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

We investigate random walks (Sn)n∈N0 on the nonnegative integers arising from isotropic random walks on distance transitive graphs. The laws of those isotropic random walks converge in distribution to the normal distribution and the transition probabilities of the Sn are closely related with a sequence of Bernstein-Szegö polynomials. We give an explicit representation for these polynomials as a sum of Chebychev polynomials of the second kind and using this representation we prove an upper bound for the rate of convergence of the laws of the Sn.