### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Large favourite sites of simple random walk and the wiener process.** / Csáki, E.; Shi, Zhan.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 3, pp. 1-31. https://doi.org/10.1214/EJP.v3-36

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TY - JOUR

T1 - Large favourite sites of simple random walk and the wiener process

AU - Csáki, E.

AU - Shi, Zhan

PY - 1998/1/1

Y1 - 1998/1/1

N2 - Let U(n) denote the most visited point by a simple symmetric random walk (Sk)k≥0 in the first n steps. It is known that U(n) and max0 ≤ xk ≤ n Sk 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.

AB - Let U(n) denote the most visited point by a simple symmetric random walk (Sk)k≥0 in the first n steps. It is known that U(n) and max0 ≤ xk ≤ n Sk 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.

KW - Favourite site

KW - Local time

KW - Random walk

KW - Wiener process

UR - http://www.scopus.com/inward/record.url?scp=85037896382&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037896382&partnerID=8YFLogxK

U2 - 10.1214/EJP.v3-36

DO - 10.1214/EJP.v3-36

M3 - Article

AN - SCOPUS:85037896382

VL - 3

SP - 1

EP - 31

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -