Polynomial ergodicity of Markov transition kernels

This paper discusses quantitative bounds on the convergence rates of Markov chains, under conditions implying polynomial convergence rates. This paper extends an earlier work by Roberts and Tweedie (Stochastic Process. Appl. 80(2) (1999) 211), which provides quantitative bounds for the total variation norm under conditions implying geometric ergodicity. Explicit bounds for the total variation norm are obtained by evaluating the moments of an appropriately defined coupling time, using a set of drift conditions, adapted from an earlier work by Tuominen and Tweedie (Adv. Appl. Probab. 26(3) (1994) 775). Applications of this result are then presented to study the convergence of random walk Hastings Metropolis algorithm for super-exponential target functions and of general state-space models. Explicit bounds for f-ergodicity are also given, for an appropriately defined control function f.