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Rate of convergence for traditional Pólya urns

Part of: Combinatorial probability Limit theorems Markov processes

Published online by Cambridge University Press:  23 November 2020

Svante Janson Open the ORCID record for Svante Janson [Opens in a new window]
Svante Janson*
Affiliation:
Uppsala University
*
*Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. [email protected]
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Abstract

Consider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.

Keywords

Pólya urnsDirichlet distributionconvergence rateWasserstein distanceminimal Lp distanceKolmogorov–Smirnov distanceLévy distance

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 60J99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1029 - 1044
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Copyright
© Applied Probability Trust 2020

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