A class of Markov chains with no spectral gap

Abstract

In this paper we extend the results of the research started in [6] and [7], 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 [5] 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 [8].

Excursions and path functionals for stochastic processes with asymptotically zero drifts

“…A now standard approach in this field is to take as an underlying model a nearest-neighbour random walk with an asymptotically zero drift: see for example [1,12,23]. Such nearest-neighbour models are amenable to explicit calculation, often via intricate algebraic methods such as Karlin-McGregor spectral theory and orthogonal polynomials [26]; other recent work on these models, not directly motivated by polymer models, includes for example [11,18,29,40]. This continued interest in asymptotically zero drift processes in the nearest-neighbour case is another motivation for the present paper, in which we present related results for a much more general class of models.…”

On Convergence of 1D Markov Diffusions to Heavy-Tailed Invariant Density

“…Indeed, this step justifies that it is possible by virtue of Moser's Harnack inequality for divergent type elliptic equation (see [12]) (cf., e.g., [30], [31] where a parabolic Harnack inequality was used for the same goal). Here it is convenient to return to an SDE on the whole line with symmetric coefficients (17) which solution is denoted by Xt . Its modulus satisfies the equation ( 5) with a new Wiener process.…”

“…This problem has certain deep relations to ergodicity and to the Perron -Frobenius theorem for Markov chains with finite state space, to spectral gap for semigroup generators, to upper and lower bounds for convergence to stationarity; yet, a spectral gap in this paper is not used. The literature in this area is huge and we only mention a few important references related to the subject of the paper more or less directly (see [3], [7], [9], [17], [19], [20], [24], [25], et al; also, see further references theiren).…”