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

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 \({L_\infty^V}\) , 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 \({L_\infty^V}\) . If the chain is reversible, the same equivalence holds with L 2 in place of \({L_\infty^V}\) . In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in \({L_\infty^V}\) but not in L 2. Moreover, if a chain admits a spectral gap in L 2, then for any \({h\in L_2}\) there exists a Lyapunov function \({V_h\in L_1}\) such that V h dominates h and the chain admits a spectral gap in \({L_\infty^{V_h}}\) . The relationship between the size of the spectral gap in \({L_\infty^V}\) or L 2, and the rate at which the chain converges to equilibrium is also briefly discussed.

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Correspondence to I. Kontoyiannis.

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I. Kontoyiannis was supported in part by a Sloan Foundation Research Fellowship, and by a Marie Curie International Outgoing Fellowship, PIOF-GA-2009-235837. S.P. Meyn was supported in part by the National Science Foundation ECS-0523620, and by AFOSR grant FA9550-09-1-0190. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Kontoyiannis, I., Meyn, S.P. Geometric ergodicity and the spectral gap of non-reversible Markov chains. Probab. Theory Relat. Fields 154, 327–339 (2012). https://doi.org/10.1007/s00440-011-0373-4

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  • DOI: https://doi.org/10.1007/s00440-011-0373-4

Keywords

  • Markov chain
  • Geometric ergodicity
  • Spectral theory
  • Stochastic Lyapunov function
  • Reversibility
  • Spectral gap

Mathematics Subject Classification (2000)

  • 60J05
  • 60J10
  • 37A30
  • 37A25