A Limit Theorem for Double Arrays

Abstract

The main result establishes that row sums $S_n$ of a double array of rowwise independent, infinitesimal (or merely uniformly asymptotically constant) random variables satisfying $\lim \sup |S_n - M_n| \leq M_0 < \infty$ a.c. (for some choice of constants $M_n$), obey a weak law of large numbers, i.e., $S_n - \operatorname{med} S_n$ converges in probability to 0. No moment assumptions are imposed on the individual summands and zero-one laws are unavailable. As special cases, a new result for weighted i.i.d. random variables and a result of Kesten are obtained.