A Comparison Inequality for Sums of Independent Random Variables

Abstract

We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X1,…,Xn be independent Banach-valued random variables. Let I be a random variable independent of X1,…,Xn and uniformly distributed over {1,…,n}. Put X̃1=XI, and let X̃2,…,X̃n be independent identically distributed copies of X̃1. Then, P(‖X1+···+Xn‖≥λ)≤cP(‖X̃1+···+X̃n‖≥λ/c) for all λ≥0, where c is an absolute constant.