Improved convergence rates for tail probabilities

Abstract

Let X₁, X₂, ... be i.i.d. random variables, and put S_n = X₁+...+X_n. We find necessary and sufficient moment conditions for ∫_δ^∞f(x)dx < ∞, δ > α, where α ≥ 0 and f(x) = ∑_na_n P(|S_n| > xb_n) with a_n > 0 and b_n is either n^1/p, 0 < p < 2, (nlogn)^1/2 or (nloglogn)^1/2. The series f(x) we deal with are classical series studied by Hsu and Robbins, Erd˝os, Spitzer, Baum and Katz, Davis, Lai, Gut, etc.