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October 2000 Precise asymptotics in the law of the iterated logarithm
Allan Gut, Aurel Spătaru
Ann. Probab. 28(4): 1870-1883 (October 2000). DOI: 10.1214/aop/1019160511

Abstract

Let $ X, X_1, X_2 \ldots$ be i.i.d. random variables with mean 0 and positive, finite variance $\sigma^2$, and set $S_n = X_1 + \cdots + X_n, n \geq 1$. Continuing earlier work related to strong laws, we prove the following analogs for the law of the iterated logarithm: $$\lim_{\varepsilon\searrow\sigma\sqrt{2}}\sqrt{\varepsilon^2−2\sigma^2}\sum_{n\ge3}\frac{1}{n}P(|S_n|\ge\varepsilon\sqrt{n\log\log n}+a_n)=\sigma\sqrt{2}$$ whenever $a_n = O(\sqrt{n}(\log \log n)^{-\gamma})$ for some $\gamma \geq 1/2$ (assuming slightly more than finite variance), and $$\lim_{\varepsilon\searrow 0}\varepsilon^2\sum_{n\ge3}\frac{1}{n \log n}P(|S_n|\ge\varepsilon\sqrt{n\log\log n})=\sigma^{2}.$$

Citation

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Allan Gut. Aurel Spătaru. "Precise asymptotics in the law of the iterated logarithm." Ann. Probab. 28 (4) 1870 - 1883, October 2000. https://doi.org/10.1214/aop/1019160511

Information

Published: October 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60024
MathSciNet: MR1813846
Digital Object Identifier: 10.1214/aop/1019160511

Subjects:
Primary: 60G50
Secondary: 60E15 , 60F15

Keywords: Davis law , Fuk–Nagaev type inequality , Law of the iterated logarithm , Tail probabilities of sums of i.i.d.random variables,

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 4 • October 2000
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