Lecture Notes ni Mathematics Edited yb .A Dold dna .B nnamkcE 709 Probability ni Banach Spaces II Proceedings of the Second International Conference on Probability ni Banach Spaces, 18-24 June 1978, Oberwolfach, Germany Edited yb .A Beck galreV-regnirpS nilreB Heidelberg New kroY 1979 Editor Anatole Beck Department of Mathematics University of Wisconsin Madison, Wl 53706 USA AMS Subject Classifications (1970): 28-XX, 28A40, 28A45, 46AXX, 46A05, 46 B10, 60B10, 60B99, 60F05, 60F99 ISBN 3-540-09242-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09242-0 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction yb photocopying machine or similar means, dna storage ni data banks. Under § 54 of the German Copyright waL where copies era made for other than private use, a fee is payable to the publisher, the amount of the fee to eb determined yb agreement with the publisher © yb Springer-Verlag Berlin Heidelberg 9791 Printed ni Germany Printing dna binding: Beltz Offsetdruck, Hemsbach/Bergstr. 0413/1412 012345- Introduction The subject of Probability in Banach Spaces has now passed the quarter-century mark and is flourishing. It is over twenty years since the first paper linking geo- metric theory and probabilistic results, and nearly twenty since the first charateri- zation of a probabilistically signifficant category of spaces by convexity conditions. Since that time, the subject has continued to grow in this direction. Theorems shown to hold in uniformly convex spaces were strengthened to show that they held either exactly in B-convex spaces or exactly in super-reflexive spaces. Some of the Hilbert- space theorems were shown to hold exactly in type-2 spaces, etc. The vision of proba- bility as being basically a subject grounded equally in geometry and in measure theory continues to be reinforced by the absence of any theorems which hold only in the real or complex numbers (except, trivially, for those which involve the multiplication of random variables). A few theorems have been found which hold exactly in finite- dimensional spaces, one for amarts and one for the strong law of large numbers, but these are not especially interesting in their own rights. Mostly, the theorems about real or complex numbers seem to flow from the fact ~hat these are Hilbert spaces. With this volume, Banach-space valued probability invades ever deeper into the territory of classical probability theory. The subject is now being accepted as being central to an understanding of many of tNe fundamental theorems there. It has truly come of age. Anatole Beck, Editor CONTENTS Ale~andro de Acosta: On the general converse central limit theorem in Banach spaces ......................................... I A. Arat~o and Stable processes with continuous sample paths ......... 9 .B Marcus: Charles R. Baker: Conditions for absolute continuity .................... 33 A. Bellow and A characterization of almost sure convergence ......... 45 A. Dvoretzky: Ren@ Carmona: Banach space valued Gausslan processes ................ 67 S.D. Chatterji: The Radon-Nikodym property ............................ 75 Manfred Denker and On B-convex 0rlicz spaces ............................. 87 Rainer Kombrink: R.M. DudleF: Lower layers in R] 2 and convex sets in 3RI are not GB classes ............................................ 97 E. Flytzanis: Invariant measures for linear operators ............... 103 Evarist Gin&, On sums of independent random variables with values V. Mandrekar, and Joel Zinn: in Lp(2~p<~) ........................................... 111 Mar~orie G. Hahn: The generalized domain of attraction of a Gaussian law on Hilbert space ...................................... 125 B. Heinkel: Relation between central-limit theorem and law of the iterated logarithm in Banach spaces ................... 145 J. Kuelbs: Rates of growth for Banach space valued independent increment processes ................................... 