On the rate of convergence of the laws of Markov chains associated with orthogonal polynomials (Q1298569)

Isotropic random walks on infinite distance-transitive graphs may be regarded as random walks on associated polynomial hypergroups without loss of information; the associated orthogonal polynomials are certain Bernstein-Szegő polynomials. The reviewer [J. Multivariate Anal. 34, No. 2, 290-322 (1990; Zbl 0722.60021)] derived a central limit theorem for random walks on polynomial hypergroups with exponential growth together with some Berry-Esséen-type estimates for the rate of convergence. This estimate for the rate of convergence is improved to the rate \(O(n^{-1/3})\) for random walks on the polynomial hypergroups associated with Bernstein-Szegő polynomials. The improved rate in this paper is a consequence of the fact that Bernstein-Szegő polynomials admit a simple well-known representation in terms of Tchebycheff polynomials of the second kind.