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April 2021 Gambler’s ruin estimates on finite inner uniform domains
Persi Diaconis, Kelsey Houston-Edwards, Laurent Saloff-Coste
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Ann. Appl. Probab. 31(2): 865-895 (April 2021). DOI: 10.1214/20-AAP1607

Abstract

Gambler’s ruin estimates can be viewed as harmonic measure estimates for finite Markov chains which are absorbed (or killed) at boundary points. We relate such estimates to properties of the underlying chain and its Doob transform. Precisely, we show that gambler’s ruin estimates reduce to a good understanding of the Perron–Frobenius eigenfunction and eigenvalue whenever the underlying chain and its Doob transform are Harnack Markov chains. Finite inner-uniform domains (say, in the square grid Zn) provide a large class of examples where these ideas apply and lead to detailed estimates. In general, understanding the behavior of the Perron–Frobenius eigenfunction remains a challenge.

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Persi Diaconis. Kelsey Houston-Edwards. Laurent Saloff-Coste. "Gambler’s ruin estimates on finite inner uniform domains." Ann. Appl. Probab. 31 (2) 865 - 895, April 2021. https://doi.org/10.1214/20-AAP1607

Information

Received: 1 July 2019; Revised: 1 May 2020; Published: April 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.1214/20-AAP1607

Subjects:
Primary: 60J10

Keywords: Gambler’s ruin , Harnack inequality , Markov chains

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2021
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