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Proceedings of the American Mathematical Society

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

 

 

The Erdős-Rényi new law of large numbers for weighted sums


Author: Stephen A. Book
Journal: Proc. Amer. Math. Soc. 38 (1973), 165-171
MSC: Primary 60F15
DOI: https://doi.org/10.1090/S0002-9939-1973-0310946-4
MathSciNet review: 0310946
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Abstract: From the partial sums $ {S_n}$ of the first $ N$ random variables of a sequence of independent, identically distributed random variables, $ N - K + 1$ averages of the form $ {K^{ - 1}}({S_{n + K}} - {S_n})$ can be constructed, one such average for each $ n$ between 0 and $ N - K$, inclusive. If we denote by $ \Sigma (N,K)$ the maximum of those $ N - K + 1$ averages, then for a wide range of numbers $ \lambda $, Erdös and Rényi (1970) proved that, as $ N \to \infty ,\Sigma (N,K) \to \lambda $ a.e. for $ K = [C(\lambda )\log N]$, where $ C(\lambda )$ is a constant depending only on $ \lambda $, not on $ N$. The objective of the present article is to extend the Erdös-Rényi theorem to the case of weighted sums. The main theorem bears a relation to the law of large numbers for weighted sums of Jamison, Orey, and Pruitt (1965) similar to the one borne by the Erdös-Rényi theorem to the ordinary strong law of large numbers.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0310946-4
Keywords: Strong limit theorems, laws of large numbers, large deviations, weighted averages
Article copyright: © Copyright 1973 American Mathematical Society