Revisit of a Diaconis urn model

TL;DR

This paper revisits a group-based urn model introduced by Diaconis, generalizes it to allow random draws, and proves almost sure convergence to uniformity along with the asymptotic distribution of the urn composition.

Contribution

It extends the Diaconis urn model by incorporating random number of draws and establishes convergence and distribution results.

Findings

Normalized urn composition converges almost surely to uniform distribution.

Asymptotic joint distribution derived using martingale CLT.

Generalization to random number of balls drawn at each stage.

Abstract

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.