On a problem of Erdos and Taylor

• 成果类型: Article
• 署名作者: Khoshnevisan, D; Lewis, TM; Shi, Z
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 761-787
• 关键词: brownian-motion

摘要

Let {S-n, n greater than or equal to 0} be a centered d-dimensional random walk (d greater than or equal to 3) and consider the so-called future infima process J(n) =(df)inf(k greater than or equal to n) \\S-k\\. This paper is concerned with obtaining precise integral criteria for a function to be in the Levy upper class of J. This solves an old problem of Erdos and Taylor, who posed the problem for the simple symmetric random walk on Z(d), d greater than or equal to 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.