Some results concerning ideal and classical uniform integrability and mean convergence

Abstract

In this article, the concept of ${\mathcal {J}}$-uniform integrability of a sequence of random variables $\left\{ X_{k}\right\}$ with respect to $\left\{ a_{nk} \right\}$ is introduced where ${\mathcal {J}}$ is a non-trivial ideal of subsets of the set of positive integers and $\left\{ a_{nk} \right\}$ is an array of real numbers. We show that this concept is weaker than the concept of $\left\{ X_{k} \right\}$ being uniformly integrable with respect to $\left\{ a_{nk} \right\}$ and is more general than the concept of B-statistical uniform integrability with respect to $\left\{ a_{nk} \right\}$. We give two characterizations of ${\mathcal {J}}$-uniform integrability with respect to $\left\{ a_{nk} \right\}$. One of them is a de La Vallée Poussin type characterization. For a sequence of pairwise independent random variables $\left\{ X_{k} \right\}$ which is ${\mathcal {J}}$-uniformly integrable with respect to $\left\{ a_{nk} \right\}$, a law of large numbers with mean ideal convergence is proved. We also obtain a version without the pairwise independence assumption by strengthening other conditions. Supplements to the classical Mean Convergence Criterion are also established.