We consider the generalization of the Pólya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under the uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.

中文翻译：

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均匀遍历性下无限色平衡瓮的强收敛

正如 [37]、[4]、[5] 和 [6] 中介绍的，我们考虑使用可能无限多种颜色的 Pólya 瓮方案的推广。对于可数的许多颜色，我们证明了 urn 配置在均匀遍历性相关马尔可夫链上的假设。该证明使用所选颜色序列的随机耦合与分支马尔可夫链在加权随机递归树如 [6]、[31] 和 [26] 中所述。使用这种耦合，我们估计任何两种选定颜色之间的协方差。特别是，我们重新证明了具有有限多种颜色的经典瓮模型的极限定理。