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 (X) over tilde (1) = X-1, and let (X) over tilde (2),...,(X) over tilde (n) be independent identically distributed copies of (X) over tilde (1). Then, P(//X-1 +...+ X-n// greater than or equal to lambda) less than or equal to cP(//(X) over bar (1) +...+ (X) over bar (n)// greater than or equal to lambda /c) for all lambda greater than or equal to 0, where c is an absolute constant. (C) 2001 Academic Press.