On the rate of convergence to equilibrium for countable ergodic markov chains

Using elementary methods, we prove that for a countable Markov chain P of ergodic degree d > 0 the rate of convergence towards the stationary distribution is subgeometric of order n^−d, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between conver- gence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Markov operator P on the Banach space \ell_1. Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed.