Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (Xn) be a positive recurrent Harris chain on a general state space, with invariant probability measure [pi]. We give necessary and sufficient conditions for the geometric convergence of [lambda]Pnf towards its limit [pi](f), and show that when such convergence happens it is, in fact, uniform over f and in L1([pi])-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, [is proportional to] [pi](dx)||Pn(x,·)-[pi]|| converges to zero geometrically fast. We also characterize the geometric ergodicity of (Xn) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,[infinity]) providing a probabilistic approach to the exponencial convergence of renewal measures.