Gambler's Ruin and the First Exit Position of Random Walk from Large Spheres

Abstract

Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random vectors in $\mathbb{R}^d$ leaves the sphere of radius $r$ in some given norm. We characterize, in terms of the distribution of the individual summands, the following probabilistic behavior: $S_{T_r}/\|S_{T_r}\|$ has no subsequential weak limit supported on a closed half-space. In one dimension, this result solves a very general form of the gambler's ruin problem. We also characterize the existence of degenerate limits and obtain analogous results for triangular arrays along any subsequence $r_k \rightarrow \infty$. Finally, we compute the limiting joint distribution of $(\|S_{T_r}\| - r, S_{T_r}/\|S_{T_r}\|)$.