PROBABELITY MEASURES 0N REAL SEPARABLE BANACH SPACES Thesis for the Degree of Ph. D. MlCHlGAN STATE UNIVERSITY JOHN MATHlESON 1974 LIBRARY Michigan Stab: University This is to certify that the thesis entitled PROBABILITY MEASURES ON REAL SEPARABLE BANACH SPACES presented by ' John Mathieson has been accepted towards fulfillment of the requirements for Ph.D. Statistics 6: Probability degree in a n ,r 3/ l c __' gibtk (Li/K CLKC ,, CCAIU____, Major professor Date 4"2 5’" 1971+ 0-7 639 ABSTRACT PROBABILITY MEASURES ON REAL SEPARABLE BANACH SPACES By John Mathieson Two fundamental problems are considered in this thesis, they can be described as follows. In Chapter I the problem of characteriz- ing the characteristic functions of probability measures is examined. When the probability measures are defined on a real Banach space with a Schauder basis we obtain general results which are applied to various sequence Spaces. In Chapter II we introduce the notion of covariance form for a Gaussian probability measure. We obtain several representation theorems for the covariance form when the Gaussian measures are de- fined on real separable Banach Space. We then apply them to sequence spaces and Spaces of continuous functions. PROBABILITY MEASURES ON REAL SEPARABLE BANACH SPACES By John Mathieson 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 1974 ACKNOWLEDGEMENTS I would like to thank V. Mandrekar, V. Fabian, V.P. Sreedharan, D.C. Gilliland and Noralee Barnes for their help in preparing this thesis. Thanks also go to the National Science Foundation for the financial Support that made this thesis possible. ii Chapter 0 II TABLE OF CONTENTS INTRODUCTION ................................ 0.1 The General Bochner Problem ............ 0.2 Representation Theorems for the Characteristic Functions of Gaussian Measures . .............................. BOCHNER THEOREMS ON BANACH SPACES WITH SCHAUDER BASIS ........ . ..................... I.l Preliminaries .......................... I.2 Topologies on Vector Spaces ............ I.3 Bochner's Theorem on Sequence Spaces .. 1.4 Probability Measures of X Type ...... . 1.5 Banach Spaces with Schauder Basis and Accessible Norm ........................ GAUSSIAN MEASURES ON BANACH SPACES .......... 11.1 Introduction .......................... II.2 Representation of the Characteristic Function of a Gaussian Measure ........ II.3 Some Applications to Sequence Spaces 11.4 Gaussian Measures on a Space of Continuous Functions .................. OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO iii 14 18 24 27 27 31 34 37 39 42 CHAPTER 0 INTRODUCTION In the development of the classical theory of probability, the concept of characteristic function has played a powerful and central role. As a consequence this concept has been extended, initally by Kolmogorov, to the study of probability measures on in- finite dimensional linear Spaces, the aim being to duplicate the results of probability theory on finite dimensional Spaces. The impetus behind this is,the fact that much of the study of stochastic processes is equivalent to the study of measures on Suitably chosen infinite dimensional linear function Spaces. In this thesis we shall consider two fundamental problems concerning characteristic functions, they can be described in the following general terms. 1. The General Bochner Problem. It is well known, see for example [4], that the characteristic functions of probability distributions can be characterized as the continuous positive definite functions on the real line. The general Bochner problem is to find analogous characterizations for the char- acteristic functions of probability measures on infinite dimensional spaces. In chapter one we shall consider initially the problem of determining sufficient conditions for a given function to be a l characteristic function. Some general theorems are derived in this direction when the Spaces under consideration are Banach Spaces with Schauder bases. Our general theorems allow us to derive the Bochner theorems of Gross [6], Sazonov [18] and to extend the theorems of A. de Acosta [2]. A general Bochner theorem is obtained utilizing the concept of x-families of measures, first introduced in Kuelbs and Mandrekar L9], [10]. The reSults of [9] and [10] are extended to Orlicz Spaces and the hypotheses are weakened. As an illustration of the power of the techniques developed in chapter one we conclude with a derivation of the Bochner theorem of Kuelbs [8]. The methods of chapter one differ substantially from those that have hitherto been employed in the solution of the Bochner problem. In all previous papers mentioned, the methods employed center on establishing the existence of a probability measure by showing that it is a limit of a compact set of probability measures. We shall establish the existence of our measure by first finding a measure on too large a Space and then finding conditions for its support to be suitable. 2. Representation Theorems for the Characteristic Functions of Gaussian MeaSures. In [11], Kumar and Mandrekar have Shown that the only possible limiting distributions of normalized Sums of independent identically distributed Banach space valued random variables are the so called stable distributions. That is to say, only the stable distributions have non empty domains of attraction. The most important stable distributions are those that are Gaussian and it is these that are studied in chapter two. An attempt to characterize the domain of attraction of a Gaussian distribution should begin by obtaining a representation of its characteristic function. Using a result of X. Fernique tin we define the concept of the covariance function for Gaussian distributions, extending the concept first introduced by Vakhania [19]. A general representation for the characteristic function of a Gaussian distribution is obtained which includes as Special cases the results of Vakhania [19], A. de Acosta [2] and Kuelbs and Mandrekar [10]. CHAPTER 1 BOCHNER THEOREMS