Quantitative Convergence Rates for Subgeometric Markov Chains

Abstract

Abstract. We provide explicit expressions for the constants involved in the characterisation of ergodicity of sub-geometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The result is fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our result accommodates also some classes of inhomogeneous chains.

On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence

“…Then θ (1) 0 + T ′ is pointwise bigger than T , and so it stochastically dominates T . So we will focus on building a dominating sequence for T ′ and estimating its expectation.…”

“…Such construction leads us to the necessity of investigating properties of the renewal proceses generated by time-inhomogeneous Markov chains. Many works of other authors are using similar coupling techniques to obtain stability estimates or convergence rates, see for example [1,7] where subgeometrical Markov chains are studied.…”

“…The main object of the study is a pair of time-inhomogeneous, independent, discrete time Markov chains defined on the probability space (Ω, F, P), with the values in the general state space (E, E). We will denote these chains as X (1) n and X…”

“…Let us recall that the chains X (1) and X (2) are independent and so we can split the probability in the formula (12) into the product:…”

Stability Estimates for Finite-Dimensional Distributions of Time-Inhomogeneous Markov Chains

“…For example, in the papers [11][12][13], the coupling method is used to obtain the ergodic properties of a Markov chain and the stability estimate of the form ||λP n (x, •) − µP n (x, •)|| for the transition probabilities of the same chain that starts from various initial distributions. In the paper [14], the coupling method is used to get the results in the time-inhomogeneous situation.…”