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 Z1 = XI, and let Z2,...,Zn be independent identically distributed copies of Z1. Then, P(||X1+...+Xn|| > t) < c P(||Z1+...+Zn|| > t/c), for all t>0, where c is an absolute constant.
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