Some results concerning ideal and classical uniform integrability and mean convergence

Al Hayek N., Ordonez Cabrera M., Rosalsky A., ÜNVER M., Volodin A.

COLLECTANEA MATHEMATICA, vol.74, no.1, pp.1-25, 2023 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 74 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.1007/s13348-021-00334-5
  • Journal Name: COLLECTANEA MATHEMATICA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.1-25
  • Keywords: Uniform integrability, Summability methods, Sequence of random variables, Weighted sums, Mean convergence, RANDOM-VARIABLES
  • Ankara University Affiliated: Yes

Abstract

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