A Class of Markov Chains with no Spectral Gap

Details A Class of Markov Chains with no Spectral Gap A Class of Markov Chains with no Spectral Gap

2011-09-27

arxiv.org/abs/1109.6050v2

Mathematics

Classical Analysis and ODEs

To appear in the Proceedings of the AMS

Scientific paper

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 $O({\log{t} \over \sqrt{t}})$ and a lower bound of order $O({1 \over \sqrt{t}})$ 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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