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 Funktsional. Anal. i Prilozhen., 2011, Volume 45, Issue 1, Pages 41–55 (Mi faa3026)

Towards Nikishin's Theorem on the Almost Sure Convergence of Rearrangements of Functional Series

Sh. Leventala, V. S. Mandrekara, S. A. Chobanyanb

a Michigan State University
b Muskhelishvili Institute of Computational Mathematics

Abstract: Necessary and sufficient conditions are found for the almost sure convergence of almost all simple rearrangements of a series of Banach space valued random variables. The results go back to Nikishin's well-known theorem on the existence of an almost surely convergent rearrangement of a numerical random series. An example is also given of a numerical random series with general term tending to zero almost surely such that this series converges in probability and any its rearrangement diverges almost surely.

Keywords: rearrangement of a series in a Banach space, almost sure convergence, ${\mathbf k}$-simple permutation, Nikishin's theorem

DOI: https://doi.org/10.4213/faa3026

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English version:
Functional Analysis and Its Applications, 2011, 45:1, 33–45

Bibliographic databases:

UDC: 519.2+517.51+517.98

Citation: Sh. Levental, V. S. Mandrekar, S. A. Chobanyan, “Towards Nikishin's Theorem on the Almost Sure Convergence of Rearrangements of Functional Series”, Funktsional. Anal. i Prilozhen., 45:1 (2011), 41–55; Funct. Anal. Appl., 45:1 (2011), 33–45

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3026
• https://doi.org/10.4213/faa3026
• http://mi.mathnet.ru/eng/faa/v45/i1/p41

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Chobanyan S., Leyental S., Mandrekar V., “Almost surely convergent summands of a random sum”, Statist. Probab. Lett., 82:1 (2012), 212–216
2. Chobanyan S., Levental S., “Contraction principle for tail probabilities of sums of exchangeable random vectors with multipliers”, Statist. Probab. Lett., 83:7 (2013), 1720–1724
3. S. Chobanyan, G. Giorgobiani, V. Kvaratskhelia, Sh. Levental, V. Tarieladze, “On rearrangement theorems in Banach spaces”, Georgian Math. J., 21:2 (2014), 157–163
4. Theory Probab. Appl., 59:4 (2015), 677–684
5. Charatonik W.J., Samulewicz A., Witula R., “Limit sets in normed linear spaces”, Colloq. Math., 147:1 (2017), 35–42
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