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

Abstract

Consider a sequence of positive integers \(\{k_n,n\ge 1\}\), and an array of nonnegative real numbers \(\{a_{n,i},1\le i\le k_n,n\ge 1\}\) 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 Hung 71(4):327–336, 1996). As a by-product, a considerable extension of a recent result of Boukhari (J Theor Probab, 2021. https://doi.org/10.1007/s10959-021-01120-6) 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.