Comparison of Sums of Independent Identically Distributed Random Variables

Let Sk be the k-th partial sum of Banach space valued independent identically distributed random variables. In this paper, we compare the tail distribution of ‖Sk‖ with that of ‖Sj‖, and deduce some tail distribution maximal inequalities. The main result of this paper was inspired by the inequality from [dP–M] that says that Pr(‖X1‖ > t) ≤ 5 Pr(‖X1 +X2‖ > t/2) whenever X1 and X2 are independent identically distributed. Such results for Lp (p ≥ 1) such as ‖X1‖p ≤ ‖X1 +X2‖p are straightforward, at least if X2 has zero expectation. This inequality is also obvious if either X1 is symmetric, or X1 is real valued positive. However, for arbritary random variables, this result is somewhat surprizing to the author. Note that the identically distributed assumption cannot be dropped, as one could take X1 = 1 and X2 = −1. In this paper, we prove a generalization to sums of arbritarily many independent identically distributed random variables. Note that all results in this paper are true for Banach space valued random variables. The author would like to thank Victor de la Peña for helpful conversations. Theorem 1. There exist universal constants c1 = 3 and c2 = 10 such that if X1, X2, . . . are independent identically distributed random variables, and if we set