ON BANACH SPACES WITH SCHAUDER BASIS §l. Preliminaries. Let X and Y be real vector Spaces in duality with respect to some bilinear form <-,-> on X X Y. For any y1,...,yn in Y and any Borel set B in the n-dimensional Euclidean Space R“, a sub-set of X of the form {x E X : (,...,) E B] is called a cylinder set in X based in the finite SubSpace generated by y1,...,yn. The class of all cylinder sets in X forms an algebra, and the class of those based on a fixed finite Subspace of Y forms a o-algebra. We shall denote by BKX,Y) the smallest o-algebra containing the algebra of cylinder sets in X. (1.1) Definition. Let u be a finite measure on (X, 51X,Y)). Then the complex valued function a defined on Y by fi(y) = £exp[i}du(x) for all y E Y is called the characteristic function g£_ &, # We shall mainly be interested in the case where X is some separable Banach Space and Y is the topological dual of X. In this case the o-algebra .B(X,Y) has some important properties summarized in the following lemma. (1.2) £2992: Let E be a separable Banach Space and let E' be its topological dual. Then 4 (i) IG(E,E') is the Smallest o-algebra containing the (norm) open sets in E. (ii) If 9 = fi on E', where u and v are finite measures on (E, B(E,E')), then v = u. (iii) Every finite positive measure u on (E, 5KE,E')) is gighg in the sense that if e > 0 there exists a (norm) COWPaCt S€t K in E such that u(E\K) < e. # We shall not prove these statements, proofs of (i) and (ii) can be found in Ito and Nisio [7], proposition 2.2, and a proof of (iii) can be found in Parthasarathy [14], theorem 3.2. In this chapter we Shall consider the question of when a function on E' is the characteristic function of a finite positive meaSure on (E, EKE,E')). We shall first demonstrate some algebraic and topological properties of characteristic functions. (1.3) £2291: Let E be a separable Banach Space with tOpological dual E' and let u be a finite positive measure on (E,48(E,E')). Then the characteristic function a satisfies the following properties: (i) Let y1,...,yn be any finite subset of E' and let a1,...,an be any finite Subset of the complex numbers, then n jEk ajékfi(yj ‘ yk) Z 0 (ii) a is ¢S(E',E) continuous at the origin. Proof. (i) can be proved by direct computation, observing that n n - 2 E a a “(y - y) = \2 a. eXp{i}\ dMX) ij k“L j k [i=1 J j and the fact that u is a positive measure. (ii) Since by lemma (1.2) part (iii) the measure u is tight there exists, given 6 > 0, a norm compact subset K Such that u(E\R) < e- Let {ya} be a net converging to O in the TS(E',E) topology. Then lim \1 - fi(ya)] $23 + lim L‘exp[i} - I‘du(x) =26 a a Hence since e is arbitrary the result follows. # The property (1) that u satisfies, is of Special interest. (1.4) Definition. Let 6 be a complex valued function on E'. If for every choice of y1,...,yn on E' and every choice of ,...,c we have that l n n n _ Z 2c.c¢(y.-y)20 j=1 k=1 3 k 3 k complex numbers c then we say that ¢ is Positive Definite. # With this terminology, lemma 1.3 can be summarized as: every characteristic function on E' is positive definite and TS(E',E) continuous at the origin. Bochner has shown, [4], that if E is of finite dimension, then every complex valued function on E' that is positive definite and continuous at the origin is necessarily a characteristic function. It is well known, Prohorov [16], that this cannot be true for all Banach Spaces of infinite dimension. The following question naturally occurs. If E is a Banach Space of infinite dimension then does there exist a topology 6 on E' such that a complex valued function on E' is a characteristic function if and only if it is 6-continuous at the origin and positive definite? Clearly if such an 6 exists then 6 is coarser than TS(E',E). In [13], Mfistari has determined the Spaces for which the above question may be answered in the affirmative. Most spaces of interest do gg£_have such an 6. As a consequence, characteristic functions on E' cannot in general be characterized as the complex valued positive definite functions on E' that are continuous at the origin in some Special topology. In sections 2 and 3 we shall consider the problem of finding a topology 2. on E' such that a complex valued positive definite function that is z-continuous at the origin is necessarily a char- acteristic function. In order for our results to be of interest we shall want 2, to be as fine as possible and compatible with the algebraic structure of E'. Dudley has shown in [3] that o(E',E) continuous positive definite functions are necessarily characteristic functions and hence we should naturally seek topologies, 24 finer than o(E',E) and, of course, coarser than TS(E',E), that is to say, topologies of the dual pair (E',E) coarser than TS(E',E). The following theorem will prove to be extremely useful. (1.5) Theorem. Let X be a real linear Space with algebraic dual X*. If ¢ is a complex valued function on X then ¢ is the characteristic function of a positive measure on (X*, 8(X*,X)) if and only if q) is positive definite and continuous on finite dimensional subspaces of X. # This theorem is well known, a reference for it is Dudley [3], theorem 1.4. As a consequence, if E is a separable Banach space with dual E' and if ¢ is a positive definite function on E' continuous in agy_topology of the dual pair (E',E), then there exists A * * a finite positive measure u on (E' , BKE' ,E')) such that ¢ = u. The linear Space E'* contains an identification of E as a proper SubSpace, so that it is clear that our aim should be to find con- ditions for u to give all its mass to E. In the following sections we shall consider the measure Space (E'*,13(E'*,E'),u) and establish estimates for the support of u in terms of prOpertieS of fl. §2. Topologies on Vector Spaces. In this chapter the topologies to be considered on vector spaces are those determined by families of semi-norms. We know, [17], theorem 3, p. 15, that if F is a family of semi-norms on the vector Space E, then there is a