The Annals of Applied Probability

The ODE method for stability of skip-free Markov chains with applications to MCMC

Gersende Fort, Sean Meyn, Eric Moulines, and Pierre Priouret

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Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation and optimization.

In this paper, some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space and the initial condition by a large constant. The resulting fluid limit is the solution of an ordinary differential equation (ODE) in “most” of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similarly to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov chain Monte Carlo methods.

Article information

Ann. Appl. Probab., Volume 18, Number 2 (2008), 664-707.

First available in Project Euclid: 20 March 2008

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 65C05: Monte Carlo methods

Markov chain fluid limit subgeometric ergodicity state-dependent drift criteria Markov chain Monte Carlo Metropolis–Hastings algorithms


Fort, Gersende; Meyn, Sean; Moulines, Eric; Priouret, Pierre. The ODE method for stability of skip-free Markov chains with applications to MCMC. Ann. Appl. Probab. 18 (2008), no. 2, 664--707. doi:10.1214/07-AAP471.

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