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

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 $c$ such that if $S_j=\sum _{i=1}^j X_i$, then \[ P\bigl (\sup _{1\le j\le 2n} \| S_j \| > \lambda \bigr ) \le c P(\| S_n \| > \lambda /c), \] for all $\lambda \ge 0$. This generalizes an inequality of Montgomery-Smith and Latała 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.