151 Walter Philipp: Almost sure invariance principles for sums of B-valued random variables ............................. 171 Hiroshi Sato: Hilbertian support of a probability measure on a Banach space .......................................... 195 ON THE GENERAL CONVERSE CENTRAL LIMIT THEOREM IN BANACH SPACES by Alejandro de Acosta Instituto Venezolano de Investigaciones Cient~ficas i. Introduction. Let B be a separable Banach space, {X .} a row-wise nj independent infinitesimal triangular array of B-valued random vectors, S =E.X . In a recent paper, de Acosta, Araujo and Gin~ [i] have ob- n j nj tained a general converse central limit theorem -necessary conditions for the weak convergence of {L(S )}-, valid for arbitrary B. In the n present work we refine this result and obtain considerably more infor- mation on the limiting behavior of {L(S )}. n Our main result (Theorem 3.3) may be roughly stated as follows. Let {X .} be an infinitesimal triangular array of B-valued random vectors nj and assume that {L(Sn) } converges weakly. If {Ak:k=0,...,m } are ap- propriately chosen disjoint subsets of B, X is the truncation of nj ,k Xnj at k A and Sn;k-E - X j nj ,k' then {L(s n;O' ... ' S n;m )} converges weakly Bm+l to a completely specified product measure on . In less formal terms, the random vectors {S k:k=O ...,m} are "asymptotically inde- n; pendent" We remark that some of the basic results of the theory of B-valued stochastic processes with independent increments (see [2], Chapter IV, Theorem 5) follow easily from Theorems 3.3 and 4.1 of the present pa- per (together with some background results contained in [i] ). We shall not carry out the details here. * Research partially supported by CONICIT (Venezuela) Grant 51-26.SI.0893. 2. Preliminaries. We refer to i for the background of definitions and theorems which we will use. If X is a B-valued random vector (r. v.) and A is a Borel subset of B, we define XA, the truncation of X at A, by XA=XI{xcA }. We will consider truncations of a r.v. at disjoint sets {Ak:k=O,...,m} ; in t~is context we shall write X k instead of XAk. If {Xnj} is a triangular array, we will denote the truncation of Xnj at A k by Xnj,k; the corresponding row sum will be written Sn;kEjXnj,k. 3. Limiting behavior of the row sums of truncated random vectors. Our first result is proved by a classical method: a comparison argument showing that the difference between two sequences of characteristic functions tends to zero. Proposition 3.1. Let {X .} be an infinitesimal triangular array such n3 that SUPnEjP{llXnjll>6}<~ for some 6>0. Let {Ak:k=l ..... m} be disjoint Borel sets in B, A kC B6, c and let A0=(U~=IAk)C . Then for every f ,...,fm~B ' , 0 m limnlfexp{i~k=Ofk(Xk)}dL(Sn;0 .... ,S n;m) (Xo,... ,Xm) - m - Hk=o/eXp{ifk(X)}dL(Sn;k)(X) I = O. Lemma 3.2. Let a k be complex numbers, lakl~l (k=O,...,m). Then m m m-i m l ak_ll la _I I IZk=Oak-m-~k=Oakl~E~=OEk=~+l m + m _Em-l~m ak_~k= m } Proof. Kk=Oak m-Ek=Oa k- ~=0 ~ k=~ ~+lak-a~ +I m-1. m i) i) =E =0(~k=~+lak - (a - . Since l~k=~+lak m -ll<Em- k=%+lJak-ll , we have m m m-i m IHk=Oak+m-~k=Oakl!~=OEk=~+llak-iI la~-ll • Proof of Proposition 3.1. Let L be the quantity the limit of which n is taken in the statement of the proposition. Then Ln=iEexp{iEk=Ofkm (Sn; k)} _ ~=oEexp{ifk(Sn;k)} I ' m m • =IEexp{iEk=OEjfk(Xnj,k)} - ~k=oEexp{iEjfk(Xnj,k)} I =lIIjEexp{iE;=Ofk(Xnj ,k ) } - I~jHk=oEexp{ifk(Xnj,k)}l SZ iEexp{ i m f (Xnj } _ m j Ek=0 k ,k ) Hk=oEexp{ifk(Xnj,k )}I" Since A k and Ak, are disjoint for k#k', it follows that for k#k' fk(Xnj,k) • fk,(Xnj,k,) = O. A simple calculation then yields exp{" m f m iEk= 0 k(Xnj,k)} = Ek=oeXp{ifk(Xnj,k ) } - m for all n,j. Using this fact and Lemma 3.2, we have L !Z m-i m n jE£=0Ek=~+llEexp{ifk(Xnj,k)}-ll IEexp{if£(Xnj,£)}-ll m-iEm =Z~= 0 k=£+iEj IEexp{ifk(Xnj,k)}-ll IEexp{if£(Xnj,£)}-ll. Now for Oi£!m-l, the infinitesimality of {X }. implies that nj limnSUpj IEexp{if£(Xnj,£)}-i =0. Also, for isk!m we have E IEexp{ifk(Xnjj ,k ) }-II!2E.P{j XnjEA k} _<2EjP{IIXnjlI>6}. It follows that for 0~£!m-l, isk~m, Ej IEexp{ifk(Xnj ,k ) }-ii IEexp{if£(Xnj,~ ) }-i I <_ isupj I Eexp{if~(Xnj,%) }-iI~ j I Eexpi{ fk(Xnj ,k ) }-II 0~- as n-~, and therefore lim L =0. nn Remark. The particular case of Proposition 3.1 corresponding to m=l, A0=B6, AI=B ; may be used in ~2 of i instead of Learns 2.6 of i ; this provides an alternative proof of the "asymptotic independence" of S n~6 and S (6) (in the notation of El)without using the L~vy decomposition. n We recall some facts from i (Theorem 2.10). Let {X }. be an infi- nj nitesimal triangular array and assume that L(S -x ) converges weakly n n to v for some sequence {Xn}~B. Then there exist a L~vy measure p and a centered Gaussian measure y such that for every T>0, one has v=6 *y,c Pois~ for some z tB;v and y are uniquely determined. More- Z T T T over, for every TtC(~) (the set of continuity radii of ~),L(S -ES ) n H,T converges weakly to y,c Pois~J; here S =E X I T n,T j nj {XnjtB~}" Theorem 3.3. Let {X }. be an infinitesimal triangular array. Suppose nj /(Sn-Xn)÷ ~ for some sequence {x }CB and let ~ be the L~vy measure W n and y the centered Gaussian measure associated to v. Let ~tC(~), {Ak:k=l .... ,m} disjoint Borel sets such that AkCB $ and ~(BdAk)=0 for k=l,...,m, and let A =(U~ 1Ak )c Then 0 = /(Sn;0-ESn,T, ;nS I ..... Sn;m)+w (y*cTPois ~( I A ) )®Pois ~( I AI)®... ~Pois ~( I Am). Proof. By (i , Theorem 2.10) E L(X ) IAk>wVlAk(k=l .... m). By the • j nj Khinchine-Le Cam lemma i , Lemma 2.7), we have for the total varia- tion norm vll'll lI(Sn.k)-Pois(E L(X ) IAk) Hv=llL(Sn k)-Pois(E L(X ,k) vI , j nj ; j nj !2Zj (P{XnjtAk}) 2 ÷0 as n÷~. Therefore L(Sn;k)÷wPois(~IAk). Let T =S -S Then (Tn)÷wPois(uIA 0 c) (arguing as in the previous n n n;0" paragraph) and since S -ES =(S 0-ES )+T it follows that n n,T n; n,T n, {(Sn;0-ES )} is relatively compact and therefore so is H,T {(Sn,0-ESn,T. ,Tn)}. Now Theorem 2.2 of i and Proposition 3.1 imply that if % is any subsequential limit of {(S -ES )}, then n;O n,T y*cTPois~=%*Pois(~ A 0). It follows that %=y,cTPois(~IA ) and therefore 0 /(Sn;0-ESn, T)÷wY*CTPois (D IA 0 ) . It follows from the preceding paragraphs that L(S ;o-ESn )~L(s )~..eL(s )÷ n ,T n;l " n;m w AIM(sioPc*Y(w÷ ))~Pois(~AI)®...@Pois(uIA ). 0 m On the other hand ' {L(Sn;0-ESn,T, Sn;l'''" ,S n;m) } is relatively com- pact because each sequence of marginal probabilities is relatively com- pact; the conclusion follows now from Proposition 3.1. Corollary 3.4. Let ~=~ ,y,c Pois~ (with ~({0})=0) be an infinitely z T divisible measure on B. (i) If {X } is an infinitesimal triangular array such that n3 L(Sn-Xn)÷w V for some sequence {Xn}CB , A is an open subset of B and P{X tA}=0 for all n,j, then ~(A)=0. n3 (2) Conversely, if C is a closed subset of B such that c CcB 6 for some 6>0 and ~(C)=O, then there exists an infinite- simal triangular array {X }such that P{X ¢C}=0 for all n,j and n3 n3 L(S )÷ v. n w Proof. (i): Follows at once from i , Theorem 2.10. (2) Let {Y }. be an infinitesimal triangular array such that nj L(Tn)+w ~, where Tn-EjXnj;- say, take {Yn3:J=l,...,n} independent and /(Y )=~ , where ~ is the n-th convolution root of ~. n 3 n n Since ~(BdC)=0 and ~=NIC c, it follows from Theorem 3.3 that if we define Xnj=YnjI{Ynj~CC} , then {Xnj} satisfies the desired condition. Corollary 3.5. Let ~ be as in Corollary 3.4. Then ~ has bounded sup- port if and only if there exists an infinitesimal uniformly bounded triangular array {X }. such that L(S )+ ~. n3 n w 4. The limiting distribution of the number of visits. For a>O, let us denote by ~(a) the Poisson distribution with parameter a on R 1 (~(a) = 6 0 if a=O). Let A be a Borel set in B satisfying d(O,A)>0 and let ~(n) be the number of visits to A by the random vectors in the n-th row of the triangular array {Xnj}; that is, ~(n)=~jIA(Xnj). If the assumptions of the general converse central limit theorem (i , Theorem 2.10) are