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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

Maximal coupling procedure and stability of discrete Markov chains. I


Authors: M. V. Kartashov and V. V. Golomozyĭ
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 86 (2012).
Journal: Theor. Probability and Math. Statist. 86 (2013), 93-104
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
Published electronically: August 20, 2013
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Abstract: Two discrete Markov chains whose one-step transition probabilities are close to each other in the uniform total variation norm or in the $ V$-norm are considered. The problem of stability of the transition probabilities for an arbitrary number of steps is investigated. The main assumption is either the uniform mixing or $ V$-mixing condition. In particular, we prove that the uniform distance 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 the transition matrices and where $ \rho $ is the uniform mixing coefficient. A number of general examples are considered. The proofs are based on the maximal coupling procedure that maximizes the one-step coupling probabilities.


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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email: nkartashov@skif.com.ua

V. V. Golomozyĭ
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine

DOI: http://dx.doi.org/10.1090/S0094-9000-2013-00891-6
PII: S 0094-9000(2013)00891-6
Keywords: Coupling theory, coupling method, maximal coupling, discrete Markov chains, stability of distributions
Received by editor(s): October 7, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society