Abstract
In this correspondence, for a nonnegative regular summability matrix B and an array \(\left\{ a_{nk}\right\} \) of real numbers, the concept of B-statistical uniform integrability of a sequence of random variables \(\left\{ X_{k}\right\} \) with respect to \(\left\{ a_{nk}\right\} \) is introduced. This concept is more general and weaker than the concept of \(\left\{ X_{k}\right\} \) being uniformly integrable with respect to \(\left\{ a_{nk}\right\} \). Two characterizations of B-statistical uniform integrability with respect to \(\left\{ a_{nk}\right\} \) are established, one of which is a de La Vallée Poussin-type characterization. For a sequence of pairwise independent random variables \(\left\{ X_{k}\right\} \) which is B-statistically uniformly integrable with respect to \(\left\{ a_{nk}\right\} \), a law of large numbers with mean convergence in the statistical sense is presented for \(\sum \nolimits _{k=1}^{\infty }a_{nk}(X_{k}-\mathbb {E}X_{k})\) as \(n\rightarrow \infty \). A version is obtained without the pairwise independence assumption by strengthening other conditions.
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22 November 2021
A Correction to this paper has been published: https://doi.org/10.1007/s11749-021-00793-9
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The authors are grateful to the reviewers for carefully reading the manuscript and for offering substantial comments and suggestions which enabled them to improve the presentation.
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The research of Manuel Ordóñez Cabrera has been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The research of Mehmet Ünver was done while he was visiting University of Regina, Canada, and the research has been supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) Grant 1059B191800534. The research of Anderi Volodin has been partially supported by a Natural Sciences and Engineering Research Council of Canada Grant.
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Ordóñez Cabrera, M., Rosalsky, A., Ünver, M. et al. 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. TEST 30, 83–102 (2021). https://doi.org/10.1007/s11749-020-00706-2
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DOI: https://doi.org/10.1007/s11749-020-00706-2