Reduction of Markov chains with two-time-scale state transitions

Abstract

In this paper, we consider a general class of two-time-scale Markov chains whose transition rate matrix depends on a parameter . We assume that some transition rates of the Markov chain will tend to infinity as . We divide the state space of the Markov chain into a fast state space and a slow state space and define a reduced chain on the slow state space. Our main result is that the distribution of the original chain will converge in total variation distance to that of the reduced chain uniformly in time as .

Keywords:

• time scale
• limit behaviour
• asymptotic behaviour
• approximation
• singularly perturbed Markov chains

AMS Subject Classification:

• 60J27
• 60J28
• 93C70

Acknowledgements

The author gratefully acknowledges Professor Da-Quan Jiang for supporting my research on the present work and gratefully acknowledges the anonymous reviewers for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.