coarsest topology on E compatible with the algebraic structure in which every semi-norm in F is continuous. We call this topology, the tOpology determined by_ E, Under this topology, E is a locally convex topological vector space and a base of closed neighborhoods of the origin is formed by the sets {x E E : sup p (x) S e} where e > O and p, E F. lSan j J We shall mainly be concerned with families of semi-norms satisfying the following property. (2.1) Definition. The family of semi-norms, F, on a vector Space E, is said to be sequentially dominated,if given any countable subfamily A, there exists p E P such that for all q 6 A there exists 0 < c(q) < a satisfying q(x) s c(q)p(x) for all x E E . # Clearly if P is sequentially dominated and A is any countable subfamily then there exists p E P such that the topology on E determined by A is coarser than that determined by p. This observation allows us to establish a useful property of such families. (2.2) LEEEE: Let V be a real valued function on the vector space E that is continuous at the origin in the topology determined by a sequentially dominated family of semi-norms P. Then there exists a semi-norm p E P such that y is continuous at the origin 10 in the topology on E determined by the Single semi-norm p. Proof. For all n 6 2+ there exists a finite family Tn<2 P such that {x: \¢(x) - ¢(O)\ < n‘l] contains a neighborhood of 0 in the topology on E determined by P“. Hence y is con- 00 tinuous at the origin in the topology determined by U P“. Since n=l F is sequentially dominated there exists p E T such that the co topology determined by U Fn is coarser than that determined by p n=1 and hence W is continuous at the origin in the t0pology determined by p. # We now introduce the notion of Gaussian summability of a semi-norm. It is well known that there exists a probability Space (0,3,?) on which we may define a sequence of independent random variables {X(j) : j E 2+} which are all normally distributed with mean zero and unit variance. (2.3) Definition. Let F be a vector space. Let + B = {ej : j 6 Z } be a subset of F. Then a semi-norm p on F is said to be Gaussian summable of order k_ with respect t 11 £29 + 16; if 1( N SUP E p { E i(j)X(j)e.} < w N21 j=l 3 where E is the expected value taken with respect to P. # It should N be observed that p[ 2 1(j)X(j)ej} is necessarily a random variable i=1 N on (0,3,P) since the map (y ,...,y ) a p( 2 y e 1 N j=1 j j map on Rn into R with the usual topologies. is a continuous (2.4) Lemma. Let {E, (-,-)} be an inner product space and let p(x) = (x,x)% for all x E E. Then p is Gaussian summable + of order 2 with respect to the countable set B = {ej : j E Z ] and 11 k E L+ Such that E 12(j)(ej,ej) < m. j=l Proof. 2N NN p ( z ixxX(j>X i=1 j=1 k=1 and 2N N2 E P ( Z k(j)X(j)ej) = 2 A (j)(e.,e.) i=1 j=1 J J and hence 2 N m 2 SUP E P ( 2 k(j)X(j)e.) S E k (j)(e..e.) < w - # N21 3:1 J j=1 J J (2.5) LEEEE: Let E be a vector Space. Let B = {ej : j E Z+} be a subset of E. Let H be a probability measure on (E*,,5(E*,E)) such that fl is continuous at the origin in the topology on E defined by a Single semi-norm p. If p Gaussian summable of order k with respect to B and A (E L+) then 31‘ u{x "MB i2(j)\\2 < a} = 1. j 1 3 Proof. From the assumption of Gaussian summability, Z (1) Epk( z k(j)x(j)ej) 5 M j 1 for some M and all. N. From the continuity of S, if e is a positive number then there is a C such that x k (2) 1 - Real p(x) < e + C p (x) for all x. For a t > 0 and a positive integer n, set n X = X = t E A X e, n.t j=1 J J J 12 and notice that by (l) and (2), with K = CMS and all \t\ S Kl/k (3) E(1 - Real Q(X)) S 23 But * * * Efi(X) = E] exp{i}du(x ) = f E{exp i}dp(x) it at E E * * by the Fubini theorem. For a fixed x , is a Gaussian random variable with mean zero and variance 2 2 2 * 2 n 2 * 2 (4) o=to(X)=t le\ n j=1 J * 2 Thus E{exp i} = exp{-% 0 } and so (3) can be rewritten as 2 2 * * (5) j eXP{-% t o (x )}dp(x ) 2 1 - 28 ‘k n E Take the limit of the left hand side, first for n a m, and then for t a 0. By the Lebesgue dominated convergence theorem the limit is the integral of the limit of the integrand, which is l on the set A = {x* : lim C§(X*) < m} and zero outside of A. It follows n—m then that p.(A)21-2€. Since 6 was arbitrary, p(A) = l, which is the assertion of the theorem. # We may now combine lemma (2.5) with the concept of a sequentially dominated family of semi-norms to obtain the main result of this section. (2.6) Theorem. Let E be a vector space. Let B = + {ej : j E Z ] be a subset of E. Let u be a probability measure 13 * * on (E , [3(E ,E)) such that f), is continuous at the origin of E in the topology determined by a sequentially dominated family of semi-norms P. Then there exists p E P such that if p is Gaussian + summable of order k with reSpect to B and X E e then * u{x* 12(j)\\2«< m] = l. uim 8 J 1 Proof. The proof follows directly from lemmas (2.2) and (2.5). # We may now apply theorem (2.6) to obtain Bochner's theorem on some sequence Spaces. 14 §3. Bochner's Theorem on Sequence Spaces. Let A and L be as in the appendix and let {uj E j 6 2+} = B be the canonical basis for A. We shall derive a Bochner theorem for positive definite functions on A and extend it to Spaces of type p. (3.1) Definition. A real finite bilinear form Y on A X A is said to b of trace class k if (1) Y is symmetric. That is to say, for all x,y E A we have Y(x,y) = Y(y,X)- (ii) Y is positive definite. That is to say, for all X E N\[0), Y(x,x) > O. (m) 2 r” 2(u,,u,) <00 . # j=1 J J We may define for any positive definite bilinear form Y a semi-norm pY by pY(X) = Y%(x,x) for x E A. Let Tk be the topology on A determined by the family of semi-norms {pY : Y is of trace class k]. (3.2) LEEEE: Let k S 2. Then {pY : Y is of trace class k} -is a sequentially dominated family. Proof, Let [Yn : n 6 2+] be a countable set of bilinear forms of trace class k. There exists c E L such that Z ck/2(n) E Y“, (uj,u.) < w . n=1 j=l 3 Let q = pY where Y(x,y) = E C(n)Yn(x,y) n=1 Since 2 c(n)Y (u,,u,) s { E ck/2(n)Yk/2(u.,u )}2/k for all j n=1 “ J J n=1 “ J j we have that Y is of trace class k for all n. 15 p (x) s 9é¥l-. # Yn c (n) We may now apply theorem (2.6) to obtain: (3.3) Lemma. Let k s 2. Let u be a probability measure on (L, 5%L,A)) such that fl is continuous at the origin. Tk Then ”(Lk) = 1. Proof. By lemma (3.2) and theorem (2.6) there exists pY such that if pY is Gaussian summable with respect to A E L+ and B then i2(j)x2(j) < w} = 1. 