## The Annals of Probability

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

### The Other Law of the Iterated Logarithm

Naresh C. Jain and William E. Pruitt

#### 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