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Theory of Probability and Mathematical Statistics

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Maximal coupling and stability of discrete non-homogeneous Markov chains


Authors: V. V. Golomozyĭ and M. V. Kartashov
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 91 (2014).
Journal: Theor. Probability and Math. Statist. 91 (2015), 17-27
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
DOI: https://doi.org/10.1090/tpms/963
Published electronically: February 3, 2016
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Abstract: We consider two time non-homogeneous discrete Markov chains whose one-step transition probabilities are close in the uniform total variation norm. The problem of stability of the transition probabilities for an arbitrary number of steps is investigated. The main assumption is the uniform mixing. We prove that the uniform difference between the distributions of the chains after an arbitrary number of steps does not exceed $ \varepsilon /(1-\rho )$, where $ \varepsilon $ is the uniform distance between transition matrices and $ \rho $ is the uniform mixing coefficient. The proofs are based on the maximal coupling procedure that maximize the one-step coupling probabilities.


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Additional Information

V. V. Golomozyĭ
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: mailtower@gmail.com

M. V. Kartashov
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: mailtower@gmail.com

DOI: https://doi.org/10.1090/tpms/963
Keywords: Coupling theory, coupling method, maximal coupling, discrete Markov chains, stability of distributions
Received by editor(s): October 11, 2011
Published electronically: February 3, 2016
Article copyright: © Copyright 2016 American Mathematical Society