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An extension of the Erdős-Rényi new law of large numbers


Author: Stephen A. Book
Journal: Proc. Amer. Math. Soc. 48 (1975), 438-446
MSC: Primary 60F15
DOI: https://doi.org/10.1090/S0002-9939-1975-0380950-0
MathSciNet review: 0380950
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Abstract: If $ {S_n}$ is the $ n$th partial sum of a sequence of independent, identically distributed random variables $ {X_1},{X_2} \cdots $ such that $ E({X_1}) = 0$ and $ E(\exp (t{X_1})) < \infty $ for some nonempty interval of $ t$'s, then, for a wide range of positive numbers $ \lambda $, Erdös and Rényi (1970) showed that $ \Sigma (N,[C(\lambda )\log N])$ converges with probability one to $ \lambda $ as $ N \to \infty $, where $ \Sigma (N,K)$ is the maximum of the $ N - K + 1$ averages of the form $ {K^{ - 1}}({S_{n + K}} - {S_n})$ for $ 0 \leq n \leq N - K$, and $ C(\lambda )$ is a known constant depending on $ \lambda $ and the distribution of $ {X_1}$. The objective of the present article is to state and prove the Erdös-Rényi theorem for the $ N - K + 1$ ``averages'' of the form $ {K^{ - 1/r}}({S_{n + K}} - {S_n})$, where $ 1 < r < 2$. This form of the Erdös-Rényi theorem arises from the extended form of the strong law of large numbers which asserts that, if $ E(\vert{X_1}{\vert^r}) < \infty $ for some $ r,1 \leq r < 2$, and $ E({X_1}) = 0$, then $ {n^{ - 1/r}}{S_n}$ converges with probability one to 0 as $ n \to \infty $.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0380950-0
Keywords: Strong limit theorems, laws of large numbers, large deviations, moment-generating functions
Article copyright: © Copyright 1975 American Mathematical Society

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