A class of Markov chains with no spectral gap

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.