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 X_1,…,X_n be independent Banach-valued random variables. Let I be a random variable independent of X_1,…,X_n and uniformly distributed over {1,…,n}. Put Z_1 = X_I, and let Z_2,…,Z_n be independent identically distributed copies of Z_1. Then, P(||X_1+…+X_n|| > t) < c P(||Z_1+…+Z_n|| > t/c), for all t>0, where c is an absolute constant.