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December, 1975 The Other Law of the Iterated Logarithm
Naresh C. Jain, William E. Pruitt
Ann. Probab. 3(6): 1046-1049 (December, 1975). DOI: 10.1214/aop/1176996232


Let $\{X_n\}$ be a sequence of independent, identically distributed random variables with $EX_1 = 0, EX_1^2 = 1$. Define $S_n = X_1 + \cdots + X_n$, and $A_n = \max_{1\leqq k\leqq n} |S_k|$. We prove that $\lim \inf A_n(n/\log \log n)^{-\frac{1}{2}} = \pi/8^{\frac{1}{2}}$ with probability one. This result was proved by Chung under the assumption of a finite third moment and under progressively weaker moment assumptions by Pakshirajan, Breiman, and Wichura. Chung posed the problem of whether it is enough to assume only the finiteness of the second moment in his review of Pakshirajan's paper in 1961. We showed earlier that $(n/\log \log n)^{\frac{1}{2}}$ is the correct normalization but were unable to show that the constant is necessarily $\pi/8^{\frac{1}{2}}$.


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Naresh C. Jain. William E. Pruitt. "The Other Law of the Iterated Logarithm." Ann. Probab. 3 (6) 1046 - 1049, December, 1975.


Published: December, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0319.60031
MathSciNet: MR397845
Digital Object Identifier: 10.1214/aop/1176996232

Primary: 60G50
Secondary: 60F15

Keywords: lim inf , lower bounds , Maxima , partial sums

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 6 • December, 1975
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