The Annals of Probability

The Rate of Escape of Random Walk

William E. Pruitt

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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

Keywords
Probability estimates stochastic compactness lim inf integral test slowly varying tails

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


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