**Stephen Montgomery-Smith and Alexander Pruss, A comparison inequality for sums of independent random variables.**
*J.M.A.A., ***254**, (2001), 35-42.
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 \(X_1,\dots,X_n\) be independent Banach-valued random variables. Let \(I\) be a random variable independent of \(X_1,\dots,X_n\) and uniformly distributed over \(\{ 1,\dots,n \}\). Put \(\tilde X_1=X_I\), and let \(\tilde X_2,\dots,\tilde X_n\) be independent identically distributed copies of \(\tilde X_1\). Then, \(P(\|X_1+\dots+X_n\| \ge \lambda)\le cP(\|\tilde X_1+\dots+\tilde X_n\|\ge \lambda/c)\) for all \(\lambda\ge 0\), where \(c\) is an absolute constant.
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