The Annals of Probability

The Other Law of the Iterated Logarithm

Naresh C. Jain and William E. Pruitt

Full-text: Open access


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

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

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Lim inf maxima partial sums lower bounds


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.

Export citation