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2020 The potential function and ladder heights of a recurrent random walk on $\mathbb {Z}$ with infinite variance
Kohei Uchiyama
Electron. J. Probab. 25: 1-24 (2020). DOI: 10.1214/20-EJP553

Abstract

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.

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Kohei Uchiyama. "The potential function and ladder heights of a recurrent random walk on $\mathbb {Z}$ with infinite variance." Electron. J. Probab. 25 1 - 24, 2020. https://doi.org/10.1214/20-EJP553

Information

Received: 17 October 2019; Accepted: 11 November 2020; Published: 2020
First available in Project Euclid: 23 December 2020

Digital Object Identifier: 10.1214/20-EJP553

Subjects:
Primary: 60G50
Secondary: 60J45

JOURNAL ARTICLE
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