Subgeometric Rates of Convergence of f-Ergodic Markov Chains

Abstract

Let Φ = {Φ n} be an aperiodic, positive recurrent Markov chain on a general state space, π its invariant probability measure and f ≧ 1. We consider the rate of (uniform) convergence of Ex[g(Φ n)] to the stationary limit π (g) for |g| ≦ f: specifically, we find conditions under which as n →∞, for suitable subgeometric rate functions r. We give sufficient conditions for this convergence to hold in terms of(i) the existence of suitably regular sets, i.e. sets on which (f, r)-modulated hitting time moments are bounded, and(ii) the existence of (f, r)-modulated drift conditions (Foster–Lyapunov conditions).The results are illustrated for random walks and for more general state space models.