  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.
