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

Authors:	Alexander R Pruss

Institution:	Department of Philosophy, Pittsburgh, Pittsburgh, Pennsylvania 15260

Abstract:	Let $X_1,X_2,\dots,X_{2n}$ 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 such that if $S_j=\sum _{i=1}^j X_i$ , then $\begin{displaymath}P\bigl(\sup _{1\le j\le 2n} \\| S_j \\| > \lambda\bigr) \le c P(\\| S_n \\| > \lambda/c), \end{displaymath}$ for all $\lambda\ge 0$ . This generalizes an inequality of Montgomery-Smith and Lata{\l}a for independent and identically distributed random variables. Our maximal inequality is apparently new even if $X_1,X_2,\dotsc$ 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.

Keywords:	Sums of exchangeable random variables maximal inequalities

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