The Annals of Probability

The Lower Limit of a Normalized Random Walk

Cun-Hui Zhang

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

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