The escape rate of favorite sites of simple random walk and Brownian motion

• 成果类型: Article
• 署名作者: Lifshits, MA; Shi, Z
• 署名单位: Saint Petersburg State University; Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2004
• 页码: 129-152
• 关键词: visited sites; LAW

摘要

Consider a simple symmetric random walk on the integer lattice Z. For each n, let V(n) denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat Surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417-436] says that V is almost surely transient, thus disproving a previous conjecture of Erdos and Revesz [Mathematical Structures-Computational Mathematics-Mathematical Modeling 2 (1984) 152-157]. More precisely, Bass and Griffin proved that almost surely, lim inf(n-->infinity)(\V(n)\)/(1/2)(-gamma)(n)((logn)) equals 0 if gamma < 1, and infinity if gamma > 11 (eleven). The present paper studies the rate of escape of V(n). We show that almost surely, the lim inf' expression in question is 0 if y < 1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.