  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 $\ell_1$. Explicit
conditions allowing to obtain the actual asymptotics for the rate of
convergence are also discussed.
