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Geometric ergodicity and the spectral gap of non-reversible Markov chains

Kontoyiannis, I and Meyn, SP (2012) Geometric ergodicity and the spectral gap of non-reversible Markov chains. Probability Theory and Related Fields, 154. pp. 327-339. ISSN 0178-8051

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Abstract

We argue that the spectral theory of non-reversible Markov chains may often be more effectively cast within the framework of the naturally associated weighted-L ∞ space L ∞V, instead of the usual Hilbert space L 2 = L 2(π), where π is the invariant measure of the chain. This observation is, in part, based on the following results. A discrete-time Markov chain with values in a general state space is geometrically ergodic if and only if its transition kernel admits a spectral gap in L ∞V. If the chain is reversible, the same equivalence holds with L 2 in place of L ∞V. In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in L infin;V but not in L 2. Moreover, if a chain admits a spectral gap in L 2, then for any h ∈ L 2 there exists a Lyapunov function V h ∈ L 1 such that V h dominates h and the chain admits a spectral gap in L ∞Vh. The relationship between the size of the spectral gap in L ∈V or L 2, and the rate at which the chain converges to equilibrium is also briefly discussed. © 2011 Springer-Verlag.

Item Type: Article
Subjects: UNSPECIFIED
Divisions: Div F > Signal Processing and Communications
Depositing User: Cron Job
Date Deposited: 08 Jan 2018 20:11
Last Modified: 07 Mar 2019 12:05
DOI: 10.1007/s00440-011-0373-4