On the convergence of series of moments for row sums of random variables

Abstract

Given a triangular array {Xn,k, 1 ? k ? n, n ? 1} of random variables satisfying E|Xn,k|p < ? for some p ? 1 and sequences {bn}, {cn} of positive real numbers, weshall prove that ??,n=1 cnE [|?n,k=1 (Xn,k-EXn,k)|/bn-?]p+ < ?, where x+ = max(x,0). Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence.