## The Annals of Probability

### The Rate of Escape of Random Walk

William E. Pruitt

#### Abstract

Let $\{X_k\}$ be an i.i.d. sequence and define $S_n = X_1 + \cdots + X_n$. The problem is to determine for a given sequence $\{\beta_n\}$ whether $P\{|S_n| \leq \beta_n \mathrm{i.o.}\}$ is 0 or 1. A history of the problem is given along with two new results for the case when $P\{X_1 \geq 0\} = 1$: (a) An integral test that solves the problem in case the summands satisfy Feller's condition for stochastic compactness of the appropriately normalized sums and (b) necessary and sufficient conditions for a sequence $\{\beta_n\}$ to exist such that $\lim \inf S_n/\beta_n = 1$ a.s.

#### Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1417-1461.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990626

Digital Object Identifier
doi:10.1214/aop/1176990626

Mathematical Reviews number (MathSciNet)
MR1071803

Zentralblatt MATH identifier
0715.60087

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60F15: Strong theorems

#### Citation

Pruitt, William E. The Rate of Escape of Random Walk. Ann. Probab. 18 (1990), no. 4, 1417--1461. doi:10.1214/aop/1176990626. https://projecteuclid.org/euclid.aop/1176990626