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May, 1983 A Simple Criterion for Transience of a Reversible Markov Chain
Terry Lyons
Ann. Probab. 11(2): 393-402 (May, 1983). DOI: 10.1214/aop/1176993604


An old argument of Royden and Tsuji is modified to give a necessary and sufficient condition for a reversible countable state Markov chain to be transient. This Royden criterion is quite convenient and can, on occasion, be used as a substitute for the criterion of Nash-Williams [6]. The result we give here yields a very simple proof that the Nash-Williams criterion implies recurrence. The Royden criterion also yields as a trivial corollary that a recurrent reversible random walk on a state space $X$ remains recurrent when it is constrained to run on a subset $X'$ of $X$. An apparently weaker criterion for transience is also given. As an application, we discuss the transience of a random walk on a horn shaped subset of $\mathbb{Z}^d$.


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Terry Lyons. "A Simple Criterion for Transience of a Reversible Markov Chain." Ann. Probab. 11 (2) 393 - 402, May, 1983.


Published: May, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0509.60067
MathSciNet: MR690136
Digital Object Identifier: 10.1214/aop/1176993604

Primary: 60J10
Secondary: 31C12 , 31C25 , 60J45

Keywords: energy , Markov chain , recurrence , Symmetric Markov chain , transience

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 2 • May, 1983
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