Perturbation theory and uniform ergodicity for discrete-time Markov chains

2020/03/16 by Yong-Hua Mao, Mao, Yonghua, Yan-Hong Song +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2003.06978

openalex publication_date 2020/03/16 · openalex created_date 2020/03/23 · openalex updated_date 2026/07/28

Abstract

We study perturbation theory and uniform ergodicity for discrete-time Markov chains on general state spaces in terms of the uniform moments of the first hitting times on some set. The methods we adopt are different from previous ones. For reversible and non-negative definite Markov chains, we first investigate the geometrically ergodic convergence rates. Based on the estimates, together with a first passage formula, we then get the convergence rates in uniform ergodicity. If the transition kernel P is only reversible, we transfer to study the two-skeleton chain with the transition kernel P². At a technical level, the crucial point is to connect the geometric moments of the first return times between P and P².

Citations

• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• Sensitivity and convergence of uniformly ergodic Markov chains

Related

• Sensitivity and convergence of uniformly ergodic Markov chains
• Perturbation bounds for the stationary distributions of Markov chains
• Equivalences of Geometric Ergodicity of Markov Chains
• Quantitative convergence rates for reversible Markov chains via strong random times
• Convergence Rates in Uniform Ergodicity by Hitting Times and L²-exponential Convergence Rates