1 u{x E L : IIMB 3 By lemma (2.4) we then have p(A) = 1 where co k/2-l 2 A = {x e L = 2 {t(j)} x (j) < m} i=1 + and where t(j) = Y(uj,uj) for all j E Z . If we now Show that ACZLk then the result follows. Let k > 2 and r = 2/k > 1 and r' = 2(2 - k)-1. Then l/r + l/r' = l and by Halder's inequality co co 2_ 0° _ _ t t 2 lx(j)\k S { z ‘X(j)t%(k/ l)(j)‘kr}l/r{ z \t(j) %(k/2 l)kr )1/r j=1 j=1 j=1 a) - (I) 2 s { 2 x2tk/2 1} - i=1 1:1 Hence if x E A then 2 ‘X(j)\k < m that is to say x E Lk. i=1 If k = 2 then A = L since t(j) = l for all j. # k (3.4) Corollary. Let ¢ be a positive definite function on A such that ¢(0) = l and ¢ is continuous at the origin. Tk Then there exists a unique probability measure u on (LknB(Lk,A)) such that S = d on A. 16 Proof. The proof follows directly from theorem 1.5 and lemma (3.3). # We may now extend this result to Spaces that generalize L Spaces. P (3.5) Definition. A Banach space X with a Schauder basis + {ej : j E 2+} and coordinate functional {e3 : j E Z } is said to + be of type B if for all x E X and n E Z we have that H :1. e.Hp s Ela,e'>\p . # 3:1 J J j=1 J This definition is less restrictive than that of A. de Acosta [ ], the motivation behind our definition is the fact, initially observed in [ ], that such Spaces contain isomorphs of LP. More precisely we have: (3.6) Lemma. Let X be'a Banach Space of type p. Then there exists a continuous linear map S from LP into X. a.) Proof. Define S for all x E LP by 8x = 2 x(j)ej- S 3:1 is clearly a continuous linear map of LP into X. # (3.7) Definition. Let A(X) be the linear SubSpace of X' spanned by the coordinate functionals and let Tk(X) be the topology on A(X) determined by the bilinear functionals on A(X) X A(X) of trace class k. # Clearly the map tS : (A,Tk) a (A(X),Tk(X)) is continuous, allowing us to prove the main theorem of this section. (3.8) Theorem. Let X be a Banach space of type p s 2. Let ¢ be a positive definite function on A(X) with ¢(0) = 1. If ¢ is TP(X) continuous at the origin then there exists a probability measure v on (X,;6(X,X')) such that D = ¢ on A(X). l7 Pgoof, The function ¢ 0 CS on A is positive definite and Tp continuous at the origin. By corollary (3.4) there exists a probability measure A on (LP,IBCLP,A)) such that B = ¢ 0 tS on A. It can be trivially verified that if v = u o (tS)-1 then 9 = ¢ on A. # 18 §4. Probability Measures of A, Type. Let E be a Banach Space with Schauder basis {ej : j 6 2+} and coordinate functionals {e3 : j E 2+]. If ¢ is a positive definite function on E', continuous in some topology compatible with the algebraic structure and ¢(0) = 1 then we have by theorem 1.5 that there exists a unique probability measure A on (E'*, 5(E'*,E')) such that (id) = (b- In this section we shall con- sider the problem of representing ¢ by a measure on E by putting certain conditions on A (4.1) LSEEE: Let E and ¢ be as above. Then there exists a unique probability measure v on (E, 5KE,E')) such that 6 = S if and only if (i) ¢ is continuous at the origin in the topology of uniform convergence on compacta. (ii) For all A > 0 m * * * lim sup A {X E E' : H 2 e.“ > A} = 0 new m>n ¢ j=n+1 j J Proof. Suppose that conditions (i) and (ii) hold. Let * * m F = {x E E' : Z e, E E]. Since E is complete we have 1:1 J J that * * m * F={x EE' :lim sup“ 2: e,\\=0} n—ooo m>n j=n+1 J J on a co * * m * .- = n U D [x 613' z“ 2‘. e,“$k1}. k=1 n=1 m=n+1 j=n+1 J J * As a consequence F E BKE' ,E') and moreover * m * * m * A (E' - F) S 2 lim sup A [x E E' : H E e \ ¢ k=l ndm m>n ¢ j=n+1 J j 19 That is to say, A¢(F) = 1. We may now consider the measurable transformation * m * Y : (F, /3(F,E')) _. (E, B(E,E')) given by ¢(x ) = 23 ej * j=1 for all x E F. By the transformation theorem we have that for all y E E' and n 2 O m(tnny) = s Since tnny converges uniformly to y on compacta, we have by condition (1) of the lemma and condition (ii) of lemma 1. that ¢(y) = $(Y) for all y E E'. Conversely suppose that there exists v on (E, B(E,E')) * such that 9 = ¢. Let us define the canonical map q : E a E' by = for all x E E, y E E'. Clearly q : (E, B(E,E')) -+ (E'*, B(E'*,E')) is a measurable transform and by the transformation theorem and the uniqueness of A¢, we have that v o q“1 = A¢. Since q-1(F) = E we have that A¢(F) = l and this clearly implies that condition (ii) holds. # Condition (ii) of lemma 4.1 suggests that we introduce the following concept of x-measure. (4.2) Definition. Let E be a Banach Space with Schauder basis [ej : j E Z+} and coordinate functionals {e5 : j 6 2+}. Let A E.A+ and let P be a probability measure on (E'*, BKE'*,E')). If there exists a real valued strictly increasing function on [0,m) with h(0) = 0 such that 20 m * * * lim sup P{x E E' : H 2 ej“ 2 h(6)} new m>n j=n+l * * m , * 2 5 lim sup P{x E E' : Z x(j)l\ 2 6] 11—10:) m>n "n+1 J then we say that P. _o_§_L-measure or 2_ io of L-tyoe. If ¢ is positive definite on E' and A¢ is a x-measure then condition (ii) of lemma (4.1) will hold if for all 6 > O * '* m , * 2 lim sup A [X E E : Z A(j) 2 6] = O . n-oco m>n (D j=n+l J This latter condition can be shown to hold if 6 has some continuity prOperties. The continuity of ¢ will be defined with respect to a topology on E' determined by a family of bilinear forms. (4.3) Definition. Let E be as in lemma 4.1. Let Q be a family of bilinear forms on E X E' such that for all Y E Q (i) Y is symmetric and positive definite. (ii) For all y E E' : 2 YLi j=1 (€3.ef)\\ < m- J J J We define T(Q) to be the topology on E' determined by the family of semi-norms {pY : Y E Q}. # The condition (ii) implies that t11n(y) converges to y in the T(Q)—topology. (4.4) Theorem. Let E be a Banach Space with a Schauder basis {ej : j E 2+] and coordinate functionals {e5 : j E 2+}. Let ¢ be a positive definite function on E' satisfying the follow- ing conditions (1) ¢ is T(Q) continuous at the origin, ¢(O) = 1. (ii) The measure Ag) on (E'*,B(E'*,E')) isa A measure + when A E L and for all Y E Q, 21 IIMB 1K(j)Y(e5393) < m ° J Then there exists a unique probability measure v Such that D = ¢. Pooof. Clearly by lemma (4.1) and definition (4.2) we need only show that for all 6 > 0 * * m 1im sup A {x E E' : Z A(J)\\ = O n-m m>n (b j=n+l Let s > 0 be arbitrary. Since A is T(Q) continuous there exist {Yj : 1 s j s p} CZQ and C < m such that for all yGE' 1 - Real ¢(y) S e + C Sup Y,(y,y). lSj Sp + Let {X(k) : k E Z } be independent identically distributed standard normal random variables and let y = E 1%(k)X(k)ek Then k=n+ l m m g g l - Real ¢(y) s e + C sup 2 E A (RJA (L)X(k)X(L)Yj (e' k’eL) ISjSp k=n+l {En-+1 On taking expected values we obtain j* 1 - exp{- -% 2 A(k)\\ 2}du¢ (X *) E'* k=n+l m S e + C sup 2 x(k)Y. (8k 8k) 1$j$p k=n+l j Hence by the Markov inequality m m *2 6 + SUP E A(S)Yj(e§3eé) * * _ A {x E E' : 2 A(S) 2 6} S 1515p S-n+l W S=n+1 (1 " exP("255)) 22 By condition (ii) of the theorem and noting that s was arbitrary we obtain the required result. # We shall now apply lemma (4.4) to the case where E is an Orlicz sequence Space. Let a and B be complementary functions in the sense of Young and let La and L8 be the Orlicz Spaces defined in the appendix such that L; = L8. (4.5) Definition. Let {uj : j E Z} be the canonical basis for La and L8. Let Qa be the set of all symmetric, positive definite, bilinear forms Y on LB X LB such that j’uj)) < co. Let Ta = 7(Qa). # (4.6) Theorem. Let La be an Orlicz Space with topological Z a(Y%(u j=1 dual LB. Let 6 be a complex valued function on LB such that (i) 6(0) = l and 6 is positive definite, (ii) 6 is Ta-continuous at the origin, and on (iii) u¢ is a x-measure where x E Lf and 2 a(]t%(j)\) < m implies that E A(j)‘t(j)‘ < m. j=1 Then thEie exists a unique probability measure v on (La, 5(La,LB)) such that Q) = 6. In the case when a6/-) is a convex function on [0,m) we have (i), (ii) and (iii) holding for every characteristic function of a probability measure on (La, 5(LG,LB)) . Pooof, If 6 satisfies (1), (ii) and (iii) then by theorem (4.4) there exists v Such that D = 6. Conversely, let us suppose that 6 = % where v is a probability measure on (La, 6(La,.(, )). B Q m Let Kn = [x E La : j§16(\x(j)\) < n] for all n. Since U K.n = La given 6 > 0, there exists n such that v(Kn) 2 l - e- 23 If Y is defined by Y(z,y) = i dv(x) for all z,y E LB n we have that \1 - 6(2)] s %Y(z,z) + 6- Since aG/') is convex we may use Jensen inequality to Show that Y is in Qa’ and hence since 6 was arbitrary 6 is Ta-continuous at 0, and condition (ii) of the theorem follows. By lemma (4.1) condition (ii) we have that H¢ is a A' measure for all A E 6+. Hence condition (iii) of the theorem will ’5 hold by choosing A such that Z 8(A (j)) < m. # If q(x) = xp we obtainjEEe following result of Mandrekar and Kuelbs. (4.7) Corollary. If 2 S p < m and 6 is defined on Lg, then 6 is the characteristic function of a probability measure on LP if and only if (i) 6 is positive definite, 6(0) = 1, (ii) 6 is Tp-continuous, (iii) A¢ is a x-measure for some A E (Lg/2)+ REESE: The Sufficiency of conditions (i), (ii) and (iii) follow: by the first part of Theorem 4.6 and the fact that x E (Lg/2)+ and Z \t(j)\p/2 < m imply that ; x(j)‘t(j)‘ < m. The necessity 3:1 / j=1 2 follows since ag/a) = xp is a convex function for p 2 2. 24 §5. Banach Spaces with Schauder Basis and Accessible Norm. For the purpose of this section E will be a real Banach + . space with Schauder basis {ej : j E Z } and coordinate luncllonnls + {e} : j E Z ]. (5.1) Definition. Let 6 be a real continuous positive definite function on E and let c : (O,m) a (0,m) be any strictly increasing function. Then if for all x E E, 6(0) - 6(x) 2 c(“xH) we say, following Kuelbs, that the norm of E is accessible withreSpect to 6. Since 6 is positive definite and continuous there exists * a positive measure P4? defined on [303 ,E) such that {>(x) = * * I exp idP (x ). * Q E (5.2) Theorem. Let E and Q be as above and let the norm of E be accessible with reSpect to 6. Then a complex valued function 6 on E' is the Fourier-Stieltjes transform of a unique probability measure on (E, BKE)) if and only if (i) 6 is positive definite, 6(0) = 1 (ii) 6 is TS(E',E) continuous. t 7‘: t ~k * (iii) lim sup I l — 6( nm(x ) - nn(x ))dPQ(x ) = 0. ham m>n * E Proof. Suppose that conditions (i), (ii) and (iii) hold * for 6. By theorem (1.5) there exists a A@ on E' such that for all x' E E' ' * * 6(x') = I exp{i]dA (x ) * m E' If J(m,n) denotes the integral in condition (iii) we then have that 25 m J(m,n) = g I l - exp{i 2 }du (x*)dPQ(x'). IE'* j=n+1 J ¢ By Fubini's theorem we may change the order of integration above, noting that the required measurability of the integrand is trivally m * * satisfied. Hence J(m,n) = I 6(0) - §( 2 ej)du¢(x ). * E' j=n+l {* \g<'* \\> s {* M0) an; '*'>><>} x : e.,x >e, x : - ,x >e c U¢ ‘j=n+1 J J e} Um j=n+1 O * m * lim sup u {x : H 2 e,H > e] = O mam ¢ j=n+1 J J Hence by lemma (4.1) the result follows. As a simple application of this theorem we shall give a proof that positive definite functions on L2 that are Tl-continuous are necessarily Fourier-Stieltjes transforms. Suppose ¢ : L2 ~»C is positive definite, ¢(O) = 1, and ¢ is Tl-continuous. For any a > 0 there exists a symmetric, positive definite bilinear form Y on L2 X L such that for x E L2 2 \1 - ¢(x)\ < e + Y(x,X) + We let (un : n 6 Z ) be the usual complete orthonormal basis for 2 L2. Let §(x) = exp{-%Hx“ }, then Q is a real valued continuous positive definite function on L2 and the norm of L2 is clearly accessible with respect to Q. Moreover if P is the corresponding Q we