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February, 1978 On the Law of the Iterated Logarithm
R. J. Tomkins
Ann. Probab. 6(1): 162-168 (February, 1978). DOI: 10.1214/aop/1176995622


Let $X_1, X_2,\cdots$ be a sequence of independent random variables, each with zero mean and finite variance. Define $S_n = \sum^n_{i=1} X_i, s_n^2 = E(S_n^2), t_n^2 = 2 \log \log s_n^2$ and $\Lambda = \lim \sup_{n\rightarrow\infty} S_n/(s_n t_n)$. Suppose that $|X_n| \leqq c_n s_n$ a.s. for all $n$ and some real sequence $\{c_n\}$ and that $s_n \rightarrow \infty$, and let $\nu = \lim \sup_{n\rightarrow\infty} t_nc_n$. By Kolmogorov's law of the iterated logarithm, $\Lambda = 1$ if $\nu = 0$. Egorov proved that $0 \leqq \Lambda < \infty$, in every case. In this paper it will be shown that, if $\nu < \infty$, then $0 < \Lambda \leqq 1 + \sum^\infty_{j=3} \nu^{j-2}/j!$. A similar result for certain classes of unbounded random variables will also be presented.


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R. J. Tomkins. "On the Law of the Iterated Logarithm." Ann. Probab. 6 (1) 162 - 168, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0376.60032
MathSciNet: MR501300
Digital Object Identifier: 10.1214/aop/1176995622

Primary: 60F15
Secondary: 60G50

Keywords: Exponential bounds , Independent random variables , Law of the iterated logarithm

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
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