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 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.

Quantitative Convergence Rates for Subgeometric Markov Chains

“…Indeed, for some applications, using the marginal convergence rate (1) to deduce a property of the type (2) may be sub-optimal for sub-geometric Markov chains; an example is briefly discussed below. The characterisation of sub-geometric Markov chains with drift and minorisation conditions has been considered in various earlier works starting with the pioneering work of Tuominen and Tweedie [20]. In the more recent works Fort and Moulines [13] and Jarner and Roberts [15] establish polynomial rates of convergence, but do not provide quantitative results.…”

Subgeometric Ergodicity under Random-Time State-Dependent Drift Conditions

“…Connor and Fort [2] and Yüksel and Meyn [3] studied ergodicity with a drift condition taking the form ( ) ( ) ≤ ( ) + 1 ( ) for some deterministic function : X → [1,∞) and a constant ∈ (0, 1). According to Theorem 2.1(ii) of [5] a deterministic sequence of functions exists : X → [1, ∞) and satisfies a Foster-Lyapunov drift condition:…”

“…It has been proved that this Foster-Lyapunov condition holds not only for every ∈ Z + but also for a sequence of stopping times {T }, ∈ Z + for some discrete time Markov chain (Φ ) ∈Z + , [4]. The results of Zurkowski [4] relied heavily on the work of Connor and Fort [2], Meyn and Tweedie [6], and Tuominen and Tweedie [5]. The aim of this study is to investigate and refine these random-time statedependent drift conditions results with our emphasis on the subgeometric ergodicity for a general Markov chain.…”

Subgeometric ergodicity of Markov chains

“…The first general results proving subgeometric rates of convergence were obtained by [NT83] and later extended by [TT94], but do not provide computable expressions for the bound in the rhs of (1). A direct route to quantitative bounds for subgeometric sequences has been opened by [Ver97,Ver99], based on coupling techniques.…”

“…This is done in two steps. The first one is Theorem 1 whose proof, based on coupling, provides an intuitive understanding of the results of [NT83] and [TT94]. The second step is the use of a very general drift condition, recently introduced in [DFMS04].…”