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Bounds on Expected Coupling Times in a Markov Chain

  • Jeffrey J. Hunter

Abstract

In the author's paper “Coupling and Mixing Times in Markov Chains” (Res. Lett. Inf. Math. Sci, 11, 1–22, 2007) it was shown that it is very difficult to find explicit expressions for the expected time to coupling in a general Markov chain. In this paper simple upper and lower bounds are given for the expected time to coupling in a discrete time finite Markov chain. Extensions to the bounds under additional restrictive conditions are also given with detailed comparisons provided for two and three state chains.

Keywords

Markov Chain Transition Matrix Independent Trial Coupling Time Stationary Probability Vector 
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References

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    Aldous, D.J., Fill, J.A.: Reversible Markov Chains and Random Walks on Graphs (Book in preparation) See http://www.stat.Berkeley.EDU/users/aldous/book.html
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    Hunter, J.J.: Mathematical Techniques of Applied Probability, Volume 1, Discrete Time Models: Basic Theory. Academic, New York (1983)MATHGoogle Scholar
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    Hunter, J.J.: Mathematical Techniques of Applied Probability, Volume 2, Discrete Time Models: Techniques and Applications, Academic, New York (1983)MATHGoogle Scholar
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    Hunter, J.J.: Mixing Times with Applications to Perturbed Markov Chains, Linear Algebra Appl. 417 108–123 (2006)MATHCrossRefMathSciNetGoogle Scholar
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    Hunter, J.J.: Coupling and Mixing times in a Markov chain, Res. Lett. Inf. Math. Sci. 11, 1–22 (2007). (Submitted to Linear Algebra Appl.)Google Scholar
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    Lovasz, L., Winkler, P.: Mixing Times. In: Aldous, D. Propp, J. (eds.) Microsurveys in Discrete Probability, DIMACS Series in Discrete Math. Theor. Comp. Sci., 85–133. AMS (1998)Google Scholar

Copyright information

© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.Institute of Information & Mathematical SciencesMassey UniversityPrivate BagNew Zealand

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