Rate of convergence for traditional P\'olya urns

TL;DR

This paper establishes the rate at which the proportions in a traditional Pólya urn converge to the Dirichlet distribution, showing a convergence rate of Θ(1/n) in various metrics, extending previous results.

Contribution

The paper provides a new proof for the convergence rate of Pólya urn proportions, using direct calculations and extending the known rate to multiple metrics.

Findings

Convergence rate is Θ(1/n) in the minimal L_p metric for all p in [1, ∞].

The same rate applies for the Lévy distance.

The Kolmogorov distance rate depends on initial urn composition.

Abstract

Consider a P\'olya urn with balls of several colours, where balls are drawn sequentially and each drawn ball immediately is 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 L\'evy 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.