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Equivalences of Geometric Ergodicity of Markov Chains

Author

Listed:
  • Marco A. Gallegos-Herrada

    (University of Toronto)

  • David Ledvinka

    (University of Toronto)

  • Jeffrey S. Rosenthal

    (University of Toronto)

Abstract

This paper gathers together different conditions which are all equivalent to geometric ergodicity of time-homogeneous Markov chains on general state spaces. A total of 34 different conditions are presented (27 for general chains plus 7 for reversible chains), some old and some new, in terms of such notions as convergence bounds, drift conditions, spectral properties, etc., with different assumptions about the distance metric used, finiteness of function moments, initial distribution, uniformity of bounds, and more. Proofs of the connections between different conditions are provided, somewhat self-contained but using some results from the literature where appropriate.

Suggested Citation

  • Marco A. Gallegos-Herrada & David Ledvinka & Jeffrey S. Rosenthal, 2024. "Equivalences of Geometric Ergodicity of Markov Chains," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1230-1256, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01240-1
    DOI: 10.1007/s10959-023-01240-1
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    References listed on IDEAS

    as
    1. Richard C. Bradley, 2021. "On some basic features of strictly stationary, reversible Markov chains," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 499-533, September.
    2. James P. Hobert, 2002. "On the applicability of regenerative simulation in Markov chain Monte Carlo," Biometrika, Biometrika Trust, vol. 89(4), pages 731-743, December.
    Full references (including those not matched with items on IDEAS)

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