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.
Citation
Andrew Rosalsky. Henry Teicher. "A Limit Theorem for Double Arrays." Ann. Probab. 9 (3) 460 - 467, June, 1981. https://doi.org/10.1214/aop/1176994418
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