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 $(\|S_{T_r}\| - r, S_{T_r}/\|S_{T_r}\|)$.
Citation
Philip S. Griffin. Terry R. McConnell. "Gambler's Ruin and the First Exit Position of Random Walk from Large Spheres." Ann. Probab. 22 (3) 1429 - 1472, July, 1994. https://doi.org/10.1214/aop/1176988608
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