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Asymptotic distributions for the Ehrenfest urn and related random walks

Part of: Equilibrium statistical mechanics Markov processes Limit theorems Abstract harmonic analysis Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Michael Voit
Michael Voit*
Affiliation:
Universität Tübingen
*
∗Postal address: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.
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Abstract

The distributions of nearest neighbour random walks on hypercubes in continuous time t 0 can be expressed in terms of binomial distributions; their limit behaviour for t, 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 for N →∞. 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.

Keywords

RANDOM WALKS ON HYPERCUBESBIRTH-AND-DEATH RANDOM WALKSKRAWTCHOUKPOLYNOMIALSUNIFORM DISTRIBUTIONTOTAL VARIATION DISTANCEFINITE HYPERGROUPS

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F05: Central limit and other weak theorems 60B10: Convergence of probability measures 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 82B05: Classical equilibrium statistical mechanics (general) 43A62: Hypergroups
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 340 - 356
Copyright
Copyright © Applied Probability Trust 1996 

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