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February, 1994 State-Dependent Criteria for Convergence of Markov Chains
Sean P. Meyn, R. L. Tweedie
Ann. Appl. Probab. 4(1): 149-168 (February, 1994). DOI: 10.1214/aoap/1177005204


The standard Foster-Lyapunov approach to establishing recurrence and ergodicity of Markov chains requires that the one-step mean drift of the chain be negative outside some appropriately finite set. Malyshev and Men'sikov developed a refinement of this approach for countable state space chains, allowing the drift to be negative after a number of steps depending on the starting state. We show that these countable space results are special cases of those in the wider context of $\varphi$-irreducible chains, and we give sample-path proofs natural for such chains which are rather more transparent than the original proofs of Malyshev and Men'sikov. We also develop an associated random-step approach giving similar conclusions. We further find state-dependent drift conditions sufficient to show that the chain is actually geometrically ergodic; that is, it has $n$-step transition probabilities which converge to their limits geometrically quickly. We apply these methods to a model of antibody activity and to a nonlinear threshold autoregressive model; they are also applicable to the analysis of complex queueing models.


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Sean P. Meyn. R. L. Tweedie. "State-Dependent Criteria for Convergence of Markov Chains." Ann. Appl. Probab. 4 (1) 149 - 168, February, 1994.


Published: February, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0803.60060
MathSciNet: MR1258177
Digital Object Identifier: 10.1214/aoap/1177005204

Primary: 60J10

Keywords: autoregressions , ergodicity , Foster's criterion , geometric ergodicity , Harris recurrence , invasion models , irreducible Markov processes , Lyapunov functions , Networks of queues , recurrence

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 1 • February, 1994
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