cmmn suacmss“or}mrlmaw mvssm’ggr'5?..iiffgji1ij Mommy msuags oN—BANACH SPACES 3- ”Thesisforthefiegrée UfPhp MICE-MAN SYATE UNWERSITY ,~ -; . ’ARUNOD KUMAR‘ 1971 VWF‘ 'i' This is to certify that the thesis entitle (Defiant?) g, (LBS O In Kn; {L :DiVlSlHkQ (Em “lax V’leiuxes (k 67'» KEQWAQCL S‘chgy presented by Av unocl Kurnaf has been accepted towards fulfillment of the requirements for Meme in §W€chfi Q, flwiuxhif'u $44M QM $212223...— 1 Date ?//g/?/ 0-7639 ABSTRACT CERTAIN SUBCLASS OF INFINITELY DIVISIBLE PROBABILITY MEASURES ON BANACH SPACES By Arunod Kumar In this thesis stable probability measures and self-decomposible probability measures on a real separable Banach space are considered. Semi-stable probability measures on a real separable Hilbert space are also considered. In Chapter 2 stable probability measures on a real separable Banach space are defined and several characterizations of these measures are established using a generalization of the convergence types theorem. These results are used to identify stable probability measures as limit laws of certain normed sums of independent, identically distributed Banach Space valued random variables. These limit laws possess a Lévy-Khinchine representation that can be char- acterized on certain Orlicz Spaces in terms of the representing Lévy-Khinchine measure. In Chapter 3 self-decomposible probability measures on a real separable Banach space are characterized as a limit law of certain uniformly infinitesimal sequence of independent Banach space valued random variables. These limit laws are infinitely divisible and possess a Lévy-Khinchine representation that can be characterized on certain Orlicz Spaces in terms of the representing I . o Levy-Kh1nch1ne measure. Arunod Kumar Finally in Chapter 4, semi-stable laws on a real separable Hilbert space are considered. These laws are also limit laws. Lévy-Khinchine representation of a symmetric semi-Stable law has also been obtained. CERTAIN SUBCLASS OF INFINITELY DIVISIBLE PROBABILITY MEASURES ON BANACH SPACES By Arunod Kumar A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1971 TO MY PARENTS ii ACKNOWLEDGEMENTS I take this opportunity to express my most sincere gratitude to Professor V.S. Mandrekar, my thesis advisor, whose constant en- couragement and guidance made this thesis possible. His discussions and comments were quite helpful. I am grateful to him for intro- ducing me to measures on metric spaces. I would like to thank Professor J.F. Hannan for his careful reading of this thesis and proposing significant changes which led to clarity and precision. My sincere thanks are due to Mrs. Noralee Barnes for her excellent typing and cheerful attitude in the preparation of the manuscript. I also thank Professor H. Salehi for his interest in the formulation of my general doctoral program. Finally I am indebted to the Department of Statistics and Probability, Michigan State University, and the National Science Foundation contract no. GP-ll626 (Summer 1970) for financial support. iii TABLE OF CONTENTS Chapter Page 0 INTRODUCTION 1 1 BASIC CONCEPTS AND RELATED KNOWN RESULTS 4 1.0 Introduction ............................. A 1.1 Basic Definitions and Results .. ......... . 5 1.2 Orlicz Spaces and Associated Hilbert Space 0.0....00...OOOOOOOIOOOOOOOOOOOI...O 10 II STABLE PROBABILITY MEASURES ON BANACH SPACES 15 2.0 Introduction 0....OOOOOOOOOCOOOOOOOOOOOOOO 15 2.1 Preliminary Results for Stable Laws ...... 16 2.2 Stable Probability Measure on a Banach Space .00....COOOCIOCOOOOOIOOC0.0.0.000... 18 2.3 Lévy-Khinchine Representation of Stable Measures on Certain Orlicz Spaces ........ 27 III SELF—DECOMPOSIBLE PROBABILITY MEASURES ON BANACH SPACES 33 3.0 IntrOduction .00...OOOOOOOOOOOOOQOOOOOOOOC 33 3.1 Preliminary Results on Self-Decomposible Laws OOOOOOOIOOOCCOOOOOOOOOOOOOOOOOOOOC0.0 34 3.2 Self-Decomposible Laws and Limit Laws .... 37 3.3 Lévy-Khinchine Representation of Self- Decomposible Probability Measures on Certain Orlicz Spaces .................... 46 IV SEMI-STABLE LAWS ON SEPARABLE HILBERT SPACES 51 4.0 Introduction ..... ..... . ........ .......... 51 4.1 The Main Theorem ......................... 52 V CONCLUDING REMARKS 56 REFERENCES 58 iv 0. INTRODUCTION In [5], Donsker proved a much stronger theorem than the classical theorem on convergence to normal ([21], p. 280) by study- ing the problem as the convergence to a limit law on a function space. This Opened up interest in the study of probability laws on abstract spaces ([19], [25], [26]). Motivated by the classical technique of Lévy [22], LeCanI [19] introduced the concept of the characteristic functional of a finite measure on a linear space and pointed out that the limit laws can be easily handled almost in the classical way if the analogues of the celebrated Lévy continuity theorem and Bochner theorem can be made available. Unfortunately, such a situation was hard to come by, as can be seen in the work of Getoor [8]. In [9], L. Gross intro- duced the idea of the extension of characteristic functions and gave complete analytic analogue of the Lévy continuity theorem on a separable Hilbert Space. Extensions of the work of Gross can be obtained for certain subclasses of Banach Spaces as shown by J. Kuelbs and V. Mandrekar in [17], but the Situation here from the point of view of-the use in limit laws is very unsatisfactory. However in some spaces J. Kuelbs and V. Mandrekar in [16] have been able to obtain complete solution to the so-called Central Limit Problem [16] and gave the form of the limit laws in terms of their Lévy-Khinchine representation. On general Banach spaces, however, the general Central Limit Problem is difficult to handle as can be seen from the work of LeCam [20]. In this thesis, we look at limit laws associated with the normed sums of Banach-Space valued random variables borrowing ideas in the scalar case from LoEve ([21], §23, p. 319). In Spite of the unavailability of the Lévy continuity theorem we have been able to obtain complete generalization of Theorem 23.3A and its corollary ([21], p. 323) for the probability measures on Banach spaces. These generalizations can also be regarded as the extensions of the work of Jajte [14] on stable probability measures on separable Hilbert space. In fact, on Banach spaces for which complete solution to the general central limit problem is available [16], we are able to give even analogues of Theorems 23.3B and Theorem 23.43 ([21], p. 324 and 327). These extensions also include and considerably generalize work of Jajte [14] on stable laws on separable Hilbert Spaces. In order to show the usefulness of the Lévy Continuity Theorem, we treat in Chapter IV the case of semi-stable laws on a real separable Hilbert space. Here we use, of course, the work of Cross [9]. The main technique in other chapters is a com- bination of methods of functional analysis and characteristic functionals. We start in the next chapter with the setting up of the preliminary ideas and notations. Chapter II is devoted to the study of stable laws, Chapter III studies the self-decomposible laws and Chapter IV the semi-stable laws. Chapter V contains re- marks leading to some unsolved problems, solution of which could be used to connect earlier work of Chapters II, III and IV to the convergence of stochastic processes. CHAPTER I BASIC CONCEPTS AND RELATED KNOWN RESULTS 1.0 Introduction In this chapter we present for the sake of completeness and easy reference some standard known results, concepts and definitions. For further details the reader is referred to [1] and [23]. We Shall denote by E a real separable Banach space with norm H'H and by R the Space of all real numbers with the usual topology. The elements of E will be denoted by x,y,z,...and of R by a,b,c,..., etc. R+ and R- will stand for the set of all positive and all negative real numbers, respectively. Rn will stand for the n-dimensional Euclidean Space. E* will denote the (tOpological) dual of E. 51E) will denote the o-field generated by the Open Subsets of E and W1 the space of all probability measures defined on ,6(E). For a 6 W2 and a e R, .a # 0, Tau is defined to be an element in 771, by Tap(B) = “(a-IE) for every B G 51E); where a B = [a-lx: x E B}, and for a = O we define Tap = 6 where for each B E EKE), 6X(B) I if x 6 B 0 otherwise. We Shall call 6x the probability measure degenerate at x. 4 1.1 Basic Definitions and Results 1.1.1 Definition. (8) For H and x: in Wb we Shall denote by u * v an element of W2 defined by a * v(B) = £p(B-x)dv(x) for every B 6,8(E), where B-x = [y-x: y E B]. We shall call u * v the convolution of a and v. (b) A probability measure a on ‘B(E) will be called infinitely divisible (i.d.) if for each positive integer n there exists an element An in W2 and an xn in E such that u = Mn)?"n * 6x , n where (An)*n denotes x“ convoluted n times with itself. 