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April 1996 On a problem of Erdös and Taylor
Davar Khoshnevisan, Thomas M. Lewis, Zhan Shi
Ann. Probab. 24(2): 761-787 (April 1996). DOI: 10.1214/aop/1039639361


Let $\{S_n, n \geq 0\}$ be a centered d-dimensional random walk $(d \geq 3)$ and consider the so-called future infima process $J_n \stackrel{\mathrm{df}}{=}\inf _{k \geq n} \|S_k\|$. 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.


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Davar Khoshnevisan. Thomas M. Lewis. Zhan Shi. "On a problem of Erdös and Taylor." Ann. Probab. 24 (2) 761 - 787, April 1996.


Published: April 1996
First available in Project Euclid: 11 December 2002

zbMATH: 0862.60068
MathSciNet: MR1404527
Digital Object Identifier: 10.1214/aop/1039639361

Primary: 60G17 , 60J65
Secondary: 60F05 , 60J15

Keywords: Bessel processes , Brownian motion , Random walk , transience

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 2 • April 1996
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