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Comparison Theory for Markov Chains on Different State Spaces and Application to Random Walk on Derangements

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  • Aaron Smith

    (University of Ottawa)

Abstract

We consider two Markov chains on state spaces $$\Omega \subset \widehat{\Omega }$$ Ω ⊂ Ω ^ . In this paper, we prove bounds on the eigenvalues of the chain on the smaller state space based on the eigenvalues of the chain on the larger state space. This generalizes work of Diaconis, Saloff-Coste, and others on comparison of chains in the case $$\Omega = \widehat{\Omega }$$ Ω = Ω ^ . The main tool is the extension of functions from the smaller space to the larger, which allows comparison of the entire spectrum of the two chains. The theory is used to give quick analyses of several chains without symmetry. We apply this theory to analyze the mixing properties of a ‘random transposition’ walk on derangements.

Suggested Citation

  • Aaron Smith, 2015. "Comparison Theory for Markov Chains on Different State Spaces and Application to Random Walk on Derangements," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1406-1430, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0559-7
    DOI: 10.1007/s10959-014-0559-7
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    References listed on IDEAS

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    1. Yuen, Wai Kong, 2000. "Applications of geometric bounds to the convergence rate of Markov chains on," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 1-23, May.
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