## The Annals of Probability

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

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

#### 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: 19 April 2007

**Permanent link to this document**

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

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176993604

**Mathematical Reviews number (MathSciNet)**

MR690136

**Zentralblatt MATH identifier**

0509.60067

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 31C25: Dirichlet spaces 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]

**Keywords**

Recurrence transience Markov chain symmetric Markov chain energy

#### Citation

Lyons, Terry. A Simple Criterion for Transience of a Reversible Markov Chain. The Annals of Probability 11 (1983), no. 2, 393--402. doi:10.1214/aop/1176993604. http://projecteuclid.org/euclid.aop/1176993604.