1.1.2 Definition. (a) A sequence {um} in W2 is said to converge weakly to H in ”Q if for every bounded continuous real-valued function f on E, lfdun « ifdu. We shall denote this convergence by “n = u. (b) A sequence [an] in W: is said to be weakly sequen- tionally compact (for short, compact or tight) if for every 6 > 0 there exists a compact set K6 in E such that pn(Ke) > 1-3 for all n. (c) A sequence {an} in W1 is said to be shift compact, if there exists a sequence {Kn} in B such that {an * 6x } is n compact. The following theorem will be used repeatedly. 1.1.3 Theorem ([23], pp. 58, 59, 153). (a) Let {kn}’ ' z: 1: {an}, [vn] be three sequences in W1 such that x“ ”n vn for each n. Then (i) if the sequences [in] and [an] are compact, then so is the sequence [vn}; (ii) If the sequence [in] is compact, then ”n and Vn are shift compact. (b) Let ”n be a compact sequence in W1 and fin(y) a m(y) for each y G E*, where fin is as defined in 1.1.7. Then ”n = u, for some u in WL Furthermore, 3(y) = ¢(y) for each y E E*, 1.1.4 Definition. Let (0,3,P) be a probability space and let X be a random variable on 0. Then X is said to be distributed as v if v = P o X-l. We now recall some ideas on measures on linear spaces. The following definitions are due to I. Segal and are taken here from [9]. 1.1.5 Definition. A weak distribution on a t0pologica1 linear space L is an equivalence class of linear mappings F from the (t0pologica1) dual space L* to the class of real-valued random variables on a probability space (depending on F) where two such mappings F1 and F2 are equivalent if for every finite set of vectors y1,...,yk in L* the sets {Fi(y1),...,Fi(yk)} have the same distribution in k-space for i = 1,2. In a finite dimensional space a weak distribution coincides with the notion of a measure, that iS, if L is finite dimensional then for any given weak distribution there exists a unique Borel probability measure on the Borel subsets of L such that the * identity map on L is a representative of the given weak dis- tribution ([11], p. 372). 1.1.6 Definition. A measure H on a locally convex topo- logical linear space L is defined to be Gaussian if for every con- tinuous linear functional y on L, y(x) has a Gaussian distribu- tion. The Gaussian measure p is said to have mean zero if y(x) has mean zero for each y. 1.1.7 Definition. The characteristic functional (ch.f. or Fourier transform) of a proability measure u on the Borel subsets of a linear topological space L is a function a on L* (the topological dual of L) given by x , * ”(Y) = Iexp[1(y,x)]du(x) , for each y 6 L L 1.1.8 Remark. One special example of a weak distribution on a real separable Hilbert space H is the canonical normal dis- tribution (with variance parameter one). This weak distribution is that unique weak distribution which assigns to each vector y in H* a normally distributed random variable with mean zero and variance HyHZ. It follows from the preceeding prOperty that the canonical normal distribution carries orthogonal vectors into independent random variables ([9], p. 4). It is known that some of the theory of integration with respect to a measure can also be carried out with respect to a weak distribution on H. For details we refer the reader to [11] and the bibliography given there. We need the following definition which is taken from ([30], p. 190). The following preliminaries are essential for the work in Chapter IV. 1.1.9 Definition. An Operator from a real separable Hilbert space H into H, which is linear, symmetric, non-negative definite, compact and having finite trace is called an S-Operator. If T is an S-Operator on H, then it is well known that T has the representation or: (1.1.10) Tx = len(x,en)en , n: where {en} is an orthonormal subset of H, x“ 2 0, and "MB < 00. I’m The S-Operator T on a real separable Hilbert space H n has a representation as an infinite symmetric, non-negative-definite matrix T = {tij} where by non-negative-definite it is meant that n . n . Z_ tikxixk 2 0 for any integer n and any (x1,...,xn) E R . 1,k—l Furthermore, if tik = (Tei,ek), where [e,] is a complete ortho- 1 normal system in H, then t.. < m . 1 1 1 "(‘18 i From the representation in (1.1.10) it is easy to verify that L (Tcx, cx)% = ‘c|(tx,x)2 for any real number c and % k «‘5 s (Tx,x) + (Ty,y) . Thus (Tx,x)15 is a semi-norm (T (xiv) .x+>') on H. Let 2 be the class of all S-operators on H. 1.1.11 Definition of T-tOpOlogy. The T-topology on H is the smallest locally convex tOpology generated by the family of semi-norms pT(x) = (Tx,x)}é on H as T varies through 2 ([29], p. 172). 1.1.12 Definition. Let H1, H2 be Hilbert Spaces with orthonormal systems {en}, [En] respectively. Then a continuous linear operator A from H1 into H2 is called Hilbert-Schmidt operator if there exists an orthonormal system {gn} in H1 such that 2 “Agnfig < m ([7], p. 34). n=1 2 1.1.13 Remark. If T is an S-operator on H then T possesses a unique non-negative, symmetric square root, which we denote by T5 ([28], Theorem, p. 265). Now using the fact (see [7], Theorem 4, p. 39) that the square roots of S-operators are Hilbert-Schmidt operators one can easily Show that the tOpology T on H is the weakest topology on H for which all Hilbert- Schmidt Operators are continuous from T to strong topology on H. Thus a basic open neighborhood of x0 is [x: “A(x-x0)“ < 5} whenever A is a Hilbert-Schmidt Operator on H. Therefore our definition of T-tOpology coincides with that of L. Cross ([9], p. 5). 1.1.14 Definition. A tame function on a real Hilbert space H is a function of the form f(x) = 6(Px) where P is a finite dimensional orthogonal projection on H and Q is a Baire function on the finite dimensional space PH. For such a function we have f(x) = Y((x,e1),..-,(X.ek)) where e1,...,ek is an orthonormal basis Of PH and Y is a Baire function of k real variables. If F is a representative of a weak dis- tribution then the random variable f~ = Y(F(e1),...,F(ek)) depends only on the function f and the mapping F while integration properties of f" such as the integral of f”, the distribution of f”, convergence in probability of sequences fg, etc. depend only on f and the fn's and on the weak distribution of which F is a representative. Let us denote by 5 the directed set of finite dimensional orthogonal projections on H directed under inclusion of the ranges. For a given continuous function f on H and a given weak distribution one may consider whether the net 10 (f o P)~ of the random variables where P ranges over directed set 3, converges in probability as P approaches the identity through 3. If so then the limit which we shall denote by f~ is a random variable whose integration prOperties are completely determined by the function f and the weak distribution. In [10] and [11] classes of continuous functions are described for which the limit defining the random variable f~ exists when the weak dis- tribution in question is the canonical normal distribution, and some explicit evaluations are also given. 1.1.15 Remark. Let H be a real separable Hilbert space, and let [Pj] be any sequence of finite dimensional projections, converging strongly to the identity operator. If a complex—valued function f on H is uniformly continuous in the tOpology T then lim in prop (f o Pj)~ exists with respect to the canonical normal j-m distribution. It will be called extension of f and will be denoted by f~ ([9], Theorem, p. 5). 