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Mat. Zametki, 2009, Volume 86, Issue 6, Pages 925–937 (Mi mz6617)  

This article is cited in 2 scientific papers (total in 2 papers)

A Generalization of the Menshov–Rademacher Theorem

P. A. Yaskov

M. V. Lomonosov Moscow State University

Abstract: For a sequence $\{X_n\}_{n\ge1}$ of random variables with finite second moment and a sequence $\{b_n\}_{n\ge1}$ of positive constants, new sufficient conditions for the almost sure convergence of $\sum_{n\ge1}X_n/b_n$ are obtained and the strong law of large numbers, which states that $\lim_{n\to\infty}\sum_{k=1}^nX_k/b_n=0$ almost surely, is proved. The results are shown to be optimal in a number of cases. In the theorems, assumptions have the form of conditions on $\rho_n=\sup_k(\mathsf EX_kX_{k+n})^+$,
$$r_n=\sup_k\frac{(\mathsf EX_kX_{k+n})^+}{(\mathsf EX_k^2)^{1/2}(\mathsf EX_{k+n}^2)^{1/2}},$$
$\mathsf EX_n^2$, and $b_n$, where $x^+=x\vee0$ and $n\in\mathbb N$.

Keywords: strong law of large numbers, random variable, second moment, almost sure convergence, Menshov–Rademacher theorem, Kolmogorov's 0–1 law

DOI: https://doi.org/10.4213/mzm6617

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English version:
Mathematical Notes, 2009, 86:6, 861–872

Bibliographic databases:

UDC: 519.21
Received: 20.12.2008
Revised: 06.04.2009

Citation: P. A. Yaskov, “A Generalization of the Menshov–Rademacher Theorem”, Mat. Zametki, 86:6 (2009), 925–937; Math. Notes, 86:6 (2009), 861–872

Citation in format AMSBIB
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\paper A Generalization of the Menshov--Rademacher Theorem
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\pages 925--937
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Korchevskii V.M., “Ob usilennom zakone bolshikh chisel dlya posledovatelnosti sluchainykh velichin bez predpolozheniya o nezavisimosti”, Vestnik Sankt-Peterburgskogo universiteta. Seriya 1: Matematika. Mekhanika. Astronomiya, 2011, no. 4, 38–41  elib
    2. V. M. Korchevsky, “On the strong law of large numbers for sequences of dependent random variables with finite second moments”, J. Math. Sci. (N. Y.), 206:2 (2015), 197–206  mathnet  crossref
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