On the concept of B-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense

Manuel Ordóñez Cabrera, Andrew Rosalsky (), Mehmet Ünver and Andrei Volodin
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Manuel Ordóñez Cabrera: University of Sevilla
Andrew Rosalsky: University of Florida
Mehmet Ünver: Ankara University
Andrei Volodin: University of Regina

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2021, vol. 30, issue 1, No 7, 83-102

Abstract: Abstract In this correspondence, for a nonnegative regular summability matrix B and an array $$\left\{ a_{nk}\right\} $$ a nk of real numbers, the concept of B-statistical uniform integrability of a sequence of random variables $$\left\{ X_{k}\right\} $$ X k with respect to $$\left\{ a_{nk}\right\} $$ a nk is introduced. This concept is more general and weaker than the concept of $$\left\{ X_{k}\right\} $$ X k being uniformly integrable with respect to $$\left\{ a_{nk}\right\} $$ a nk . Two characterizations of B-statistical uniform integrability with respect to $$\left\{ a_{nk}\right\} $$ a nk are established, one of which is a de La Vallée Poussin-type characterization. For a sequence of pairwise independent random variables $$\left\{ X_{k}\right\} $$ X k which is B-statistically uniformly integrable with respect to $$\left\{ a_{nk}\right\} $$ a nk , a law of large numbers with mean convergence in the statistical sense is presented for $$\sum \nolimits _{k=1}^{\infty }a_{nk}(X_{k}-\mathbb {E}X_{k})$$ ∑ k = 1 ∞ a nk ( X k - E X k ) as $$n\rightarrow \infty $$ n → ∞ . A version is obtained without the pairwise independence assumption by strengthening other conditions.

Keywords: Sequence of random variables; Uniform integrability; Statistical convergence; 40A35; 60F25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s11749-020-00706-2

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