1.2 Orlicz Spaces and Associated Hilbert Space We shall now recall definition of Orlicz Spaces and some results from [16] for use in later chapters (cf. §2.3 and §3.3). Let F be a function satisfying the following conditions (a) P is defined on [0,m) into [0,m), (b) F(O) = 0, F(S) > 0 for S > 0, (1.2.1) (c) F is convex and strictly increasing on [0,m) (d) F(ZS) s M F(s) for all s E [0,m), where M is a positive constant independent of s. I 11 Then it follows that S F(S) = & p(X)dX where p(0) = 0 and p(s) is non-decreasing on [0,m). We assume without loss of generality that p(s) is left-continuous. Let m(x) be the inverse function of p as defined in ([32], p. 76). We now define S (1.2.2) Ms) = g cp(x)dX Then T and A are complementary in the sense of Young ([32], * p. 77). By S? we mean all real sequences [xi] such that G) (1-2-3) Zl“(\x.\) <00 . 1 l * Similarly, SA is all real sequences [xi] such that (I) Z [\(‘Xip < co . 1 If x = [xi] is a sequence we define IIM8 CD SUP {Z \xiyi\ = y 1 i “X“? A(\yi\) S 1} 1 and oo oo HXHA = SUP {2 \xiyil : 2 F(\yi\) s 1} . y 1 i=1 1.2.4 Definition. The Orlicz space Sr (SA) is the collection of all real sequences such that “X“? (“X“A) is finite. 12 1.2.5 Remarks. (a) Let a be a function satisfying 2 (1.2.1) and F(s) = a(s ). Then clearly F satisfies (a), (h), (c) in (1.2.1) and since F(2s) = 9(432) s M 0(232) S M2(82) = M2f(s) , F satisfies (1-2-1), hence we know ([32], Corollary, p. 81) that * S (and, therefore E ) contains the same sequences as S . Further, P a F it is known that S is a real separable Banach space in the norm T qur and since F(25) s M2F(s) for s 2 O we also have ([32], Lemma 0, p. 83) that {pn} C ST converges to p E S in norm pro- F vided . m _ . m 2 _ 11m .2 P(]xi,n-xi]) — 11m .2 Q/[(xi n-xi) ] — O , n 1=l n i=1 ’ th . where x. n.X. mean the i elements of pn’ p respectively. 3 ) is a Banach space F . . th with a basis [bn] where bn is the vector with one as the n (b) By Theorem 6.2 of [16], (EasH'“ coordinate and other coordinates zero. Let a be a function satisfying (1.2.1) and assume that aC(-), the complementary function of a(o) in the sense of Young ([32], p. 77), satisfies (1.2.1). Notice that if Ea = L2 then a natural choice for the function a is 0(5) 5 5. Hence UC(S) = 0 on [0,1] but ac(s) = m for s > 1. Thus aC(-) does not Satisfy (1.2.1) when Ea = L and this is a Special case 2 which is easily handled. Following [16], we shall denote by E either the Hilbert space L2 or an E space where a and a a QC satisfy (1.2.1). 13 In terms of the notations we have used in this section, Ed is equivalent (isometrically isomorphic) to the Orlicz space SF 2 where F(s) = a(s ). We will let Sq, 80 denote the Orlicz c spaces given by a(.) and OC(°), reSpectively. Then the dual space of SO can be identified as S and since a(-) also a c satisfies (1.2.1), except when Ea = L2, it follows that the dual f S ' S 32 , . 150 . 0 ac is O(([]p ) For each A = (11:A2:---) in the positive cone Of Sq , c we define the Space H1 as all sequences x = (x1,x2,...) such a: that E 1.x, < m. Then H is a Hilbert space with HXHZ = 1 1 1 x 1 9° oo 2 2 1.x and the inner product (x,y) = Z x,x,y,. In the Special 1 1 i 1 1 1 1 case E = L we have 8 = L and for simplicity we take 0’ 2 are a) k = (1,1,...). Then HI = L2 and we shall assume without loss of generality that 0(3) 2 s. The following lemma is proved in [16]. 1.2.6 Lemma. Ea is a Borel subset of H1 for each X in the positive cone of S . Furthermore, every Borel subset of ac E is a Borel subset of H . O l 1.2.7 Definition. A linear Operator from E: into Ed is an a-operator if the matri: of the Operator, [tij], is symmetric, non-negative definite with .2 a(tii) < m. The proof of the follOwing lemma is included in ([12], p. 42). 1.2.8 Lemma. Let T be an infinite dimensional matrix {tij} such that T is symmetric, non-negative-definite and * E a(tii) < m. Then T is an a-Operator on E into Ea. a 14 We know (Lemma 1.2.6) that Ea is a Borel subset of H and the H-Hr-topology is stronger than H-Hx-topology on Ea. * * Hence it follows that HA is a subset of Ea. We now identify * H1 by HA and state the following lemma whose proof is in ([12], p. 43). * 1.2.9 Lemma. Every O-operator T on E is a trace class a Operator on H1. QMNERII STABLE PROBABILITY MEASURES ON BANACH SPACES 2.0 Introduction In this chapter, we consider stable probability measures (laws) on a real separable Banach space. Using a generalization of the convergence types theorem ([3], p. 174) we establish several characterizations of stable probability measures and deduce as corollaries extensions to Banach space of known results on stable laws ([3], p. 199, [14], p. 64, [21], p. 327). These results allow us to identify stable probability measures on Banach space as the limit laws of certain normed sums of independent, identically dis- tributed Banach space valued random variables. Finally, we char- acterize stable probability measures on certain Orlicz Spaces in terms of their Lévy-Khinchine representation given in [16]. In Section 2.1, following the preliminaries, the convergence type theorem is established. Section 2.2 presents the characteriza- tions of stable laws and final section characterizes the Lévy- Khinchine representation of the stable laws on Orlicz spaces. The lemmas in Section 2.2 are suggested by some recent work of Jajte [14]. The proofs in [14] in the Hilbert case treated there are incomplete and the main theorem in ([14], p. 64) which iS extended here to certain Orlicz Spaces, contains a lacuna ([14], p. 70). 15 16 2.1 Preliminary Results for Stable Laws In this section we present basic results needed in this chapter. Throughout this chapter E will denote a real separable Banach Space with norm H'H' 2.1.1 Lemma. Let [an] and u be probability measures on 5(1‘3) and [an], aER. Then un=u and an-oa implies Tanun = Tap. Proof of the lemma is immediate from ([1], p. 34). Before we prove the main theorem of this section, we need the following lemma. 2.1.2 Lemma. Let fi(-) be the ch.f. of a probability measure on ETE) such that for some 6 > 0, ‘fi(y)] = 1 whenever “YHE* S 5- Then u = 6X for some x E E. Proof:1) Let A = {y E E*: “y“ * s 6]. Consider the random variable (-,y) defined on E for each fixed y in E*. Then by ([21], p. 202), (-,y) is degenerate say at 9(y). Hence fi(y) = eie(y). Let Ay = [x: (x,y) = 9(y)]. Then Ay is closed and ”(Ay) = l, for every y in E*. Consequently, the support cu of u (see [23], p. 27) is contained in H A . Suppose there exist two points x1 and x2 in Cu. Then (x1,y) = (x2,y) for every y E E*. Hence x1 = x2. Thus the support of p contains only one point. This completes the proof of the lemma. ‘ 2.1.3 Theorem (Convergence of types theorem). Let [an] and p be probability measures on BTE) Such that ”n = u, and 1)I thank Professor J.F. Hannan for pointing out this proof. l7 and for positive constants an's and a sequence {xn] in E T u * 5 n u' where p and a' are non-degenerate probability a n x n n measures on I303). Then there exist an a 6 R and an x E E such that p' = Tau * fix, an a a and Hxn - x“ a 0 as n a m Proof: Suppose lim an = m. Then there exists a sub- -1 sequence [m] C [n] such that am a m. Let cm = 3m . Then “m = {Tc (Ta pm * 5x )} * 5_C x 2 u- Since Ta “m * 5x = p', m m m mm m m therefore by Lemma 2.1.1 T (T u * 6 ) = 6 . Hence by Theorem c a m x 0 m m m 1.1.3(a), {9-c x } being compact converges to 5 for some m m x 0 x0 belonging to E by ([1], p. 37). Hence a is degenerate, contradicting the hypothesis. Hence, lim an < m. Suppose now {am} and {aL} are two subsequences of {an}, such that am‘d a, 8L a a', where a f a'. We note that neither a nor a' can be zero, since u' is non-degenerate. Also, we have = * * M {T (Ta u'm 6x )} 6-x C m m [Tl mm H = {T (T u * 6 >1 * 6- 2 u - Ct at L XL XLCL Now by Lemma 2.1.1 and the hypothesis we get = ' * = T ' * P. T U! 5x1 CIIJ' 6X2 where x = lim - x c and x = lim - x c , 1 m m ’ 2 (“in Law 6 L Therefore (2.1.4) \fi'(ay)\ = \fi'(a'y)l for every y E E*. 18 a Without loss of generality we can assume b = ‘1'< 1. Hence, by m iteration ‘fi'(y)\ = lfi'(by)\ =...