° (p([j' ' 82( )d (u) A5) a. . . Zk=1 LC? k y P 1 < and if a zj,k!lsjkfjfk (f €.L¢) is a norm continuous function on it? where sjk'jl. Bj(y)5k(y)dp,(y) (for all j and k), then {nu} converges weakly to the Gaussian measure v on L“, with Fourier transform x(f) - exp{-1/2 } for all f €.L'. W 4.3.3 REMARK. Noting that S I (s is a symmetric positive jk’ definite matrix and that {stk} E.L¢,*we have the hypothesis of theorem 4.2.1 satisfied; thus, we conclude that X is the Fourier transform of a Gaussian measure on L . Now we prove the theorem. M. For f 61,4) define 5,03) = E{exp{i}} = [apexwi }duN(y) and x(f) = E{exp[i}} = IL exp{i }dp(y) . w Then for all f 6 Lc'p’ xN(f) = [x(f/,/N)]N. Using the series expansion 46 for the exponential function we write, X(f//N) = L. [1 4‘ i - 1/2 2 ¢ +’0(1/N)]du(y). but 21 having zero mean implies xzdu(Y) + o(1/N) W II H I (l/ZN)}:j ,k=lsjkfjfk + o(1/N) ; thus, limN xN(f) whim/mu" exp{-1/2 } = x(f)- Hence x must be a positive definite function on .Lé with x(0) = 1. Now we show that the sequence of measures (”N3 is conditionally com- pact and must converge weakly to a measure v. Take e,6 > 0. Since m €‘K(2,0), by proposition 2.2.19 (p o f is concave, implying, , L m My 6 cc? . p (y) > a} s (1/6)Ej=L+1Ichcp(Bj(Y))duNW) 52 jj). s (1/5)2:éb+1o(s 2 2 Since ILJJWN‘NW) - It Bj(y)du-(y) for j - 1.2.--.. and by assumption 4.3.2 as in the proof of theorem 4.2.1, we conclude that the sequence of measures {“N} is conditionally compact. Therefore {”N} converges weakly to a measure v on qu and since limth(f) = 'x(f) for all f E Lé, the Fourier transform of v is x. Then by 47 theorem 4.2.1 (remark 4.3.3) v is a Gaussian measure on L . Thus we know u“ a v, and v is the Gaussian measure with the required Fourier transform. 48 §4 .4 CONCLUDING REMARKS The question of the realization of Orlicz function Spaces can be discussed following these definitions taken from [3] and [26]. Let E denote a topological vector Space of Borel measurable functions defined on the interval [0,1]. 4.4.1 DEFINITION. A random linear form (r.l.f.) X, defined on E is a linear mapping of E into a space of measurable functions ”((0 ’4’?) ' 4.4.2 DEFINITION. A r.l.f. X defined on E is said to have independent increments if for all finite collections of elements {f1,. . "fn} C; 1?. having disjoint support, the random variables X(f1),...,X(fn) are independent; furthermore, fn -+ O a.e. implies X(fn) —~ 0 in probability. 4.4.3 DEFINITION. A r.l.f. X defined on E is said to be homogeneous if for congruent Borel subsets E1,E2 of [0,1], manly) - Axum». 4.4.4 DEFINITION. A r.l.f. X defined on E is said to be symmetric if the random variable X(f) is synmetric about zero for all f GE. A t.v.S. E is said to be realizable as a Space of random variables if there exists a probability space (n,d,P) and a linear homeomorphism T mapping E into 7I((fl,d,P) such that the map T when considered as a r.l.f., is symmetric, homogeneous with independent increments. The problem as formulated above is considered by Urbanik and Woyczynski [26], and Bretagnolle and Dacunha-Castelle [3] when the topological vector Space E is an Orlicz function space L¢[0,1]. Their results can be used to Show that as in the sequence space case (p G K(2,0) 49 is necessary and sufficient for the function space LCP to be realizable. The following problems are presently under study: 1) Harmonic analysis on realizable Orlicz function Spaces. 2) The domain of attrac- tion problem for Gaussian measures on realizable Orlicz sequence Spaces. 