论文标题
重新访问DiConis urn模型
Revisit of a Diaconis urn model
论文作者
论文摘要
令$ g $为有限的阿贝尔（Abelian）订单$ d $。我们认为最初有一个标记的球会产生组$ g $。用更换从urn中选择两个球，观察其标签，并在相应的组元素上执行组乘法以获得组元素。然后，我们将带有该结果元素标记的球放入urn中。该模型是由Diaconis提出的，同时研究了一种称为Meataxe（Holt and Rees（1994））的群体理论算法。 Siegmund和Yakir（2004）部分研究了该模型。在本文中，我们进一步调查并概括了该模型。更具体地说，我们允许在DIACONIS urn模型的每个阶段从urn中抽出随机数的球。对于这种情况，我们验证了归一化的URN组成几乎可以肯定地汇合到组$ g $的均匀分布。此外，我们通过使用Martingale Central limem定理获得了URN组成的渐近关节分布。
Let $G$ be a finite Abelian group of order $d$. We consider an urn in which, initially, there are labeled balls that generate the group $G$. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees (1994)). Siegmund and Yakir (2004) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group $G$. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem.
