Subgeometric Rates of Convergence of f-Ergodic Markov Chains

Let . = {. n} be an aperiodic, positive recurrent Markov chain on a general state space, . its invariant probability measure and f . 1. We consider the rate of (uniform) convergence of Ex[g(. n)] to the stationary limit . (g) for |g| . f: specifically, we find conditions under which https://static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aartic le%3AS0001867800026549/resource/name/S0001867800026549_eqn1.gif?pub-stat us=live 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.