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

TL;DR

This paper introduces a new stochastic domination concept, applies it to strengthen laws of large numbers, and extends recent results with novel methods, even for independent summands.

Contribution

It develops the concept of _{n,i}-stochastic domination and uses it to improve existing laws of large numbers, extending prior results and exploring related integrability concepts.

Findings

Removed an assumption in a strong law of large numbers.

Extended a recent result by Boukhari with a different proof.

Established relationships between stochastic domination and uniform integrability.

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.