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Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Part of: Limit theorems Special processes Distribution theory

Published online by Cambridge University Press:  30 January 2018

Larry Goldstein  and
Gesine Reinert
Larry Goldstein*
Affiliation:
University of Southern California
Gesine Reinert*
Affiliation:
University of Oxford
*
∗ Postal address: Department of Mathematics, University of Southern California, KAP 108, Los Angeles, CA 90089-2532, USA. Email address: larry@math.usc.edu
∗∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK. Email address: reinert@stats.ox.ac.uk
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Abstract

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Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

Keywords

Stein's methodurn modelbeta distributionarcsine distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1187 - 1205
Copyright
© Applied Probability Trust 

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