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.

中文翻译：

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传统 Pólya 骨灰盒的收敛速度

考虑一个带有多种颜色球的 Pólya 瓮，其中球是按顺序抽取的，每个抽取的球会立即与固定数量的相同颜色的球一起替换。众所周知，不同颜色的球的比例在分布上收敛于狄利克雷分布。我们证明收敛速度是$\Theta(1/n)$在最小的$L_p$任何指标$p\in[1,\infty]$，扩展了 Goldstein 和 Reinert 的结果；我们进一步展示了 Lévy 距离的相同比率，而 Kolmogorov 距离的比率取决于参数，即取决于瓮的初始组成。这里使用的方法不同于 Goldstein 和 Reinert 使用的方法，它使用基于已知精确分布的直接计算。