Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory |
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Authors: | Esa Nummelin Pekka Tuominen |
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Affiliation: | Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00100 Helsinki 10, Finland |
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Abstract: | Let (Xn) 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 λPnf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L1(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pn(x,·)-π6 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,∞) providing a probabilistic approach to the exponencial convergence of renewal measures. |
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Keywords: | 60J10 60K05 Markov chain geometric egodicity splitting rate of convergence non-singular renewal processes |
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