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

摘要

Let X-1, X-2,...,X-2n 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 = Sigma(i=1)(j) X-i, then P(sup(1 less than or equal to j less than or equal to 2n)parallel to S-j parallel to>lambda) less than or equal to cP(parallel to S-n parallel to>lambda/c), for all lambda greater than or equal to 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 X-1, X-2,... 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. [References: 11]