Coefficients of ergodicity for stochastically monotone Markov chains

In this paper we show that to each distance d defined on the finite state space S of a strongly ergodic Markov chain there corresponds a coefficient ρ d of ergodicity based on the Wasserstein metric. For a class of stochastically monotone transition matrices P , the infimum over all such coefficients is given by the spectral radius of P – R , where R = lim k P k and is attained. This result has a probabilistic interpretation of a control of the speed of convergence of by the metric d and is linked to the second eigenvalue of P.