Equivalences of Geometric Ergodicity of Markov Chains

M. A. Gallegos-Herrada; D. Ledvinka; J. S. Rosenthal

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arXiv · 2203.04395

Equivalences of Geometric Ergodicity of Markov Chains

M. A. Gallegos-Herrada·D. Ledvinka·J. S. Rosenthal

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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 just 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 the different conditions are provided, mostly self-contained but using some results from the literature where appropriate.

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M. A. Gallegos-Herrada, D. Ledvinka, J. S. Rosenthal. 2022-03-08. Equivalences of Geometric Ergodicity of Markov Chains. https://arxiv.org/abs/2203.04395

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