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.

Metropolis–Hastings Algorithms with acceptance ratios of nearly 1

“…These lemmas are closely related to Theorem 11.3.9 of Meyn and Tweedie (1993), Proposition 3.1 of Tuominen and Tweedie (1994) and Theorem 3.2 of Jarner and Roberts (2002b). First, lemma is merely a modification of Theorem 3.2 of Jarner and Roberts (2002b).…”

“…Sub-geometric rate of convergence is studied in, for example, by Tuominen and Tweedie (1994), Fort and Moulines (2000), Jarner and Roberts (2002a,b) and Douc et al (2004). In Jarner and Roberts (2002b), they proved the following theorem.…”

Uniform ergodicity of continuous-time controlled Markov chains: A survey and new results

“…Subexponential ergodicity for continuous-time processes, for some particular models, has been studied also in, e.g., Dai and Meyn (1995), Ganidis et al (1999). The reader interested in the discrete-time case, in which subexponential ergodicity is named subgeometric ergodicity, is referred to the papers Fort and Moulines (2002), Jarner and Roberts (2002), Tuominen and Tweedie (1994), and Chapters 15 and 16 in Meyn and Tweedie (1993a).…”

On exponential ergodicity of multiclass queueing networks

“…Under this bound, polynomial rates of convergence are obtained on the rate of convergence to steady state. Using different methods, polynomial bounds on the steady-state buffer lengths and polynomial rates of convergence were obtained in [2] for stochastic networks based on a general result of [20]. The proof is simplified considerably in this countable state space setting.…”