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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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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 L2= L2(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 L2in 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 Linfin;Vbut not in L2. Moreover, if a chain admits a spectral gap in L2, then for any h ∈ L2there exists a Lyapunov function Vh∈ L1such that Vhdominates h and the chain admits a spectral gap in L∞Vh. The relationship between the size of the spectral gap in L∈Vor L2, and the rate at which the chain converges to equilibrium is also briefly discussed. © 2011 Springer-Verlag.

Item Type: Article
Divisions: Div F > Signal Processing and Communications
Depositing User: Cron Job
Date Deposited: 08 Jan 2018 20:11
Last Modified: 01 Nov 2018 03:00