have that I dPQ(x ) 5k . Hence J 26 m m {I ' ¢( 2 ej)dP§(X') S e + E ‘1’(J,j) 2 j=n+1 J j=“+1 m Now 2 Y(j,j) < m since Y is of trace class 1 and hence i=1 m lim sup I 1 - ¢( 2 <8.,X>ej)dP®(x') S e . n-aao m>n {,2 j=n+1 J But 6 was arbitrary and the conditions of the theorem are satisfied. Hence the function ¢ is a Fourier-Stieltjes transform. We may similarly obtain the results of §3 by careful choice of Q, we shall not, however, include the details here. CHAPTER 11 GAUSSIAN MEASURES ON BANACH SPACES §l. Introduction. There are many possible equivalent definitions of Gaussian mea- sures on vector Spaces; in this chapter we will use that of X. Fernique [5]. (1.1) Definition. Let E be a real vector space and let B be a o-algebra of Subsets of E. We say that Q _i_s_ compatible with the algebraic structure of. E_ if (i) The map (x,y) _. x+y of (E X E, [3 X5) into (E,EO is measurable. (ii) The map (x,)() _. )(x of (E X R, B X B(R)) into (E,B) is measurable. Let (0,3,P) be a probability Space and X a measurable map from (0,?) into (E98), we say that X is a random variable with values _ig E. The law Q_f_ )_(_ is the measure P o X_1 induced on (E ,6) by X. The concept of independence of random variables with values in E may be defined as for real valued random variables with 6 replacing 6(R). We shall say that X is a Gaussian random variable with values _11 E_ if the following condition is satisfied: For all pairs (X1,X2) of independent random variables with the same law as X and for all pairs (s,t) of real numbers with 82 + t2 = l, the random variables (sX1 + th) and (tX1 - 3X2) 27. 28 are independent and have the same law as X. A measure u on (EMS) is said to be Gaussian if there exists a probability Space (0,3,?) and a Gaussian random variable X with values in E such that u is the law of X. # The following result of X. Fernique generalizes a well known result for Gaussian measures on R and is of fundamental importance in this chapter. (1.2) Theorem (X. Fernique [5]). Let (EnB) be as above and let u. be a Gaussian measure on (E,B). If “u is a B- measurable norm on (EnB) then there exists a > 0 such that £exp[aux“2}du(x) < m. # It is clear that for such u the integral of any power of u-“ is finite, this is the property that will prove to be most useful. In this chapter we shall extend the results of Vakhania [19] to arbitrary separable Banach Spaces and hence we must Show that our definition of Gaussian measure is equivalent to the definition of [19]. It is well known, for example [4], theorem 2, p. 526, that a Egal_valued random variable is Gaussian in the sense of definition (1.1) if and only if it has a normal distribution with zero mean. It is this fact that allows us to establish the equivalence of definition (1.1) and the definition of Gaussian measures to be found in [ ]. (1.3) Lemma, Let E be a Banach Space. Then (i) B(E,E') is a o-algebra, compatible with the algebraic structure, for which the norm of E is measurable. 29 (ii) A probability measure u on (E, 6KE,E')) is Gaussian if and only if for all x' E E there exists c(x') 2 0 such that x' is a real valued random variable on the probability Space (E, B(E,E') ,p) with distribution N(O,c52(x')) . M. (i) Since B(E,E') is generated by sets of the form {x : ~< a} in order to prove (i) of definition (1.1) we need only Show that {(x,y) : < a} E /3(E,E') X B(E,E'). Let Q = {r E R : r is rational]. Then we have {(x,y) : ‘< a} = U {x : < r} X {y : < a-r} rEQ e B are Gaussian for all x' E E'. Suppose that for all x' E E' the random variable w q is Gaussian with values in R. Let X1 and X2 be independent and have the same law as X and let (s,t) be real 2 2 . with s +'t = 1. Then clearly . and are 2 independent and have the same law as for all x' E E'. Hence the real valued random variables ’ and are independent and have the same law as for all x' E E'. Now since B(E,E‘) is generated by sets of the form {x : E B} where B E 6(R) we have that 5X1 + tX2 and tX1 - 5X2 are independent with the same law as X. 30 Conversely suppose that X is Gaussian with values in E. Let X1 and X2 be independent random variables with values in E having the same law as X. Then for all (s,t) with 32 + t2 = l and any x' E E' we have that (S’+ t) and (t - s) are independent. Hence by [ ], theorem 2, p. 526 has a normal distribution and hence has a normal distribution. The mean of is clearly zero. # Gaussian measures on finite dimensional Spaces are uniquely determined by their covariance matrices. We may extend this result to infinite dimensional Spaces by introducing the notion, following Vakhania, of the covariance (1.4) Definition. Let p be a Gaussian measure on the Banach Space E. For x',y',I,E' we define Y by u vu = j«,x'>du is a real Gaussian random variable on the probability Space (E, 5KE,E'),H), moreover it is easily seen that A p(x') = exp{-% Yu(x',x')} for all x' E E' In this chapter we shall determine properties that a covariance form must have and for some special case we shall characterize such covariance forms by means of operators. 31 §2. Representation of the Characteristic Function of a Gaussian Measure. In this section we shall proceed to an operator type repre- sentation of fl by means of an initial Lévy-Khinchine type repre- sentation. Let E be a separable Banach Space, and let S = {x E E : “X“ = l}, S will be a complete separable metric Space under the induced norm topology. Let 5K8) be the o-algebra of subsets of 3 generated by the open sets. (2.1) LEEEE. (Lévy-Khinchine type representation of a). Let E be a real separable Banach Space and let u be a Gaussian measure on (E, 5KE,E')) with covariance operator T. Then there exists a unique positive, finite, symmetric measure P on (S, 6(5)) such that for all x',y' E E' = idI‘(x) 2322;. By lemma (1.4) we have that for all x',y' E E' = £dp(x). Let k be the finite measure on (E, 5KE,E')) defined by dx(x) = Hxnzdp(x) and let j : E {0] a S by the continuous map j(x) = x/nx“. By the transformation theorem we have that for all xI’yI 6 El I£.dp(x) = idF(X) where F = A o j-1. The result then follows since F is clearly finite and Symmetric. 