Abstract
Abstract In this paper we give Banach fixed point theorems and explicit estimates on the rates of convergence of the transition function to the stationary distribution for a class of exponential ergodic Markov chains. Our results are different from earlier estimates using coupling theory, and from estimates using stochastically monotone one. Our estimates show a noticeable improvement on existing results if Markov chains contain instantaneous states or nonconservative states. The proof uses existing results of discrete time Markov chains together with h-skeleton. At last, we apply this result, Ray-Knight compactification and Itô excursion theory to two examples: a class of singular Markov chains and Kolmogorov matrix.
Highlights
Throughout this paper, unless otherwise specified, let {Xt; t ∈ [, ∞)} be a time homo-T geneous, continuous time Markov chain with an honest and standard transition function pij(t) on a state space E = {, . . .}, and its density matrix is Q =, qi = –qii
Let X = (Ω, F, Ft, Xt, θt, Px) be the right process associated with pij(t)
Our goal is to find out the computable bounds of the constants Ri and α, R especially α
Summary
Throughout this paper, unless otherwise specified, let {Xt; t ∈ [ , ∞)} be a time homo-. The authors (see [ – ]) gave the bounds of convergence rates for Markov chains. Their main methods are based on renewal theory. In [ – ], the authors gave the convergence rates of stochastically monotone Markov chains Their results and methods have the advantages of being applicable to some Markov chains or processes. R In this paper we shall first develop the methods in [ ] to the continuous time situation, which leads to considerable improvements of convergence rates. Our result and the Itô excursion theorem to compute two examples in Section , R which will show the advantages of our result
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
