Subgeometric Rates of Convergence of f-Ergodic Markov Chains

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 E x [ 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.