32 In order to verify that F is unique, it suffices to Show that if F is a finite, symmetric measure on (S,.6(S)) such that 2 (l) £\‘ dF(x) = O for all x' E E'. There exists k > 0 such that for all b E R, (l - cos br)r-3dr = 2 a) k‘b‘ and hence if (1) holds, i 81 - cos rdrdg(x) = 0. If r we define the meaSure Q by 0(A) = I j l - cosr-3dr dF(x) U NA k>0 we obtain that (2) £1 - exp[i}dn(x) for all x' e E' By using exactly the method of Parthasarthy [14], p. we see that (2) implies that 0 = 0 and hence that F = O. # It is not clear whether a can be characterized in terms of measures on S, however in §4 we shall see that in some special cases (Orlicz and LP Spaces), such a characterization is possible. We shall now use lemma (2.1) to obtain an operator repre- sentation of a that directly generalizes the results for Hilbert Spaces. In the Hilbert space case fl is represented in terms of Hilbert-Schmidt (for definition see [6]) operators, such operators are generalized by the following: (2.2) Definition. Let X and Y be normed Spaces and let T : X a'Y be a linear map. T i§_said t b absolutely p + summing if there exists a constant C such that for any N E Z and (x1,...,xN) CZE we have that 33 N p N 2 “Tx.“ S C sup{ 2 \\p : x' E X', Hx'“ s l} . # -= J -_ J J 1 3-1 Such operators were first introduced by Pietch [153 who showed that if X and Y are Hilbert Spaces then T is absolutely p summing if and only if T is Hilbert—Schmidt. (2.3) Theorem. Let E be a real Banach space and H a Gaussian probability measure on (E, B(E,E')) . There exists an absolutely 2-summing operator, A, on E' into L2(S,T) such that Yu(x',y') = . The transpose map, tA, is a map on L2(S,F) into E defined by the Bochner integral tAf =£xf(x)d1"(x) for all f e L2(s,r). Proof. Define the bounded linear operator A : E' a L2(S,F) by Ax' = <-,x'>. If we define B on L2(S,F) by Bf = £xf(x)df(x) V f E L2(S,F) then since £“Xf(X)HdF(X) = £\f(X)\dF(x) < m we have that Bf E E. Moreover = t£f(x)dr(x) = L = . 2 Hence B = tA. For any x',y' E E' we have that ‘i’(x',y') = idu(x) =£ooc,y'>d1"(x) = L = (AtAx',y'> . 2 The proof is now completed by observing that A is absolutely 2-summing as a consequence of corollary l, p. 187 of Wong [20]. ill-III] 34 §3. Some Applications to Sequence Spaces. In this section we shall characterize the covariance operators of Gaussian measures on various sequence Spaces, in particular we shall consider the Orlicz space La defined in the appendix. The basic theorem, from which we will derive some Special cases, is the following: (3.1) Theorem. (a) The covariance operator of a Gaussian probability measure on La is a bounded linear operator T : L; a La satisfying: (i) T is symmetric and positive. 00 (ii) 2 a(<.Tuj j=1 (b) Conversely, if 0 satisfies condition (2) of the appendix ,Uj>%) < co . then a bounded linear Operator T on L' into L satisfying con- a/ (Y ditions (i) and (ii) is the covariance operator of a Gaussian measure on L . a Proof. (a) In order to prove (ii) it suffices (see appendix) to Show that for all y E L; 00 . % (1) E A(J) < co . j=1 J J + For all d E Z the real valued function x(j) defined on the probability Space (L ,p) (where p is Gaussian) is a normally a 2 distributed random variable with mean zero and variance Oj = . As a consequence { \x(j)\dp. =/Z71; O'j =/§-/-n. «;é d Then if y E LB 35 N ).(j)<1‘u.,u,>$5 s/n/Zj‘ 2: ‘x(j)y(j)\dp.(x) 1 J 3 La d=l utntz s x175 M, { uxuadm Since the latter term is finite and independent of N the condition a: 1 holds, imply that Z a(%) < m. j=1 (b) Let T : L' «.L be a bounded linear operator satisfying a 0! conditions (i) and (ii). If ¢ is defined on A by ¢(x) = exp{-% }, then, by theorem 1.5, chapter one, there exists a probability measure u on (L, 61L,A)) Such that fi(x) = ¢ for all x E A. ,uj> and as a consequence J of condition ( ) in the appendix there exists C < a such that For all j 6 2+, {\x(j)\2du(x) = )- Utilizing condition (ii) we obtain that N sup I E a(\X(j)\) < m N21 L j=1 and by the monotone convergence theorem on J“ 2 a(\X(j)\)du(X) < co . 4, j=1 This implies that u(Lq) = l. The proof is completed by observing that u is Gaussian with covariance operator T. Corollary (A). A bounded linear operator T : L; a Lp (l s p < m) is the covariance operator of a Gaussian probability measure on Lp if and only if (i) T is symmetric and positive. 36 (ii) 2 <,Tuj,uj>p < m . j=1 Egggf. For p > 1 the proof follows directly from theorem (3.1) on taking q(x) = xp/p. For p = l the proof is essentially of the same form as the proof of theorem (4.1) and will hence be omitted. Corollary (Bl, Let E be a Banach Space of type p where l S p < m. A bounded linear Operator T : E' a E is the covariance operator of a Gaussian probability measure on E if (i) T is symmetric and positive. 2 >p/ m (ii) 2de3,ej' (CD . j=1 Proof. Let A E L6. Using Hfilder's inequality and the fact . k I t I i ' that drej,ek> s dek,ek> we obtain that °° °° , 2 °° , /2 2 2 \ I: x(j)\P/ :2 mm 1: p } < co k=l j=1 J j=1 j 3 As a consequence we may define the map C : L' a Lp by P C(X)(k) = E A(j)T . j=1 Noting that C clearly satisfies conditions (i) and (ii) of corollary (A) and we see that there exists a Gaussian measure p on LP with covariance operator C. If S is the map from Lp into E defined in chapter one we may trivially verify that u o 8-1 is a Gaussian measure on E with covariance operator T. 37 §4. Gaussian Measures on a Space of Continuous Functions. Let C be the space of real valued continuous functions on [0,1]. Under the usual uniform norm, C[0,l] becomes a Banach Space with a Schauder basis that we shall exhibit later. As observed by de Acosta [ ], lemma 8.3, Gaussian measures on C are the measures induced by continuous Gaussian processes. Our main representation theorem is essentially that of [ ], theorem 8.4; we repeat it here, with proof, Since the proof given in [ ] is incomplete. (4.1) Theorem. Let P be a Gaussian measure on