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

Autor(en): Stadje, Wolfgang 
Wuebker, Achim
Stichwörter: bounds for the spectral radius; countable state space; geometric ergodicity; isoperimetric constant; Markov chain; Mathematics; reversibility; spectral gap property; Statistics & Probability
Erscheinungsdatum: 2011
Herausgeber: UNIV WASHINGTON, DEPT MATHEMATICS
Journal: ELECTRONIC JOURNAL OF PROBABILITY
Volumen: 16
Startseite: 1001
Seitenende: 1019
Zusammenfassung: 
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.
ISSN: 10836489
DOI: 10.1214/EJP.v16-900

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