Open Access
April, 1992 On the Position of a Random Walk at the Time of First Exit from a Sphere
Philip S. Griffin, Terry R. McConnell
Ann. Probab. 20(2): 825-854 (April, 1992). DOI: 10.1214/aop/1176989808


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 $\|S_{T_r}\|/r \rightarrow_p 1$.


Download Citation

Philip S. Griffin. Terry R. McConnell. "On the Position of a Random Walk at the Time of First Exit from a Sphere." Ann. Probab. 20 (2) 825 - 854, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0756.60060
MathSciNet: MR1159576
Digital Object Identifier: 10.1214/aop/1176989808

Primary: 60J15
Secondary: 60G50 , 60K05

Keywords: moments , multivariate renewal theory , overshoot , Random walk

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
Back to Top