Journal of Applied Probability

Stein's method for the Beta distribution and the Pólya-Eggenberger urn

Larry Goldstein and Gesine Reinert

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 50, Number 4 (2013), 1187-1205.

Dates
First available in Project Euclid: 10 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1389370107

Digital Object Identifier
doi:10.1239/jap/1389370107

Mathematical Reviews number (MathSciNet)
MR3161381

Zentralblatt MATH identifier
1304.60033

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 60K99: None of the above, but in this section

Keywords
Stein's method urn model beta distribution arcsine distribution

Citation

Goldstein, Larry; Reinert, Gesine. Stein's method for the Beta distribution and the Pólya-Eggenberger urn. J. Appl. Probab. 50 (2013), no. 4, 1187--1205. doi:10.1239/jap/1389370107. https://projecteuclid.org/euclid.jap/1389370107


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