### Abstract

Let U(n) denote the most visited point by a simple symmetric random walk (S_{k})_{k≥0} in the first n steps. It is known that U(n) and max_{0 ≤ xk ≤ n} S_{k} satisfy the same law of the iterated logarithm, but have different upper functions (in the sense of P. Lévy). The distance between them however turns out to be transient. In this paper, we establish the exact rate of escape of this distance. The corresponding problem for the Wiener process is also studied.

Original language | English |
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Pages (from-to) | 1-31 |

Number of pages | 31 |

Journal | Electronic Journal of Probability |

Volume | 3 |

DOIs | |

Publication status | Published - Jan 1 1998 |

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### Keywords

- Favourite site
- Local time
- Random walk
- Wiener process

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty