## The Annals of Probability

### A Simple Criterion for Transience of a Reversible Markov Chain

Terry Lyons

#### Abstract

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 $\mathbb{Z}^d$.

#### Article information

Source
Ann. Probab. Volume 11, Number 2 (1983), 393-402.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176993604

Digital Object Identifier
doi:10.1214/aop/1176993604

Mathematical Reviews number (MathSciNet)
MR690136

Zentralblatt MATH identifier
0509.60067

JSTOR