= ‘fi'(bnu)‘. Letting n a m * we get \fi'(y)‘ = l, for every y E E . Hence by Lemma 2.1.2 u' is degenerate, contradicting the hypothesis. This proves that an a a and O < a < m. Now it follows that Ta ”n = T u by Lemma 2.1.1, and from a n hypothesis Ta un * 6x = H'° Therefore, by Theorem 1.1.3(a) n n [5x } is compact and hence by ([1], p. 37) xn converges to some n - * * x in E. Hence, Ta ”n 6x = Tau 5X. Thus p n n which completes the proof. = * Tau 6x, 2.2 Stable Probability Measure on a Banach Space We define a stable probability measure on a real separable Banach Space following Loéve ([21], p. 326). (See also ([14], p. 64).) 2.2.1 Definition. Let u be a probability measure on the Borel subsets 61E) of a real separable Banach space B. We say that u is a stable probability measure if for each pair of posi- tive real numbers a and b there exist a positive real number c and an x E E, such that * = * . (2.2.2) Tau pr Tcu 6x Our main effort in this section will be to prove various characterizations of the stable probability measures which will be useful in studying stable probability measures as limit laws of the sums of independent random variables. For this we need the follow- ing lemmas. 19 2.2.3 Lemma. If u is a stable probability measure on STE), then there exists a sequence {an} of positive numbers and * a sequence {xn} of elements of E such that Ta u n * 5 = a. X n n Proof: We shall prove the lemma by showing that for each * n, there exist a and x such that a = 5 * T u n. n n xn an For n = 1, take x1 = 0, and a1 = 1. Suppose that we have x1,...,xm_1 and a1,a2,...,amu1 such that *i U; =6X *Tapa for i. =1,2,...,m-1 o i i m-l Then u ’ T _1 (p * 5_ ) Hence, a Xm-l m-l *m u = T -1 u * 5 _1 * u - am-l -xm-lam-l Now we use the fact that u is stable to conclude that u = TC u* 6 _1 for some cm> 0 and x E E. m X am-le-l Consequently, *m u ’ T ~1” * 5 -1( -1 X) Cm Cm am-lxm-l -l -l -l m cm (am_1xm_1-x).‘Thus we have Shown by induc- Define, am = cm , x * tion that p = 6x * Ta a m for every m. This completes the proof m m of the lemma. 2.2.4 Lemma. If for some sequence of positive real numbers {an} and a sequence {Xn} of elements of the space E, we have v = lim (5 * T ”*n) , new Xn an a where v is non-degenerate, then an a 0, a a l as n a m. n+1 20 Proof: Suppose an-tL 0. Then there exists a subsequence {am} of {an} such that a;1 a a < m. Therefore by Lemma 2.1.1 * * 5 *um=T (6 *Tam)_.Tv y ‘1 x a a m am m m x where ym = gfl'. This implies ‘fi(h)\m « [6(ah)‘ for every m * h 6 E . Since v is continuous at the origin, therefore |%(ah)\ > o for those h with Uh“ < a for some 5 > 0. Hence |fi(h)‘ = l on “h“ < 6 and hence by Lemma 2.1.2, H is degenerate. Consequently, v is degenerate which contradicts the assumption and hence an a 0. a Suppose a n ,L. 1. Then there exists a subsequence [m] n+1 a of [n] such that m a a where a # 1. am+1 a If a = m. Then c = *mil-4 0, and m a m * * 6x * Ta u m = {T <6, * T u ”)1 * 6 _C x . m+1 m+1 Cm m 8m Xm-l-l m m \“_m 1(X 9h) (2.2.5) 5x * Ta u (y) = —' 901) m+1 m+1 Nauru?!) i(xm*1,h) m+l because for every h, fi(am+1h) H l and e (fi(amdlh)) a 9(h). * Since Tc (6x * Ta u m) a 50 by Lemma 2.1.1, we conclude that m m m * ‘9(h)‘ = l for every h E E . Hence by Lemma 2.1.2, v is de- generate which contradicts the assumption. 21 Now suppose a < m. Then dm = "mb'a a, and *m+1 m+1 6 * T p a = [Td (5X * Ta H )} * 6x -d x m m m m+1 m+1 m m m+1 i(x ,h) /*\“T+1 e m * [fi(a h)]m * _ m x 6 T u- (y) - -* v01) Xm am filamh) by the reasoning similar to one following (2.2.5). But * Td (6x * Ta u m+1) a T v, hence [9(h)‘ = ‘9(ah)| for every m m+1 m1 3 h E E . Without loss of generality we can assume a < 1. Hence by the same argument as in (2.1.4) we conclude that v is degen- a erate which contradicts the hypothesis. Hence a l. a n+1 2.2.6 Lemma. Let for every positive integer n, xn E E, + an E R , and where p. and v are probability measures on [3(E). Then there exists an r > 0 and a function z of two variables defined for every pair of non-negative numbers a and b with values in E, such that r 1/r ' * e1(z(a,b),h)%((ar + b ) h) for every h E E . $(ah)%(bh) = Hence in particular, v is stable. Proof: If v is degenerate, then there is nothing to prove. So assume v is non-degenerate. Now by Lemma 2.2.4, for any arbitrary pair of positive numbers a and b, there exist subsequences {ank] and [amk] of [an] such that 22 a n = __k. .2 ~ wk a e a (Loeve [21], p. 323) . mk a ___. nk Suppose lim ;——————— = s = m. Then the sequence [ck] + “k "‘k ank + where ck ='——::———‘, will have a subsequence {Ck'} such that n k "'.00 Ck (2.2.7) 6 * T u = {T (6 * T u )} * Xnk' ' ank .4111. Ck. Xk' ank' .. *mk' * {Tck'°wkséxm Ta 6 )} 62k. k' mk' for suitable z By the hypothesis we conclude that k" 5x * Ta nk.+m]u “kl+nk' of (2.2.7) converge weakly to 60. Hence, \%(y)\ = l for every * y E E . Thus by Lemma 2.1.2, v is degenerate, which is a con- * +n I I u mk k a v. The terms in the parenthesis tradiction. Hence 5 < w. . -l -1 Suppose there ex1sts two subsequences [ck,], [ck"} Of [cLl] converging to b' and b" reSpectively where b' # b". Making use of the following equality * 7!: (2.2.8) [13(6x Ta 6 )3 * [Tam (6x Ta 6 )1 “k “k k mk mk *n *n = T u k+mk * 5 = [T -1(6 * T H k+mk)]* 6 aan zk ack x +m a z , k “k k nk+mk k for suitable 2k and 2; we conclude that 23 \6(ah)‘.\6(bh)\ = |6(xb'h)\ = [9(xb"h)‘ for every h e E* . Since b' # b" and both are finite, we can conclude by the same reasoning as in (2.1.4), that v is degenerate. This contradicts an the assumption. Hence, 5 = lim “Er—L— . “k + mk Now we make use of equation (2.2.8) and Theorem 1.1.3 to conclude that there exists a function z(a,b) which is the limit a a n n of zfi = a x + a -—h'x - a -————E——'- x and satisfies n a a + n mk nk+mk mk k the equation ' * $(ah)$(bh) = e1(z(a’b)’h)%(ash) for every h in E . Define a function g(.,.) on [0,m) x [0,m) as follows g(X.y) = X~s. X.y > 0; g(X.0) = X. X 2 0; g(O.y) = y. y 2 0; then the equality (2.2.9) D(xh)%(yh) = ei(z(x’y)’h)v(g(x,y).h) holds for all h in E* 9 and for all x,y 2 0. We shall prove in this part that g is the only function which satisfies (2.2.9). Suppose not. Then there exist g1 and g2 satisfying (2.2.9) and for some x0 and yo, g1(x0,y0) < g2(x0,y0). Let - 31(X0 ayo) — —-?—-———7 Then u < 1. g2 x0’yo U ei(z(x0,yo),h/g2(xo.y0)) i(Z(x0.y0)h/32(X0.yo)) 9(uh) = e 6(h). Thus ‘%(uh)‘ = ]%(h)\. Hence the same argument as in (2.1.4) 24 yields that v is degenerate. This contradicts the assumption. Hence the uniqueness of g has been proved. The function g so defined is continuous. Let xn a x; yn d y. Then we shall prove that t = II; g(xn,¥p is finite. Suppose not, then there exists a subsequence [n'] of {n} such that g-l(xn,,yn,) d 0. Therefore en, = xn,g-1(xn,,yn,) and -1 kn. = yn,g (xn.,yn.) a 0. Hence from (2.2.9) we get [0(en..h)‘-\$(kn,.h)] = \§(h)\ for every h in E* By letting n a m, in view Of Lemma 2.1.2 we get v is degenerate. Thus t < m. To conclude that t = lim g(xn,yn), we shall Show that no two distinct subsequences of g(xn,yn) can converge to two dif- ferent limits. If not, let t' = lim g(xn.,yn,) and t" = lim g(xn",yn”), where t' f t". Consequently, from (2.2.9) we get \§(t'h)‘ = \9(t"h)| for every h in E*. Since t' f t", therefore from the same reasoning as in (2.1.4) we conclude that v is degenerate. Hence, t = lim g(xn,yn). Now we make use of (2.2.9) again to conclude that ' * Q(xh).§(yh) = e1(z(x’y)’h)%(t,h) for every h in E . Since g is unique, therefore t = g(x,y). Thus g is continuous. It can be verified that the function g satisfies the hypothesis of Theorem 4.1 Of ([2], p. 632). Hence by ([2], p. 632) g(x,y) = (xr + yr)l/r for some 0 < r < w, and D(ah)§(bh) = ei(z(a’b)’h)0((ar + br)1/rh) for every h 6 E* 25 where a,b are positive real numbers. This completes the proof. 2.2.10 Theorem (Characterizations of stable probability measures). Let E be a real separable Banach space and u be a probability measure on 51E). Then the following are equivalent. (a) u is stable. (b) There exists a sequence an of positive real numbers *n * and [xn} C E such that 6X Ta u = a. n n (c) For each integer n, there exists a yn E E and *n Cn > 0 such that p = 6 * T p. C yn n Proof: The equivalence of (a) and (b) follows from Lemma 2.2.3 and Lemma 2.2.6. We note that for each n, (c) implies, ( *n * *n = = * o p T -1 u O-y ) T _1U O _1 c n c -c y n n n n Hence (c) implies (b). Also if u is degenerate clearly (b) implies (c). Assume that (b) holds and a is not degenerate. Then for every k = 1,2,... 