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 convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Markov operator $P$ on the Banach space ${\mathrm{\ell }}_1$. Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed.