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 {\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.