3) Representation of Fourier transforms of infinitely divisible dis- tributions on realizable Orlicz sequence Spaces. The results of this study will be presented elsewhere. BIBLIOGRAPHY [1] C2] [3] [4] [5] [6] [7] L8] L9] [10] [11] [12] [13] BIBLIOGRAPHY Billingsley, Patrick (1968). Convepgence pf Probability Measures. John Wiley & Sons, New'York. Breiman, Leo (1968). Probability; AddisonJWesley, Reading. Bretagnolle, J. and Dacunha-Castelle, D. (1969). Application de l'étude de certaines formes linéairés aléatoires au plongement d'eSpaces de Banach dans des espaces LP. Ann. scient. Ec. Norm. Sup., a 437-480. Dunford, N. and Schwartz, J.T. (1957). Linear Operators Vol. 1, Interscience, New York. Fernique, M. Xavier (1968). Intégrabilité des vecteurs Gaussiens, C.R. Acad. Sc., Paris, 219' 1698-1699. Gnedenko, B.W. and Kolmogorov, ASN. (1954). Limit Distribu- tions for Sums 2; Independent Random‘12giables. Addison- Wesley, Reading. Gribanov, Y. (1957). On the theory of Ln-Spaces. Uc. zap. Kazansk. un-ta 111 62-65. Gross, L. (1963). Harmonic Analysis gg_gilpgrt Spaces. Mem. Amer. Math. Soc. No. 46. Halmos, Paul R. (1950). Measure Theory. D. Van Nostrand, Princeton. Krasnosel'skii, M.A. and Rutickii, Ya. B. (1961). Convex Functions and Orlicz Spaces. Gordon and Breach, New‘York. Kucharczak, J. and Woyczynski, W.A. (1973). Bochner's theorem for Orlicz sequence Spaces. Bull. Acad. Polon. Sci., §§; 551-558. Kuelbs, J. (1973). Fourier analysis on linear metric Spaces. Trans. Amer. Math. Soc., 18; 293-311. Kuelbs, J. and Mandrekar. V. (1970). Harmonic analysis on certain vector Spaces. Trans. Aner. Math. Soc., 142‘ 213-231. 50 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] .Urbanik, K. and Woyczynski, WkA. (1967). 51 Kuelbs, J. and Mandrekar, V. (1972). Spaces with a basis. Trans. Amer. Math. Soc., 162 113-152. Kuelbs, J. and Mandrekar, V. (1968). Report #902, The University of Wisconsin, Madison, Wisconsin. LeCam, L. (1970). dans les espaces localement convexes, les Probabilities sur Harmonic analysis on F- MRC Techn ical Sumry Remarques sur le théoremé limite central les Structure Algébrigues, Colloq. C.N.R.S., Paris. 233-249. Lindberg, Karl (1973). Studia mth., M 119-146. Lindenstrauss, J. and Tzafriri, L. (1971). Spaces. Israel Journal of Mathemtics, .l_0_ 379-390. Loéve, Michel (1963). Princeton. Matuszewska, W. (1960). On generalized Orlicz Spaces. Acad. Polon. Sci., ELL; 349-353. Mazur, J. and Orlicz, W; (1958). On some classes of linear Studia Math., Ell; 97-119. Mustari, D.H. (1969). the realization of LP Spaces by Spaces of random variables. Theor. Prob. Appl., 14' 699-701. Mustari, D.H. (1973). On subSpaces of Orlicz sequence spaces. On Orlicz sequence Probability Theopy, D. Van Nostrand, Bull. spaces. On a problem of Laurent Schwartz and on Some general questions of the theory of (Russian) Teoriia Veroiatnostei i ee Primeneniia, l§_ 66-77. Rudin, Walter (1973). Functional_Analysis. Schilder, Michael (1970). Some structure theorems for the symmetric stable laws. Ann. Math. Statist., 41, 412-421. probability measures in linear Spaces. McGraw-Hill, New York. A random integral and Bull. Acad. Polon. Sci., X__Y_ 161-169. Vakhania, Mm Nicolas (1965). tions normales de probabilités dans les espaces L (1 s p < w), C.R. Acad. Sc., Paris, ggg, 1334-1336. Vakhania, Mg‘NicolaS (1965). Orlicz Spaces. Sur une prOpriété des réparti- Sur les répartitions de proba- bilities dans les eSpaces des suites numériques. C.R. Acad. Sc., Paris, 299 1560-1562. Zaanan, A.C. (1953) . Linear Analysis. and Interscience, 13 New York. P. Noordhoff, Groninger HIIUIWIH 196 3386 l 293 03 "I "I'll R I" u I" ”I H H. 3