A maximal inequality for partial sums of finite exchangeable sequences of random variables

Abstract

Let X1, X2,... , X2. be a finite exchangeable sequence of Banach space valued random variables, i.e., a sequence such that all joint distributions are invariant under permutations of the variables. We prove that there is an absolute constant c such that if S j-j=1 X_, then P( sup JISjll > A) A/c), 1 0. This generalizes an inequality of Montgomery-Smith and Latala for independent and identically distributed random variables. Our maximal inequality is apparently new even if X1,X2,... is an infinite exchangeable sequence of random variables. As a corollary of our result, we obtain a comparison inequality for tail probabilities of sums of arbitrary random variables over random subsets of the indices. Montgomery-Smith [8] and Latala [7] have independently proved that if X1,-... Xn are independent and identically distributed Banach space valued random variables, then (1) P( sup ZX, > AZ A/C I 0 and 1 < k < n, where c is an absolute constant. It is obvious that this cannot hold for arbitrary independent random variables; as MontgomerySmith [8] notes, we need only let k -= n = 2, X1 1 and X2 -1 to see this. Levy's inequality says that (1) also holds for arbitrary independent symmetric random variables Xi (not necessarily identically distributed). For positive random variables, (1) is trivial, of course. A natural and much-studied extension of the concept of independent and identically distributed random variables is that of exchangeable random variables. We say that a finite sequence X1, ... , Xn of (not necessarily independent) random variables is exchangeable if the n-tuples (XI, ... , Xn) and (X(I), ... , Xr(n)) both have the same distribution whenever ir is a permutation of [n] If{,... ,n}. Evidently an exchangeable sequence of independent random variables is precisely a sequence of independent and identically distributed random variables. Received by the editors August 2, 1996 and, in revised form, December 2, 1996. 1991 Mathematics Subject Classification. Primary 60E15.