## 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) ^{-γ} 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 language | English |
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Pages (from-to) | 129-152 |

Number of pages | 24 |

Journal | Annals of Probability |

Volume | 32 |

Issue number | 1 A |

Publication status | Published - 1 Jan 2004 |

## Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty