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.
