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

Research output

4 Citations (Scopus)

Abstract

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 Erdös and Révész [Mathematical Structures-Computational MathematicsMathematical Modeling 2 (1984) 152-157]. More precisely, Bass and Griffin proved that almost surely, lim inf n→ |V(n)/n| 1/2(logn) 0 if γ < 1, and is infinity if γ > 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 γ 1, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.

Original languageEnglish
Pages (from-to)129-152
Number of pages24
JournalAnnals of Probability
Volume32
Issue number1 A
Publication statusPublished - 1 Jan 2004

Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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