## The Annals of Probability

### Three favorite sites occurs infinitely often for one-dimensional simple random walk

#### Abstract

For a one-dimensional simple random walk $(S_{t})$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Tóth [Ann. Probab. 29 (2001) 484–503], and disproves a conjecture of Erdős and Révész [In Mathematical Structure—Computational Mathematics—Mathematical Modelling 2 (1984) 152–157] and of Tóth [Ann. Probab. 29 (2001) 484–503].

#### Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2545-2561.

Dates
Revised: September 2017
First available in Project Euclid: 24 August 2018

https://projecteuclid.org/euclid.aop/1535097634

Digital Object Identifier
doi:10.1214/17-AOP1232

Mathematical Reviews number (MathSciNet)
MR3846833

Zentralblatt MATH identifier
06964343

Subjects
Primary: 60J15 60J55: Local time and additive functionals

Keywords
Random walk favorite sites

#### Citation

Ding, Jian; Shen, Jianfei. Three favorite sites occurs infinitely often for one-dimensional simple random walk. Ann. Probab. 46 (2018), no. 5, 2545--2561. doi:10.1214/17-AOP1232. https://projecteuclid.org/euclid.aop/1535097634

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