Rates of convergence for everywhere-positive Markov chains

Abstract

We generalize and simplify a result of Schervish and Carlin (1992) concerning the convergence of Markov chains to their stationary distributions. We prove geometric convergence for any Markov chain whose transition operator is compact and has everywhere-positive density functions (with respect to some reference measure). We also provide, without requiring compactness, a quantitative estimate of the convergence rate, given in terms of the stationary distribution.