# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## An extension of the Erdős-Rényi new law of large numbersHTML articles powered by AMS MathViewer

by Stephen A. Book
Proc. Amer. Math. Soc. 48 (1975), 438-446 Request permission

## 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 $E(|{X_1}{|^r}) < \infty$ 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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