## The Annals of Probability

### An Integral Test for the Rate of Escape of $d$-Dimensional Random Walk

Philip S. Griffin

#### Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables taking values in $\mathbb{R}^d$ and $S_n = X_1 + \cdots + X_n$. For a large class of random variables, which includes all of those in the domain of attraction of a type $A$ stable law, an integral test is given which determines whether $P\{|S_n| \leq \gamma_n \mathrm{i.o.}\} = 0 \quad\text{or}\quad 1$ for any increasing sequence $\{\gamma_n\}$. This result generalizes the Dvoretzky-Erdos test for simple random walk and the Takeuchi and Taylor test for stable random walks.

#### Article information

Source
Ann. Probab., Volume 11, Number 4 (1983), 953-961.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993444

Digital Object Identifier
doi:10.1214/aop/1176993444

Mathematical Reviews number (MathSciNet)
MR714958

Zentralblatt MATH identifier
0524.60068

JSTOR

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

#### Citation

Griffin, Philip S. An Integral Test for the Rate of Escape of $d$-Dimensional Random Walk. Ann. Probab. 11 (1983), no. 4, 953--961. doi:10.1214/aop/1176993444. https://projecteuclid.org/euclid.aop/1176993444