Some results concerning ideal and classical uniform integrability and mean convergence

In this article, the concept of J-uniform integrability of a sequence of random variables { Xk} with respect to { ank} is introduced where J is a non-trivial ideal of subsets of the set of positive integers and { ank} is an array of real numbers. We show that this concept is weaker than the concept of { Xk} being uniformly integrable with respect to { ank} and is more general than the concept of B-statistical uniform integrability with respect to { ank}. We give two characterizations of J-uniform integrability with respect to { ank}. One of them is a de La Vallée Poussin type characterization. For a sequence of pairwise independent random variables { Xk} which is J-uniformly integrable with respect to { ank} , 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