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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 3, Pages 401–423 (Mi tvp2640)

Convergence and limit theorems for subsequences of random variables

V. F. Gaposhkin

Moscow

Abstract: It is shown that if $X_n$ $(n=1,2,…)$ are random variables and $X_n\to0$ weakly in $L_2(\Omega)$, $X_n^2\to1$ weakly in $L_1(\Omega)$ then there exists a subsequence $X_{n_k}$ which is equivalent to $\{Y_k\}$, and $\sum_1^na_kY_k$ is a martingale (see Lemma A).
This fact is used in the rest of the paper to prove some results about subsequences of random variables: in section 2 — convergence and the strong law of large numbers; in section 3 — the central limit theorem; in section 4 — the law of the iterated logarithm.

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English version:
Theory of Probability and its Applications, 1973, 17:3, 378–400

Bibliographic databases:

Citation: V. F. Gaposhkin, “Convergence and limit theorems for subsequences of random variables”, Teor. Veroyatnost. i Primenen., 17:3 (1972), 401–423; Theory Probab. Appl., 17:3 (1973), 378–400

Citation in format AMSBIB
\Bibitem{Gap72} \by V.~F.~Gaposhkin \paper Convergence and limit theorems for subsequences of random variables \jour Teor. Veroyatnost. i Primenen. \yr 1972 \vol 17 \issue 3 \pages 401--423 \mathnet{http://mi.mathnet.ru/tvp2640} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=310948} \zmath{https://zbmath.org/?q=an:0273.60010} \transl \jour Theory Probab. Appl. \yr 1973 \vol 17 \issue 3 \pages 378--400 \crossref{https://doi.org/10.1137/1117049} 

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• http://mi.mathnet.ru/eng/tvp/v17/i3/p401

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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. S. V. Astashkin, “Identification of subsystems “majorized” by the Rademacher system”, Math. Notes, 65:4 (1999), 407–417
2. S. V. Astashkin, “Rademacher functions in symmetric spaces”, Journal of Mathematical Sciences, 169:6 (2010), 725–886