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 ρd
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
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.