The Annals of Probability

The Other Law of the Iterated Logarithm

Naresh C. Jain and William E. Pruitt

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Abstract

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 $\pi/8^{\frac{1}{2}}$.

Article information

Source
Ann. Probab., Volume 3, Number 6 (1975), 1046-1049.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996232

Digital Object Identifier
doi:10.1214/aop/1176996232

Mathematical Reviews number (MathSciNet)
MR397845

Zentralblatt MATH identifier
0319.60031

JSTOR
links.jstor.org

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F15: Strong theorems

Keywords
Lim inf maxima partial sums lower bounds

Citation

Jain, Naresh C.; Pruitt, William E. The Other Law of the Iterated Logarithm. Ann. Probab. 3 (1975), no. 6, 1046--1049. doi:10.1214/aop/1176996232. https://projecteuclid.org/euclid.aop/1176996232


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