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Dynamic Pólya–Eggenberger urns

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  • Feng, Yarong
  • Mahmoud, