On a new concept of stochastic domination and the laws of large numbers

Abstract

Consider a sequence of positive integers $\{k_n,n\ge1\}$, and an array of nonnegative real numbers $\{a_{n,i},1\le i\le k_n,n\ge1\}$ satisfying $\sup_{n\ge 1}\sum_{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty).$ This paper introduces the concept of $\{a_{n,i}\}$-stochastic domination. We develop some techniques concerning this concept and apply them to remove an assumption in a strong law of large numbers of Chandra and Ghosal [Acta. Math. Hungarica, 1996]. As a by-product, a considerable extension of a recent result of Boukhari [J. Theoret. Probab., 2021] is established and proved by a different method. The results on laws of large numbers are new even when the summands are independent. Relationships between the concept of $\{a_{n,i}\}$-stochastic domination and the concept of $\{a_{n,i}\}$-uniform integrability are presented. Two open problems are also discussed.