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Subgeometric Rates of Convergence of f-Ergodic Markov Chains

Pekka Tuominen and Richard L. Tweedie
Advances in Applied Probability
Vol. 26, No. 3 (Sep., 1994), pp. 775-798
DOI: 10.2307/1427820
Stable URL: http://www.jstor.org/stable/1427820
Page Count: 24
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Subgeometric Rates of Convergence of f-Ergodic Markov Chains
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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 $r(n)\underset |g|\leq f\to{\text{sup}}|\mathbf{\mathit{E}}_{x}[g(\Phi _{n})]-\pi (g)|\rightarrow 0$, 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.

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