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January, 1986 Exact Convergence Rate in the Limit Theorems of Erdos-Renyi and Shepp
Paul Deheuvels, Luc Devroye, James Lynch
Ann. Probab. 14(1): 209-223 (January, 1986). DOI: 10.1214/aop/1176992623

Abstract

The original Erdos-Renyi theorem states that $U_n/(\alpha k) \rightarrow 1$ almost surely for a large class of distributions, where $U_n = \sup_{0\leq i \leq n - k} (S_{i+k} - S_i), S_i = X_1 + \cdots + X_i$ is a partial sum of i.i.d. random variables, $k = \kappa(n) = \lbrack c \log n\rbrack, c > 0$, and $\alpha > 0$ is a number depending only upon $c$ and the distribution of $X_1$. We prove that the $\lim \sup$ and the $\lim inf$ of $(U_n - \alpha k)/\log k$ are almost surely equal to $(2t^\ast)^{-1}$ and $-(2t^\ast)^{-1}$, respectively, where $t^\ast$ is another positive number depending only upon $c$ and the distribution of $X_1$. The same limits are obtained for the random variable $T_n = \sup_{1\leq i \leq n}(S_{i+\kappa(i)} - S_i)$ studied by Shepp.

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Paul Deheuvels. Luc Devroye. James Lynch. "Exact Convergence Rate in the Limit Theorems of Erdos-Renyi and Shepp." Ann. Probab. 14 (1) 209 - 223, January, 1986. https://doi.org/10.1214/aop/1176992623

Information

Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0595.60033
MathSciNet: MR815966
Digital Object Identifier: 10.1214/aop/1176992623

Subjects:
Primary: 60F15
Secondary: 60F10

Keywords: Erdos-Renyi laws , large deviations , Law of the iterated logarithm , laws of large numbers , moving averages

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 1 • January, 1986
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