Electronic Communications in Probability

The mean number of sites visited by a random walk pinned at a distant point

Kôhei Uchiyama

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This paper concerns the number $Z_n$ of sites visited up to time $n$ by a random walk  $S_n$ having zero mean and moving on the two dimensional square lattice ${\bf Z}^2$.    Asymptotic evaluation of  the conditional expectation of  $Z_n$ for large  $n$ given that $S_n=x$ is carried out under some exponential moment condition. It gives an explicit form of the leading term valid uniformly in $(x, n)$, $|x|< cn$ for each $c>0$.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 17, 9 pp.

Accepted: 24 February 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Range of random walk pinned random walk Cramel transform local central limit theorem

This work is licensed under a Creative Commons Attribution 3.0 License.


Uchiyama, Kôhei. The mean number of sites visited by a random walk pinned at a distant point. Electron. Commun. Probab. 20 (2015), paper no. 17, 9 pp. doi:10.1214/ECP.v20-4027. https://projecteuclid.org/euclid.ecp/1465320944

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