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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A remark on the strong law of large numbers


Author: Robert Chen
Journal: Proc. Amer. Math. Soc. 61 (1976), 112-116
MSC: Primary 60F15
DOI: https://doi.org/10.1090/S0002-9939-1976-0420802-1
MathSciNet review: 0420802
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Abstract: Let $ {X_1},{X_2}, \ldots $ be mutually independent random variables such that $ E({X_n}) = 0$ and $ E(X_n^2) = \sigma _n^2 = 1$ for all $ n = 1,2, \ldots $. For each $ n = 1,2, \ldots $ let $ {S_n} = \sum\nolimits_{j = 1}^n {{X_j}} $; then, by the Kolmogorov criterion for mutually independent random variables, $ {S_n}/{n^{1/2 + \alpha }} \to 0$ almost surely as $ n \to \infty $ for any positive constant $ \alpha $. A deeper understanding of this theorem will be facilitated if we know the order of magnitude of $ E\{ {N_\infty }(\alpha ,\varepsilon )\} $ as $ \varepsilon \to {0^ + }$, where $ {N_\infty }(\alpha ,\varepsilon )$ is the integer-valued random variable defined by $ {N_\infty }(\alpha ,\varepsilon ) = \sum\nolimits_{n = 1}^\infty {{\chi _{(\varepsilon {n^{1/2 + \alpha }},\infty )}}(\vert{S_n}\vert)} $. The present note does the work for a wide class of random variables by using Esseen's theorem and Katz-Petrov's theorem.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0420802-1
Keywords: Esseen's theorem, Euler-Maclaurin sum formula, Katz-Petrov's theorem, Kolmogorov's criterion, mutually independent random variables, strong law of large numbers
Article copyright: © Copyright 1976 American Mathematical Society