A Class of Markov Chains with no Spectral Gap

TL;DR

This paper explores a family of Markov chains without a spectral gap, demonstrating polynomial convergence rates to stationarity and extending previous diagonalization techniques using orthogonal polynomials.

Contribution

It introduces a new method to generate Markov chains lacking spectral gaps and provides asymptotic bounds on their convergence rates.

Findings

Polynomial convergence rates of order O(log t / sqrt t) and O(1 / sqrt t)

Chains exhibit no spectral gap and are outside geometric ergodicity scope

Extension of Karlin-McGregor diagonalization to new family of chains

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 $O({\log{t} \over \sqrt{t}})$ 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.