6 * T *nk “ = where = 6 * T *n x a H — u'nk H H'n x a H . nk nk n n Hence, * * (a *Tun)*---*(6 *Tun) x a x a * 43 n n n = O * T “k k factors k.x a H n n = * (T n unk) 6 an ._—— kx —-——-x ank n ank nk 26 a a where d =';fl- and c = kx - -fl-x k° “k nk “k “ ank “ *k Then, Td unk * 62 = u . Since u is not degenerate, therefore “k “k by Theorem 2.1.3 we conclude that d ~ 6 , z * Z 6 E, as n r m, n k n *k k k and p = T * 6 . Thus (b) = (c) which completes the proof d” z k of the theorem. Let n a m above. 2.2.11 Corollary (Proposition 9.25 of ([3], p. 199)). Let X1,X2,... be identically distributed, non-degenerate, independent, random variables taking values in E. Then a = lim “S where x1+...+x “ n _ n . . Sn An yn, for some sequence An of p031t1ve real numbers and {yn} CZE iff for each non-negative integer n, there * exists a cn > 0 and 2n 6 E such that u n = 5 * TC p. Z n 11 The proof follows from the equivalence of (b) and (c) in Theorem 2.2.10. In particular this Shows that limit laws of the normed sums given in (b) of Theorem 2.2.10 are infinitely divisible. 2.2.12 Corollary ([21], p. 327). Class of stable prob- ability measures on. 51E) coincides with the limit laws of normed sums of independent and identically distributed random variables taking values in E. The proof follows from the equivalence of (a) and (b). The following corollary is now obvious. 2.2.13 Corollary ([14], p. 64). Every stable law on a real separable Banach space is infinitely divisible. Corollaries 2.2.11, 2.2.12, and 2.2.13 relate stable prob- ability measures to a certain subclass of infinitely divisible measures. Recently, J. Kuelbs and V. Mandrekar [16] have obtained Lévy-Khinchine representation for infinitely divisible 27 measures on certain Orlicz spaces extending the work of S.R.S. Vardhan ([31], p. 227) on Hilbert Space. In the next section we obtain a characterization of stable probability measures as a subclass of of these infinitely divisible measures in terms of the Levy- Khinchine representation of their ch.f.‘s. This result will gen- eralize the recent work of Jajte ([14], p. 64) to these Orlicz spaces. 2.3 Lévy-Khinchine Representation of Stable Measures on Certain Orlicz Spaces The Lévy-Khinchine representation for the characteristic functional of a stable probability measure on Hilbert spaces has been studied by Jajte in [14]. In this section we Shall obtain similar representation for stable probability measures on certain Orlicz spaces. We remark that the proof of the main theorem in [14] is incomplete and contains a locuna which can be corrected (Cf. Lemma 2.3.6) . We recall for easy reference some notation and results on Orlicz Spaces [16] from Chapter I. The function a used in this section will have the following properties. r(a) a is defined on [0,m) into [0,m), (b) a(0) = 0, 0(8) > 0 for s > 0, (c) a is convex and strictly increasing on [0,m) (2.3.1) (d) 0(23) 5 Ma(s) for all S E [0,m), where M is a A finite positive constant independent of s, (e) I a(uz)dv(U) S ca [J u2du(u)] for all Gaussian measures v on (-m,m) with mean zero, where c is a constant. 28 2.3.2 Definition. The space of real sequence so 2 x = (x ,x ,x ,...) satisfying 2 g(x.) < m is denoted by E . l 2 3 1‘1 1 a - 2 The Orlicz space SF given by F(t) = g(t ), t E [0,m), is isomorphically isometric to Ed. Throughout this section we use this identification for Ea, [16]. Let ac be the function complementary to a in the sense of Young ([32], p. 77) and SO be the Orlicz space corresponding c to ac ([32], p. 79). Then for each A in the positive cone of 8 (except when E = L2), whose norm is less than or equal to QC 0 , 2 m 2 , 2 one half, def1ne; HXH = Z 1.x, and if E = L , then “x“ = 2 A 1:11]- CY 2 A Z Xi' The space Of sequences with property that “x“x < w, will be denoted by HA. Obviously Ea C H1 by Young's inequality ([32], p. 77). In fact, H1 is a Hilbert Space containing Ea as its measurable subset [16]. 2.3.3 Theorem. Let H be a probability measure on the Orlicz space Ea’ where 0 satisfies (2.3.1). Then u is stable on Ea iff either A . l *- (2.3.4) ”(y) = exp[1(x0,y) - §'(Ty,y)] for all y E E0 , where x0 6 E and T is an a-operator (i.e. u is a Gaussian a measure) OR (2.3.5) fi(y) = exp[i(x0,h) + g(ei(x,y) - l - iié‘xg9dF(x) + Huxu, ]‘ (Sm-V) - 1 - M)dF(x)] -u 0' where x0 6 EO’ “X“? is the norm of x in SF’ U = Q 2 [x E E : 2 g(xi) s 1], F is a c-finite measure on Ea’ finite on i=1 29 on the complement of every neighborhood of zero in Ed and such so that Z a(]xidF(x)) < m, and there exists a r (0 < r < 2) such i=1 U that TaF = arF for every positive real a (Stable probability measure of index r). Proof: Let (2.3.4) hold. Then . x , 1 2 2 u(ay)-u(by) = eXp[1(XO.y)(a + b) -'§CTy.y)(a + b )] for every a and b positive and therefore A A A 2 2 % , 2 2 % u(ay)-u(by) = u((a + b ) y) exP[1(x0.y)((a + b) - (a + b ) )1 n b ([13 37) T * T = T * 6 wh = ( 2 + b2)% ence y ], p. , a” b“ Cu x ere c a k and x = ((a + b) — (a2 + b2) )x0 E E . Consequently, u is stable. 0 If (2.3.5) holds, then by ([16], p. 71), u is infinitely divisible on Ea. Hence, there exists a sequence of finite measure Fn on E such that F 1 F and a sequence {x ] CIE , such that a n n a * I e(Fn) 6x = u on Ea. We can regard Fn s and u as measures n on HA (see Lemma 1.2.6). Since every bounded and continuous function on HA is also bounded and continuous when restricted to Ed by [16], we conclude that e(Fn) * 6x = u on HA. Hence by ([31], p. 224) n ] (ei(x,y) - 1 - ijz‘l%)dF(x)] for Hx l+\\x\\x every y E H*, where x E H , “x“zdF(x) < m and F is the we get fi(y) = eXp[i(X1:Y) + o-finite measure as before. Since TaF = a on H , therefore 1 by ([14], p. 64) a is Stable on HA. Consequently, for every a,b > 0 there exists a c > 0 and 2 € H1 such that T * T = T * . To rove is stable on E it would be an bu Cu 62 p u a’ 30 enough to show that z 6 EU. Denote z = (21,22,23,...). . _ '1 _ Define ”n - uPn where Pn(x) — (x1,x2,...,xn,0,0,0,...) on Ea' Obviously ”n a H on Ed by argument Similar to one . .22 . ' ° * = * in ([17], p 1) Now it is easy to see that Tau“ T Tcun 67 n bun where Tn = (zl,zz,...,zn,0,0,0,...). We note that Tn E EU for every n and ”n = u implies for any real d, Tdun = Tdu. Hence by Theorem 1.1.3(a), [Tn] is compact on E . Consequently, a a 0 by Tn * 20 E E, by ([1], p. 37). Hence “Tn _ [l6] , therefore 2 = 2 Hence, for all a and b > 0 there 0° 't > rE.lthatT*T =T* . ex1s s a c 0 and suc1 a” bu Cu 62 2O 1 a 0 This completes the proof of sufficiency. Suppose u is stable on Ea. Then by Corollary 2.2.13, p is infinitely divisible on Ea. Consequently, by [16], v * B where B is the Gaussian part of M on E and u a v lim e(Fn) * 6 where Fn's are increasing sequence of finite x n-m [1 measures on Ea and xn E E for all n. We can regard F 's, a n u, v and B as measures on H [16]. Since an a-operator A on Ea is also a trace class Operator on HA (see Lemma 1.2.9), therefore by ([31], p. 226) B is Gaussian on HA. Thus p = v * 6 where B is Gaussian on H , and v = lim e(Fn) * 6 on H . k ham xn A Since u is stable on HA, therefore by ([14], p. 64), u = B or u = v where $(y) = exp[i(z,y) + g (ei(x,y) - 1 - i£§4x%)dF(x)] , Huxn, * for all y 6 H)’ and where 2 C H1, F = lim F“, 2 “am r quxdF(x) < m and for some 0 < r < 2, Tap = a F for all positive a. 31 Since v = lim e(Fn) * 6 on E , the result follows from n4» xn a ([16], p. 66). This completes the proof of the theorem. To correct the proof Of the theorem in ([14], p. 64) we need the following lemma. 