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

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Abstract

We consider two Markov chains on state spaces \(\Omega \subset \widehat{\Omega }\). 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 \(\Omega = \widehat{\Omega }\). 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.

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  1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Drive, Ottawa, ON, K1N 7N5, Canada

    Aaron Smith

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  1. Aaron Smith
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Smith, A. Comparison Theory for Markov Chains on Different State Spaces and Application to Random Walk on Derangements. J Theor Probab 28, 1406–1430 (2015). https://doi.org/10.1007/s10959-014-0559-7

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  • DOI: https://doi.org/10.1007/s10959-014-0559-7

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