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. (tex, dvi, ps, pdf, actual article.)

 

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