Probability Estimates for the Small Deviations 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 distributions we obtain estimates for the probability that $S_n$ is in a ball centered at the origin. Such an estimate would follow from a local limit theorem if $X_1$ were in the domain of attraction of a stable law.