## The Annals of Probability

- Ann. Probab.
- Volume 14, Number 2 (1986), 560-581.

### The Lower Limit of a Normalized Random Walk

#### Abstract

Let $\{S_n\}$ be a random walk with underlying distribution function $F(x)$ and $\{\gamma_n\}$ be a sequence of constants such that $\gamma_n/n$ is nondecreasing. A universal integral test is given which determines the lower limit of $S_n/\gamma_n$ up to a constant scale for $\lim \sup \gamma_{2n}/\gamma_n < \infty$. The generalized LIL is obtained which contains the main result of Fristedt-Pruitt (1971). The rapidly growing random walks and the limit points of $\{S_n/\gamma_n\}$ are also studied.

#### Article information

**Source**

Ann. Probab., Volume 14, Number 2 (1986), 560-581.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992531

**Digital Object Identifier**

doi:10.1214/aop/1176992531

**Mathematical Reviews number (MathSciNet)**

MR832024

**Zentralblatt MATH identifier**

0603.60065

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60J15 60F15: Strong theorems 60F20: Zero-one laws

**Keywords**

Normalized random walks lower limits generalized law of the iterated logarithm exponential bounds truncated moments

#### Citation

Zhang, Cun-Hui. The Lower Limit of a Normalized Random Walk. Ann. Probab. 14 (1986), no. 2, 560--581. doi:10.1214/aop/1176992531. https://projecteuclid.org/euclid.aop/1176992531