2.3.6 Lemma. Let H be a Hilbert Space and F a o-finite measure on H satisfying for every a and b positive (2.3.7) T F + T F = T a b x+bA F . (a 1/1 ) Then F satisfies the equation TaF = axF for every positive a. Proof: Since F is o-finite, therefore it is enough to prove the above result for finite measure. So assume without loss of generality that F is a finite measure. Let B E ETH) such that 5(8), the boundary of B, has F measure zero. Then TaF(B) is a continuous function on (0,m), by Lemma 2.1.1 and ([23], p. 40). From (2.3.7) we get F(a-lB) + F(b-IB) = F((ax + bx)-1/xB) for all a and b > 0. Since the above is true for all a and b > 0, therefore, we get F(a-l/XB) + F(b-l/XB) = F((a+b)-1/)‘B) Let F(a-I/KB) = g(a). Then g is continuous on (O,w) and g(a) + g(b) = g(a+b), for all a and b > 0. Therefore g(a) = c.a for a > 0, where c is a constant depending on B. Hence g(ax) = axc. Thus F(a-lB) = axe. Let a = 1. Then c = F(B). Hence F(a-IB) = akF(B). Since the class {B: B 616(H), F(BB) = 0] 32 is a field by ([1], p. 16), therefore by Caratheodory extension theorem F(a-IB) = axF(B) for every B E 51H). CHAPTER III SELF-DECOMPOSIBLE PROBABILITY NEASURES ON BANACH SPACES 3.0 Introduction In this chapter, we consider self-decomposible probability measures (laws) on a real separable Banach space. In Section 3-1: a necessary and sufficient condition for a self-decomposible law to be stable has been obtained in terms of its component. Section 3-2 presents a characterization of a self-decomposible law Similar to ([21], p. 323). This result allows us to identify self-decom- posible laws as the limit laws of certain normed sums of inde- pendent, uniformly infinitesimal Banach Space valued random vari- ables. We also show that the self-decomposible laws are subclass of infinitely divisible laws. This result is interesting on gen- eral separable Banach Spaces Since it is not known whether limit laws of uniformly infinitesimal triangular arrays of the Banach space valued random variables are infinitely divisible (see [20]). So in particular, from this result we can say that limit laws of certain subclass of triangular arrays are infinitely divisible. Finally, we characterize self—decomposible laws on certain Orlicz spaces in terms of their Lévy-Khinchine representation given in [16]. 33 34 3.1 Preliminary Results on Self-Decomposible Laws In this section we establish some preliminary results used in this chapter. We start with the following proposition. 3.1.1 Proposition. Let u be a probability measure on STE). If there exists a number c > O and a non-degenerate prob- ability measure ”e such that u = Tc“ * ”C, then c < 1. Proof: Since (1C is non-degenerate on 803), we have ([13], p. 37), c # 1. Now suppose c > 1. Then * fi(y) = fi 0. Also note that if Hy” < a, then g(y) # 0. There exists a sequence [yn} of elements of SO such that Hynu a a. Hence the sequence [yn} is norm bounded. There- 2 fore byAlaoglu's Theorem ) ([29], p. 202) {yn : n = 1,2,...] is * * weak compact. Consequently, there exists a y0 E E such that * for some subsequence [ym] of {yn], ym a yO in weak sense. Let 2x0 = yo. Then by the dominated convergence theorem fi(ym) 4 ”(y0)' Hence fi(2x0) = 0. Since HchOH < “2x0“ 5 a, therefore fi(c2x0) # 0, for every 0 < c < l, and hence by (3.1.3) fic(2x0) = 0 for every 0 < c < 1. Therefore by ([13], p. 37), A 2 . (3.1.5) \pC(x0)\ s 2‘1 - ”C(x0)\ . 110:) Since fic(x0) = EIEEET a l as c a 1, therefore by (3.1.5), * l s 0, which is a contradiction. Thus fi(y) # 0 for any y E E . Remark A. From the above proof it follows that if a is * any probability measure on 513) and there exists a z E E such that fi(z) = 0, then there exists a y0 such that fi(y0) = 0 and “yo“ = $2§OHYH- 2) I thank Professor P.K. Pathak for suggesting the use of Alaoglu's theorem in this context. EFL 36 3.1.6 PrOposition. Let p be a non-degenerate probability measure on 6(5). Then p. is self-decomposible with its component ”c for 0 < c < 1, given by 6_ * T x , where x E E and (1_c1)1/1 0 < A S 2, iff u is stable. Proof: Let a and b be two positive real numbers. Then by letting c = a X 1/ , we get (a)\ + b ) A U=Ta ”*Tb ”*5-X (a*+bx)1/* (511+thx Consequently, Tu*Tp.=T [1*6 . a b (ax + bk)1/A (ax + bk)1/kx Hence a is stable by (2.2.2). On the other hand if a is stable, then for every pair of positive numbers a and b, there exists an x E E by Lemma 2.2.6 and Theorem 2.2.10, such that (3.1.7) T a * T U = T a * 6 for some 0 < X S 2 . a b (81 + b1)1/1 x _ 1 1/1 Take b - (1 - c ) , where 0 < c < 1. Then from (3.1.7) we get T u * T u = u * 6 - c (1 _ cl)l/x x Hence, c x (1 _ C1,)1/1, This completes the proof of the proposition. 37 3.2 Self-Decomposible Laws and Limit Laws In this section we Show that the class of self-decomposible measures coincides with certain limit laws of sums of independent Banach space valued random variables. Let us denote by WCE) the class of probability measures p on ENE) with the property that there exist sequences {xn} CLE and {bn} C1R+, and a sequence [an] of probability measures on 18(E) such that sup sup ‘fik(ybn) - 1‘ a 0 as n a w for every compact set k=1,ooo,n YES * * 'k SCE,and 6X [I prkap. n k=l n 3.2.1 Proposition. If u is a non-degenerate probability b measure on BCE), and p. E WCE), then b -+ O, n .. 1. n bn+l Proof: If bush 0, then there exists a subsequence [n'] of [n] such that b;} a b, b finite. By the fact that u E NCE), , * we get by letting yn, = %—— for y E E I n fik(y) = fik(yn'°bn') a l as n a m for each k. Thus uk is degenerate for each k. Hence, u is degenerate which is a contradiction. Thus bn a O. b Suppose now bn -#vl. Then there exists subsequences [n'] b n+1 I of [n] such that n a b where b # l. b I n +1 I If b = m, then c , = -E;tl a 0, and n b , n n' n' (3.2.2) 6 * n *T u =T (6 * II *T u)* xn'+l =1 bn'+l k Cn' Xn' =1 bn' k 6x c ,x ° 38 1( y)!“+1 Since, Ox * H *rb Ulk(Y) = " (b y) '7' 1 9 n'+l k=1 n'+1 LLn'+1 n'+1’ and Cn' a 0, therefore from (3.2.2) we get [C(y)‘ = l for * every y E E . Hence, by Lemma 2.1.2 H is degenerate, which con- tradicts the assumption. Thus b is finite. Let C“, = :n: . Then en, a b and n +1 n'+l n'+l (3.2.3) 6 n. * kgl * Tbn'uk = Tcn'(6xn'+1 * kg1 Tbn'+1uk) * x .-c ,x ,+1 ° . * n'+1 i(xn,,y) n' A A S1nce 6Xn' k21*Tbn'uk(Y)‘= ‘0 k21 uk(bn..y)\~|unu+1(bn..y)l r \fi(y)"1 9 because Tbn'un.+1 = Tbn' (Tbn'+1un,+1) a 60 by Lemma 2.1.1, bn'+l and Cn' e b, we conclude by (3.2.3), that \fi(by)] = \fi(y)\ for every y E E* Since b # 1 and is finite, therefore a is degenerate by Lemma b n bn+1 2.1.2, which contradicts the assumption. Hence, a 1. This completes the proof of the proposition. 3.2.4 Theorem. If H E W(E), then fi(y) # 0 for any y E E n o = * 91' Proof. Let v 6 H Tb uk. Then n k—l n b y b r i(x ,L) m b i(x ,y>-i(x ,—“y) n x _ m b . ,JQ n m b . vn(y) - [e m U “k(b ~y bm)}°[e m H uk(y bn)} k'l m k=m+l (3.2.5) < Tbn = —-— * vn b vm km , L m n where to every integer n there corresponds an integer m < n b such that gfl-a c E (0,1), m, n-m a m as n a m ([21], p. 323), m and km is the probability measure corresponding to the term in n the second bracket on the right hand Side of (3.2.5). * Suppose there exists a z E E such that fi(z) = 0, then * E E such that fi(y0) = 0, and by Remark A, there exists a yO * “yo“ = inf Hy“, where S0 = [y E E : fi(y) = 0]. Hence from (3.2.5) y€S0 we conclude that an(y0) a 0. Since vn(y0) a O, Tbn um a Ten E—' Y m (Lemma 2.1.1) and HC°§QH < “you, therefore from (3.2.5) y fi O , n k=l, . . . ,n yES HyHSt and hence, sup sup ]§k(fi) - l‘ a 0 . k=l,2,...,n yES Thus u E W(E). Conversely, suppose u E W(E). Then there exist two sequences [xn] C E and {bn} C R+, and a sequence [pk] of n probability measures such that v = 6 * H * T u = u and n xn k=l bn k 41 [Tb pk, k = 1,2,...,n] is a uniformly infinitesimal sequence. n b Hence, by Proposition 3.2.1 bn a 0, gn——'« 1. Consequently, n+1 given a c (0 < c < l), we can correspond to every integer n b . n an 1nteger m < n such that B—'~ c and m, n-m w m as n a m 3 ([21], p. 323). Note that where Am is the probability measure given by n n 6 * H * T u . Xn bnxm k=nrtl bn k Since Vn a u, therefore by Lemma 2.1.1, Tb n —_ :T b vm cu ’ m and hence by Theorem 1.1.3, [Km ] is compact. In view of Theorem A n 3.2.4, X (y) « %£Xl-, and hence by Theorem 1.1.3(c), mn u(Cy) (3.2.63) A 3 U' 9 where uc is given by * A “C(y) = fi(y)/fi(cy), for every y E E Thus u = Tcu * uc. This completes the proof of the theorem. 3.2.7 Theorem. Let H be a self-dceomposible probability measure on a real separable Banach E. Then u and for each (0 < c < l),the associated measure ”c are (see (3.1.3)) infinitely divisible. 42 Proof: Let 0 < c < 1. From (3.1.3) it follows by itera- tion that for each n, = * * ‘k * * u p TC” T ZMC ... T n-IHC T nu C C H y * H F n = * * k...* . ° H where A , u T p T 2” T 1“ Since C 0 as n a m, it follows from Lemma 2.1.1, T . nu 2 60 c Consequently by Theorem 1.1.3(a) and (c), (3.2.8) In“: =9 p, as n -+ co . Let m be a positive integer. Let = * * * * vn,c u'c Tcmuc Tc2mpc '°° TC(n-l)mPc ° Then * * * * (3.2.9) Vn c T v T 2Vn,c ... TCm-lvn,c = * T *...