Asymptotic distributions for the Ehrenfest urn and related random walks

Abstract

The distributions of nearest neighbour random walks on hypercubesin continuous timet0 can be expressed in terms of binomial distributions; their limit behaviour fort, N →∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity forN →∞.Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.