Let ${S_n, N \geq 0}$ be a centered d-dimensional random walk $(d
\geq 3)$ and consider the so-called future infima process $J_n =^{df} \inf _{k
\geq n} \|S_k\|$. This paper is concerned with obtaining precise integral
criteria for a function to be in the Lévy upper class of J. This
solves an old problem of Erdös and Taylor, who posed the problem for
the simple symmetric random walk on $mathbb{Z}^d, d \geq 3$. These results are
obtained by a careful analysis of the future infima of transient Bessel
processes and using strong approximations. Our results belong to a class of
Ciesielski-Taylor theorems which relate d-and $(d-2)$-dimensional Bessel
processes.
Publié le : 1996-04-14
Classification:
Brownian motion,
Bessel processes,
transience,
random walk,
60J65,
60G17,
60J15,
60F05
@article{1039639361,
author = {Khoshnevisan, Davar and Lewis, Thomas M. and Shi, Zhan},
title = {On a problem of Erd\"os and Taylor},
journal = {Ann. Probab.},
volume = {24},
number = {2},
year = {1996},
pages = { 761-787},
language = {en},
url = {http://dml.mathdoc.fr/item/1039639361}
}
Khoshnevisan, Davar; Lewis, Thomas M.; Shi, Zhan. On a problem of Erdös and Taylor. Ann. Probab., Tome 24 (1996) no. 2, pp. 761-787. http://gdmltest.u-ga.fr/item/1039639361/