Resumen de Some results concerning ideal and classical uniform integrability and mean convergence

Nour Al Hayek, Manuel Hilario Ordóñez Cabrera , Andrew Rosalsky, Mehmet Ünver, Andrei Volodin

• 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.