C. Then there exists a continuous map K : [0,1]2 a R such that (i) K(s,t) = K(t,s) for all s,t E [0,1]. (ii) K is positive definite, that is to say for all n E [0,1] we have 2 CiCjK(Si’Sj) 2 0. ,cn E R and 1,1 51,...,s (iii) For all v E C' . l l P(v) = exp{-%g g K(s,t)dv(s)dv(t)] Egggf, We define K directly by %K(s,t) = gx(t)x(s)dP(x) for all s, t E [0,1]. K is well defined since g\X(t)X(S)\dP(X) < guxuz dP(x) < m and clearly is continuous. Conditions (i) and (ii) are trivial to verify. Since P(v) = exp{-% i]‘2dP(x)} and = 1 &x(s)dv(s) we observe that: 1 1 -2 log P(\)) = g {E £x(s)x(t)dv(s)dv(t)dP(x). Since 1 l H[WWW\dvdvdp s No, 1]} WW dP(x) < o, 38 we may use Fubini's theorem to obtain A l l -2 log P(v) = & £K(s,t)dv(s)dv(t) and the result follows. # A partial converse of theorem (4.1) is possible. (4.2) Theorem. Let K be a real continuous function on [0,1]2 satisfying condition (1) and (ii) of theorem 4.2. If for some a > 2, K satisfies a Lipshitz condition Of order a in each of its variables then the operator T from C' into C defined by (Tv)(s) = $K(s,t)dv(t) is the covariance operator of a Gaussian measure on C. 2393;. 'Let {uj : j E 2+] be the coordinate functionals correSponding to the Schauder basis for C constructed in [ ], p. 69. The following conditions are satisfied = x(b - Mano) - wan» u > 2 where a(n) < b(n) < c(n) and \c(n) - a(n)‘ = 0(l/n). Let ¢(v) = exp{-% ]. m is positive definite by con- dition (ii) Of theorem 4.1 and using the fact that K satisfies Lipshitz condition we obtain by direct computation that = 0(n-a). Since a > 2 we have that T is an Operator of trace class 1 and hence by theorem (3.8), chapter one, there exists a measure P on C such that P = ¢ on the SubSpace of C' spanned by the coordinate functionals. The result then follows. APPENDIX APPENDIX List Of symbols R The real numbers. + . . Z The p081t1ve integers. Z+ L = R The space of real valued functions on Z + + L The space of positive real valued functions on Z . L 0 < p < m The subset Of L consisting of the functions A with z \i(n)\p < m . n=1 A The subset of L consisting Of the functions that + are zero except at a finite number of elements of Z . L The Orlicz Space {x E L : Z a(\x(j)\) < m}. j=1 (1) Vector Spaces. Our basic reference throughout this thesis will be Robertson and Robertson [17]. We will only be considering vector Spaces Over the scalor field of real numbers. If E is such a vector space we will denote its algebraic dual by E* and define it to be the set of all real linear forms on E. If E has a topology then we shall denote its topological dual by E'. The concept of duality may be found in [17], p. 31, definitions Of the polar topologies may be found on p. 46. Of particular interest are the weak (c(E,E')) topology and the topology of uniform con- vergence on strongly compact subsets of E. We denote this latter topology TS(E',E) and note that if E is a Banach space then the strongly compact sets are the norm compact sets. 39 4O (2) Banach spaces with Schauder basis. If B is a real Banach Space ([ l], p. 60), a Schauder + basis {ej : j E Z } is a sequence of elements in B such that for each x E B there is a unique sequence of real numbers x(j) n such that lim “x - E x(j)e,“ = 0. The mappings x « x(j) are n—coo i=1 J continuous real linear forms on x and hence there exists a sequence + [e5 : j E Z } of elements in B' such that x(j) = . These are called the coordinate functionals of the basis. We shall use n n for all n to denote the maps given by n (x) = 2 e, n n j=1 J J and observe that “x - nn(x)“ « 0. The transpose map tnn may n similarly be defined by tn (x') = 21e'. n j=1 j j It is known, [8], §l, that nh(x) converges uniformly to t x on norm compact subsets Of B and hence nn(x') converges to x in the TS(B',B) topology. (3) Sequence Spaces. Let uj E L be the map uj(k) = 1 if j = k, uj(k) = 0 if k # j. The set [uj : j E 2+} will be called the canonical basis for subSpaces Of L. If p 2 l the space Lp becomes a Banach Space with Schauder basis {uj : j E 2+} under the norm “A“ = ( El‘x(n)\p)1/p. If p < l, Lp can be topologies with the invariant metric oo d(11.12) = z \x1(n) - x2(n)\p . n=1 (4) Orlicz Spaces. We take as our basic reference Zaanen [21]. Definition. If the non-decreasing functions v = ¢(u) and u = w(v) are inverse to each other then the functions on [0,m) 41 x x q(x) = £¢(x)dx and a(x) = g¢(x)dx are termed complementary in the sense of Young. We know that x,y S a(x) + a(y), a most useful inequality. We shall assume that there exists an M such that a(2x) S Ma(x) and 5(2x) S MB(x) for all x > 0. We define La to be [2 La={XEL: Ea(\x(j)\)y\ : fawn-n s 1} . j=1 j=1 Under this norm. LG is a Banach space with dual L . {uj : j E Z+] will be a Schauder basis for La since m “x - finxua = n§1a(\x(j)\) a 0 . It is sometimes Of interest to know when an element 2 E L is an element Of La, and the uniform boundedness principle establishes a useful criteria. m Criteria l. 2 E L is in La if for all y E L; the Sum j21\y(j)z(j)| is finite. The function a is said to satisfy condition (2) if there exists a constant C such that for any Gaussian distribution v on R jq(\x\)dv(x) s c a({Ixzdv(x)}%) REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) REFERENCES Day, M.M. Normed Linear Spaces. Springer-Verlag, Berlin (1962). De Acosta, A.D. (1970). Existence and convergence Of probability measures on Banach Spaces. Trans. Amer. Math. Soc., 273-298. Dudley, R.M. (1969). Random linear functionals. Trans. Amer. Math. Soc. 136, 1-24. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley & Sons, New York (1970). Fernique, X. Intégrabilité des vecteurs Gaussians. C.R. Acad. Sci. Paris, Ser. A-B 2199 A1698-A1699. Gross, L. (1963). Harmonic Analysis on Hilbert Space. Mem. Amer. Math. Soc. 46 . Ito, Kiyosi and Nisio, Makiko (1968). On the convergence Of sums of independent Banach Space valued random variables. Osaka J. Math. 5, 35-48. Kuelbs, J. Fourier analysis on metric Spaces. 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