State-Dependent Criteria for Convergence of Markov Chains

Meyn, Sean P. ; Tweedie, R. L.

Ann. Appl. Probab., Tome 4 (1994) no. 4, p. 149-168 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1994-02-14
Classification:  Foster's criterion,  irreducible Markov processes,  Lyapunov functions,  ergodicity,  geometric ergodicity,  recurrence,  Harris recurrence,  invasion models,  autoregressions,  networks of queues,  60J10