A Comparison Inequality for Sums of Independent Random Variables

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 X1 = XI , and let X2 · · · Xn be independent identically distributed copies of X1. Then, P X1 +· · ·+Xn ≥ λ ≤ cP X1 +· · ·+ Xn ≥ λ/c for all λ ≥ 0, where c is an absolute constant.