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A class of Markov chains with no spectral gap


Authors: Yevgeniy Kovchegov and Nicholas Michalowski
Journal: Proc. Amer. Math. Soc. 141 (2013), 4317-4326
MSC (2010): Primary 33C45, 60G05
Published electronically: August 16, 2013
MathSciNet review: 3105873
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Abstract: In this paper we extend the results of the research started by the first author in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to estimate the rate of convergence to a stationary distribution.

We use a method of Koornwinder to generate a large and interesting family of random walks which exhibits a lack of spectral gap, and a polynomial rate of convergence to the stationary distribution. For the Chebyshev type subfamily of Markov chains, we use asymptotic techniques to obtain an upper bound of order $ O\left ({\log {t} \over \sqrt {t}}\right )$ and a lower bound of order $ O\left ({1 \over \sqrt {t}}\right )$ on the distance to the stationary distribution regardless of the initial state. Due to the lack of a spectral gap, these results lie outside the scope of geometric ergodicity theory.


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Additional Information

Yevgeniy Kovchegov
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: kovchegy@math.oregonstate.edu

Nicholas Michalowski
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: Nicholas.Michalowski@math.oregonstate.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11697-7
Received by editor(s): September 26, 2011
Received by editor(s) in revised form: September 28, 2011, and February 9, 2012
Published electronically: August 16, 2013
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.