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April, 1989 On a Problem of Csorgo and Revesz
Qi-Man Shao
Ann. Probab. 17(2): 809-812 (April, 1989). DOI: 10.1214/aop/1176991428

Abstract

Suppose $\{X_n\}$ is an i.i.d. sequence of random variables with mean 0, variance 1 and $S_n = \sum^n_{i = 1}X_i$. Let $0 < r < 1$. It is well known that $S_n - W(n) = O((\log n)^{1/r}) \mathrm{a.s}.$ when $Ee^{t_0|X_1|^r} < \infty$ for some $t_0 > 0$, where $\{W(t), t \geq 0\}$ is the standard Wiener process. We prove that $O((\log n)^{1/r})$ cannot be replaced by $o((\log n)^{1/r})$.

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Qi-Man Shao. "On a Problem of Csorgo and Revesz." Ann. Probab. 17 (2) 809 - 812, April, 1989. https://doi.org/10.1214/aop/1176991428

Information

Published: April, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0684.60020
MathSciNet: MR985391
Digital Object Identifier: 10.1214/aop/1176991428

Subjects:
Primary: 60F15

Keywords: increments of partial sums , invariance principle

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 2 • April, 1989
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