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 $\|S_{T_r}\|/r \rightarrow_p 1$.
Publié le : 1992-04-14
Classification:
Random walk,
overshoot,
multivariate renewal theory,
moments,
60J15,
60G50,
60K05
@article{1176989808,
author = {Griffin, Philip S. and McConnell, Terry R.},
title = {On the Position of a Random Walk at the Time of First Exit from a Sphere},
journal = {Ann. Probab.},
volume = {20},
number = {4},
year = {1992},
pages = { 825-854},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989808}
}
Griffin, Philip S.; McConnell, Terry R. On the Position of a Random Walk at the Time of First Exit from a Sphere. Ann. Probab., Tome 20 (1992) no. 4, pp. 825-854. http://gdmltest.u-ga.fr/item/1176989808/