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

Abstract

Let ( X n ) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λP nf towards its limit π( f), and show that when such convergence happens it is, in fact, uniform over f and in L 1 (π)-norm. As a corollary we obtain that, when ( Xn) is geometrically ergodic, ∝ π(dx)‖ P n ( x,·)-π‖ converges to zero geometrically fast. We also characterize the geometric ergodicity of ( X n ) 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,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.