* T u'c cuc Cn m-luc ’ and the right hand side in (3.2.9) converges weakly to u as n a m by (3.2.8). Consequently by Theorem 1.1.3(b) Vn C is 9 shift compact. Therefore there exists a sequence xn c in E ’ such that vn c * 6 is compact. Hence there exists a sub- (3) n,c * . ] of [n] such that VL,c 6X =9 )(m’c 1n W1 L.C sequence [L as L(c) a m. From (3.2.9) it follows that * (3.2.10) (v 6X ) * Tc(vL c * 6 ) *...* L c x ’ Lac , Lac T (v * 6 ) * 6 CID-1 L,C XL,C -C -1 .x c-l L,c converges weakly to n as‘L(Cx»an Since vL c * 6x converges 3 L:C weakly, therefore 6 m is compact. Consequently for C -1 c-l XL,C m some in E - C -1 x converges to in norm Hence ym,c ’ c-l L,c ‘ yc,m ' ° from (3.2.10) one gets, 3.2. * T * T *...* T * — . ( 11) xm,c cxm,c 2xm,c Cm-lxm,c 6ym C u 3 Let ck be a sequence in (0,1) converging to 1. Then from (3.2.11) 302. 2 * *ooo* * = o ( 1 ) xm,ck TCkxm’Ck Tcm_1xm,Ck bym C u for each R k ’ k Consequently, Km c is shift compact. Hence there exists a sub- 3 k sequence {p} of {k} and a sequence zm p in B such that ) * ' A . km,C 6z 2 Am in WI as p m P m,p Thus from (3.2.12) we get (3.2.13) (Km,c * 52 ) * TC (km * a ) *...* T m-1(*m,c * a ,c z 2 P m,p P P m,p Cp P m,p * 6 = for each . ym C -m.zm n u p , p , Since Km c * 62 converges weakly, therefore by Theorem ’ 1) mm 3.1.3(a) 6 -mz is compact. Hence y - mz con- m,cp map m,cp m:p verges in norm to some element ym in E. Thus from (3.2.13) 44 we get by letting p a m, Km * xm *...* Km * 6y = u . m Since m was an arbitrary positive integer, therefore p is infinitely divisible. To prove that ”c is infinitely divisible for each (0 < c < l),consider the Hilbert space H containing E such that every Borel subset of E is a Borel subset of H ([15], p. 355). We can regard we as a measure on H. Then uc is i.d. by Theorem 3.2.6, (3.2.6a) and ([23], p. 199). Hence for each m there exists a probability measure vm on H such that We will be done once we show that vm is concentrated on E. Since u is infinitely divisible on E, therefore there exists a probability measure km on E such that Since H is self-decomposible, therefore, 1 = T A * v on H Hence A A * [km(cy)]m * [vm(y)]m for every y 6 H , [it"(y) 1m [1m.;m(mm . 45 Therefore 2n n(y) i m * for every y E H , Xm(y) = Kn(cy)%m(y) e where n(y) is an integer valued function of y. Note that 2" Bill 1 (y) m m e = X (c )9 ( ) , since the denominator does not vanish m y my 2"“9221 m A (see Theorem 3.1.4). Therefore e is a continuous func- h7—5 tion of y taking only countably many values. Hence it must be 2“ n(y) i degenerate. Since at y = 0, e m = 1, therefore n(y) must take values which are multiples of m. Thus Xm(y) = im(cy)§m(y) for each y E H*. Consequently, km = Tcxm * vm on fl. Since km and Tcxm are concentrated on E, therefore vm is also concentrated on E. This completes the proof of the theorem. Remark. The above theorems generalize the classical re- sults about the self-decomposible laws (LOEVe [21], p. 323, Theorem 23.3A and corollary). We note that both these theorems are very easy to handle in the finite-dimensional case because of the powerful Lévy continuity theorem available. However in the case of the Banach space, no complete analogue of the Lévy con- tinuity theorem is available and hence the methods used here are combinations of the methods of characteristic functionals and functional analysis. To bring out the simplicity of the method of characteristic functional in the context of the availability of the full force of the Lévy continuity theorem we treat in 46 Chapter IV, semi-stable laws whose very definition depends on characteristic functional ([22], p. 92). In the next section we obtain results for self-decomposible laws analogous to results of §2.3 for stable laws. More Specifically we characterize self-decomposible laws in terms of their Levy- Khinchine representation. 3.3 Lévy-Khinchine Representation of Self-Decomposible Probability Measures on Certain Orlicz Spaces In view of Theorem 3.2.7, we know that H is infinitely divisible. Hence on Orlicz spaces it has a Lévy-Khinchine repre- sentation given by J. Kuelbs and V. Mandrekar [16]. In Loéve ([21], p. 324), self-decomposible laws are char- acterized in terms of the "Levy-measure". The purpose of this section is to obtain analogue of Theorem 23.33 of ([21], p. 324) for Orlicz spaces Ea discussed in §1.2. To prove this theorem, we shall first characterize self-decomposible probability measures on Hilbert spaces. The functions 0, ac, F, and spaces Ea’ SF’ H S , Sq , carry the same meaning as in §2.3. 3.3.1 Theorem. A functional m(-) is a ch.f. of a self- decomposible probability measure on a real separable Hilbert space H iff (3.3-2) cp(y) = eXP[i(x0.y) - ';'(Dy.y) + IK(X.y)dM(X)] * for every y E H , where x0 6 H, K(x,y) = e1(x’y) - 1 - l£§;1% , D is an S-operator, HHXH M is a o-finite measure on 51H), finite on the complement of 47 2 every neighborhood of zero in H, I “x“ dM(x) < m and for each HXH31 0 < c < l, M = TCM.+ MC, where MC is a measure on 61H). The representation (3.3.2) of m(y) is unique. Proof: Suppose @ is a ch.f. of a self-decomposible probability measure on EKH). Then for each c C (0,1), there exists a ch.f. me on H, such that (3.3 3) ¢ ~ ¢C(y) Now by Theorem 3.2.7, m and wc are infinitely divisible. Therefore by ([31], p. 227), ¢(y) = exp[i(x0,y) - %(Dy,y) + fK(x,y)dM(x)] for every y E H*, where x0 6 H, D is an S-operator, and M is a o-finite measure on iBCH), finite on the complement of every neighborhood of zero in H and such that “KszM(x) < m. Hence, x 51 m(cy) = exp[i(x0,cy) - %-c2(ny,y) + jK(x.cy)dM(x)], ei(cx,y) _ 1 _ igcx,y) + igcx,y) _ i(x,cy) , MW) z 1+ucxu2 1+ucxu2 Huxuz = cx i(cx,y)uxflz (1 - oz) K( ,y) + (1+HXH2)(1+CZHXH2) ince 53,3)flxfl2 (x S x 2 (X) < m, s , j (1+“X“2)(1+c2“xH2) dM ) HYH fix‘sl“ H + HY“ fixH>ldM therefore, y(cy) = 9XP[i(§.y) - %'02(Dy.y) + [K(X.y)dM(c'1x)] . 2 2 where (x,y) = (c x0,y) + I C(l-C%(X’Y)¥Xfl2 dM(x). Since, <1+uxu mm M > me is infinitely divisible, therefore by ([31], p. 227), 48 - . l, * cpc(y) - exp[1(x0.y) - 2 (Dewy) + [K(X.y)nd(X)] for every y E H . where xC E H, DC is an S-operator, MC is a o-finite measure on 51H), finite on the complement of every neighborhood of zero in H, and such that “X“sz (x) < m. x $1 c If we denote m(y) = [x0, D, M], then we have from (3.3.3) - 2 - 2 [xO,D,M] = [x,c D,TCM]-[xc,DC,MC] = [x + xc,c D + DC,TCM + MC] . By the uniqueness of the representation of infinitely divisible ch.f., we have M = T M +'M . C C To prove sufficiency, we make use of Theorem 4.10 of ([23], p. 181) to conclude that m is a ch.f. of an infinitely divisible probability measure p, on 501), and note that [x,D,M] = - 2 _ 2 - [x,c D,TCM].[xO-x,(1-c )D,MC],wmere x is same as before and 0 < c < 1. Now by the one-to-one correspondence between the proba- bility measures on 6TH) and their ch.f., we conclude that = ‘k u Tau uc : where u, ”c are the probability measures corresponding to [x0,D,M], and [x0 -.§, (1-c2)D,MC] respectively. Uniqueness follows from Theorem 5.10 of ([31], p. 227). This completes the proof of the theorem. 4 Remark B. The main result of Jajte in [14] can be obtained as a corollary of the above theorem in the following manner. 49 Let p be a stable probability measure on 13(H). Then by Proposition 3.1.6 H is self-decomposible and hence for each c E (0,1), M = T M . c (1_Cx)1/x Hence M = T M.+ T M. c (l-cx)1/x Now by the same argument as in Lemma 2.3.6, we get TcM = cxM for every c 6 (0,1) . To conclude the remark, note that TaM = axM for every a C (0,m) iff TaM = axM for every 3 ( (0,1). 