Combinatorics, Probability and Computing Pólya Urns via the Contraction Method P ´ Olya Urns via the Contraction Method
Combinatorics, Probability and Computing Pólya Urns via the Contraction Method P ´ Olya Urns via the Contraction Method
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组合学、概率和通过收缩法计算 Pólya Urns
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通讯作者：
Margarete Knapeand
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作者：
Margarete Knape;Ralph Neininger;Margarete Knapeand
我们提出了一种方法来分析的P'olya瓮的渐近行为的基础上的压缩方法。为此，一个新的组合离散时间嵌入到随机根树的骨灰盒的演变。这些树的分解导致一个递归分布方程系统，该系统捕获每种颜色的球的数量分布。收缩法的思想被用来研究这种系统的递归分布方程的渐近。我们将我们的方法应用到一对夫妇的具体P'olya瓮，导致正常的极限分布，非正常的极限分布和渐近周期性分布行为的极限法律。
We propose an approach to analysing the asymptotic behaviour of P ´ olya urns based on the contraction method. For this, a new combinatorial discrete-time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each colour. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete P ´ olya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behaviour.