Some results concerning ideal and classical uniform integrability and mean convergence

• Al Hayek, Nour ^[1] ; Ordóñez Cabrera, Manuel ^[2] ; Rosalsky, Andrew ^[3] ; Ünver, Mehmet ^[4] ; Volodin, Andrei ^[5]
  1. [1] Kazan Federal University

     Kazan Federal University

     Rusia

     
  2. [2] Universidad de Sevilla

     Universidad de Sevilla

     Sevilla, España

     
  3. [3] University of Florida

     University of Florida

     Estados Unidos

     
  4. [4] Ankara University

     Ankara University

     Turquía

     
  5. [5] University of Regina

     University of Regina

     Canadá

     

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• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 1, 2023, págs. 1-25
• Idioma: inglés
• DOI: 10.1007/s13348-021-00334-5
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.

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