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July, 1994 Gambler's Ruin and the First Exit Position of Random Walk from Large Spheres
Philip S. Griffin, Terry R. McConnell
Ann. Probab. 22(3): 1429-1472 (July, 1994). DOI: 10.1214/aop/1176988608


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 $(\|S_{T_r}\| - r, S_{T_r}/\|S_{T_r}\|)$.


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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.


Published: July, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0820.60055
MathSciNet: MR1303650
Digital Object Identifier: 10.1214/aop/1176988608

Primary: 60J15
Secondary: 60G50 , 60K05

Keywords: Gambler's ruin , multidimensional renewal theory , overshoot , Random walk

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 3 • July, 1994
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