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A class of Markov chains with no spectral gap

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  • 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 (log t/√t) and a lower bound of order O (1/√t) 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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  • Kovchegov, Y., & Michalowski, N. (2013). A class of Markov chains with no spectral gap. Proceedings of the American Mathematical Society, 141(12), 4317-4326. doi:10.1090/S0002-9939-2013-11697-7
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  • 141
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  • 12
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  • American Mathematical Society
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