On the Position of a Random Walk at the Time of First Exit from a Sphere

Abstract

Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random vectors leaves the sphere of radius $r$. The spheres are determined by some given norm on $\mathbb{R}^d$ which need not be the Euclidean norm. As a particular case of our results, we obtain, for mean-zero random vectors and each $0 < p < \infty$ and $0 \leq q < \infty$, necessary and sufficient conditions on the distribution of the summands to have $E(\|S_{T_r}\| - r)^p = O(r^q)$ as $r \rightarrow \infty$. We also characterize tightness of the family $\{\|S_{T_r}\| - r\}$ and obtain other related results on the rate of growth of $\|S_{T_r}\|$. In particular, we obtain a simple necessary and sufficient condition for $\|S_{T_r}\|/r \rightarrow_p 1$.