3.3.4 Theorem. Let u be a probability measure on the Orlicz space Ea, where a is as before. Then H is self- decomposible iff (3.3.5) my) = exp[i(x ,y> - -1- + fem”) - 1 - Mme) 0 g 2 2 U 1+“ka +.J (ei(x,y) - 1 - il§41%)dM(x) for every y 6 E*: 0;” Huxur 6’ a: 2 where x G E , T is an a-operator, u = {x Q E : 2 g(x.) S 1}, O (y 0’ 1 1 and M is a o-finite measure on 18(Eu), finite on the complement 00 2 of every neighborhood of zero in Ea, 2 g(f xidM(x)) < w and for i=1 U each c C (0,1), M = TCM + MC, where MC is a measure on ETEQ). For each fixed k, the representation (3.3.5) of fi(y) is unique. Proof: Suppose H is self-decomposible on Ea. Then by Theorem 3.2.7, u is infinitely divisible. Consequently, by [16], u = v * e where B is the Gaussian part of p and e(Fn) * 5 a v where Fn's are increasing sequence of x n 50 finite measures on Ed and xn e E . Using Theorem 7.2 of a [16] , we get that fi(y) has the form given in (3.3.5) except M = TcM + MC. We can regard Fn's, v and B as measures on H 1 by [16] . Since an a-Operator on Ed is also a trace class operator on Hl’ therefore by ([31], p. 226), B is Gaussian on H . Thus u = v * e on Hk. Since every bounded and con- k tinuous function on HA is also bounded and continuous when restricted to Ed by [16] we get e(Fn) * 6x 2 v on n H . Since T u and H are concentrated on E , therefore i C C a * ' = * Ten ”c is concentrated on Ea’ and hence u Tcu pc on Hk. Therefore U is self-decomposible on HR. Consequently by Theorem 3.3.1, M = lim F and M 8 T M + M . n C c nam Conversely, suppose (3.3.5) holds. Then u is infinitely divisible on Ea by [16] . Hence there exists a sequence of finite measures Fn's on Ea such that Fn 1 M, and a sequence xn 6 Ed such that e(Fn) * 6x“ a v on Ed and u = v * B where B is Gaussian on Ea. By the same reasoning as in the proof of the necessary part of this theorem, we conclude e(Fn) * 6x a v on HX’ and B can be regarded as Gaussian on HX. Since n u = v * B on H) and hence by [16], u is infinitely divisible on HA. Thus by Theorem 3.3.1, u is self-decomposible on Hk. Hence for every c 6 (0,1), ._ * H . (1, TC”, pc , on k Since 0 and Ten give all its mass to Ea’ therefore ”c will also. Thus u is self-decomposible on Ea. The uniqueness follows from Theorem 7.2 of [16] . CHAPTER IV SEMI-STABLE LAWS ON SEPARABLE HILBERT SPACES 4.0 Introduction The main purpose of this chapter is to study semi-stable laws which arose classically only through the method of characteristic functions (Levy [22], p. 95). Thus the results of this section are valid when complete analytic analogues of Lévy continuity theorem and Bochner theorem are available. In case of Hilbert space, L. Cross [9] has established such results. We can therefore use his results to study semi-stable laws on a separable Hilbert space. Let p be a probability measure on a real separable Hilbert space. Then its characteristic functional fl is T-continuous ([9], p. 7) and hence in view of Remark 1.1.15 it has an extension H". We note that if fl is T-continuous and a is any positive real number, then (fia) is T-continuous. Hence (fla)~ is well defined as an extension. But in view of the definition of extension explained in Remark 1.1.15, we obtain ({1")8 = (fia)". This observa- tion will be used in Theorem 4.2.6. For the sake of completeness we recall here the Lévy continuity theorem of Gross ([9], p. 8). Theorem. Let p“ be a sequence of probability measures on a real separable Hilbert Space H with reSpective characteristic functionals qh' Let m be a uniformly T-continuous functional on H such that m(0) = 1. If ”n converges weakly to a measure 51 52 u whose characteristic functional is m, then wn converges to m on H and q; converges to m” in probability. Conversely if m; converges to m" in probability then ”n converges weakly to a probability measure u with characteristic functional m. Now we are ready to obtain generalization of Theorems of ([24], p. 780). We also obtain here the Lévy-Khinchine representa- tion of a symmetric semi-stable law on a real separable Hilbert space in the same spirit as Jajte in ([14], p. 64) or alternatively we characterize the symmetric semi-stable laws on a real separable Hilbert space in terms of its Vardhan representation. 4.1 The Main Theorem 4.1.1 Definition. A probability measure u on a real separable Banach space E is said to be semi-stable if there exist real numbers a > 0 and b such that the characteristic functional fl satisfies * (4.1.2) g(y) = [g(by)]a for every y E E . Remark. We can assume a > 1, otherwise u would be de- generate. By the similar argument as in PrOposition 3.1.1, it would follow that ‘b‘ < 1, provided H is non-degenerate. From now on we shall, without loss of generality, assume a > 1, \b‘ < 1. 4.1.3 Theorem. Let u be a probability measure on 51E) and for x > 0, [x] denote the greatest integer contained in x. If there exist two numbers a > 0, and b real such that [T “MM ] =° u b 53 then u is semi-stable. ll‘ Proof: Let vn = [T nuJ‘a J. Then +1 n n+1 n _ [an 1 _ [a [a 1-[a 1 (4.1.4) vn+l - [T n+lu1 — [T n+lu] .[T n+1u] ' b b b n+1 [an+1 [an n+1 a“(a 1) ["(bn+1 )19“ (4.1.5) La1 1 1 = [a(b y>1 “ y e . - n+1 + [n(b y)] n 1 , n n+1 . where an, 9n+1 are fractional parts of a , a reSpectively. Now it is clear from (4.1.5), that as n a m, n+1 [a-[a1a'1 = [g(by)]a . 4.1.6 Main Theorem. Let u be a semi-stable probability n measure on a real separable Hilbert space H. Then [T nu][a ] a u. n Proof: Let v = [T u][a ]. Then to show that v 2 u, n bn n it is enough by ([9], p. 8) to show that 9; = fl" in probability. n ~ ~ A A n Since (My)~ = [p.(bny)][a ] -[u(b y)j n, therefore ~ \[fi(bny)][a 3"\ \1 - (a(bny)) “\ M (4.1.7) My)" - a"\ n 9~ \1 - La 0 is a characteristic functional of a probability measure on H. Proof: In View of ([23], p. 181), (1) follows from Theorem 4.1.6 and (2) is obvious from ([23], p. 181). 4.1.9 Theorem. A functional m(-) is a characteristic function of a symmetric-semi-stable probability measure on a real separable Hilbert space H iff (4.1.10) ¢ = exp[-% + f(cos O, b XHSI 2 such that ab = l and aTbM = M. Proof: Suppose m is a characteristic function of a semi— stable law. Then if we denote the Lévy-Khinchine representation of a symmetric infinitely divisible characteristic function by [0,D,M], we get by Theorem 4.1.8 55 m = [0, D, M] . Using (4.1.2) we get 2 a 2 ‘ i0,D,M] = [0,b D,TbM] = [0,b aD,ale]. Now by the uniqueness of the representation we get ab = 1 and aTbM = M . 2 Conversely, suppose there exist two numbers a > 0 and b 3 ab and aTbM = M. From ([23], p. 181), it follows that m is infinitely divisible. Thus, _ _ 2 _ - 2 a a - [0,D,M] - [0,ab D,aTbM] - [0,b D,TbM] . * Hence m(y) = [qp(by)]a for every y E H . Remark. In case the symmetric semi-stable law u is actually stable, then in view of Theorem in ([14], p. 64), we get that abx = l for 0 < k < 2. Even non-symmetric Gaussian law is not semi-stable. CHAPTER V CONCLUDING REMARKS In this part we state some problems which arise out of the work of previous chapters. 1. The limit theorems proved in §2.2 and §3.2 can be used to obtain certain invariance principles for stochastic processes in the same spirit as D'Acosta [4] used his work on convergence to Gaussian measures. In fact in view of the work of Lamperti [18] it seems that certain limit processes associated with branching processes are symmetric stable processes. So the following two problems raised by Rajput and Cambanis in [27] for Gaussian measure become of interest for stable measures. 2. Given a stable stochastic process with sample paths in a Banach function space, is there a stable measure on the Banach space which is induced by the given stochastic process? 3. Given a stable measure on a Banach function space, is there a stable stochastic process with sample paths in the Banach space which induces the given measure? Some partial progress has been made on problem 2 and it is hoped that 3 can be handled by similar methods. 4. One can introduce, following Doob's ideas on infinitely divisible processes [6], the idea of self-decomposible process and study its sample path properties. Also problems 2 and 3 can 56 57 again be asked about self-decomposible process. The above problems are under investigation and progress on them will be reported elsewhere. REFERENCES [1] [2] [3] [4] [5] [6] [71 [8] [9] [103 [11] [12] [13] REFERENCES Billingsley, P., Convergence of Probability Measures, John Wiley and Sons, New York, 1968. Bohnenblust, F., An axiomatic characterization of L Spaces, Duke Math. J. 6 (1940), pp. 627-640. 9 Breiman, L., Probability, Addison-Wesley Publishing Company, Reading, Massachusetts, 1968. 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Zaanen, A.C., Linear Analysis, P. Noordhoff, Groningen and Interscience, 1, New York, 1953. IIIIIIIIIIIIIIIIIIIIIII IIIillllflllmlllfllifllflflw 14511 RRRRRRRRR