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

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.