Equivalences of Geometric Ergodicity of Markov Chains

TL;DR

This paper consolidates 34 equivalent conditions for geometric ergodicity of time-homogeneous Markov chains, covering convergence, drift, spectral, and other properties, with detailed proofs and various assumptions.

Contribution

It provides a comprehensive collection of both old and new conditions equivalent to geometric ergodicity, enhancing understanding of Markov chain behavior.

Findings

34 equivalent conditions for geometric ergodicity

Includes conditions for both general and reversible chains

Provides detailed proofs connecting different criteria

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.