## Electronic Journal of Probability

### Three Kinds of Geometric Convergence for Markov Chains and the Spectral Gap Property

#### Abstract

In this paper we investigate three types of convergence for geometrically ergodic Markov chains (MCs) with countable state space, which in general lead to different `rates of convergence'. For reversible Markov chains it is shown that these rates coincide. For general MCs we show some connections between their rates and those of the associated reversed MCs. Moreover, we study the relations between these rates and a certain family of isoperimetric constants. This sheds new light on the connection of geometric ergodicity and the so-called spectral gap property, in particular for non-reversible MCs, and makes it possible to derive sharp upper and lower bounds for the spectral radius of certain non-reversible chains

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 34, 1001-1019.

Dates
Accepted: 20 April 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820203

Digital Object Identifier
doi:10.1214/EJP.v16-900

Mathematical Reviews number (MathSciNet)
MR2820067

Zentralblatt MATH identifier
1231.60068

Rights

#### Citation

Stadje, Wolfgang; Wübker, Achim. Three Kinds of Geometric Convergence for Markov Chains and the Spectral Gap Property. Electron. J. Probab. 16 (2011), paper no. 34, 1001--1019. doi:10.1214/EJP.v16-900. https://projecteuclid.org/euclid.ejp/1464820203

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