# Dynamic Pólya–Eggenberger urns

*Yarong Feng* and
*Hosam M. Mahmoud*

*Statistics & Probability Letters*, 2021, vol. 174, issue C

**Abstract:**
We study types of Pólya-Eggenberger urn schemes with a dynamic replacement matrix. We consider two-color schemes (white and blue) in this class. The nature of the governing limit stochastic relation exhibits a phase change when the addition function meets a regularity condition. We show that when the condition is met, the proportion of white balls in such an urn converges to a Bernoulli limit distribution. This is to be contrasted with dynamic Pólya-Eggenberger urns with slow addition function, where the limit distribution is believed to be continuous on the interval [0,1]. We demonstrate that the condition is satisfied when the addition function, along a subsequence of draws, grows exponentially (the exponential Pólya-Eggenberger urn) or faster (the super exponential Pólya-Eggenberger urn).

**Keywords:** Urn model; Random structure; Martingale; Limit distribution (search for similar items in EconPapers)

**Date:** 2021

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**DOI:** 10.1016/j.spl.2021.109089

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