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Rate of convergence for traditional Pólya urns
- Part of
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- Journal:
- Journal of Applied Probability / Volume 57 / Issue 4 / December 2020
- Published online by Cambridge University Press:
- 23 November 2020, pp. 1029-1044
- Print publication:
- December 2020
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- Article
- Export citation
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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.
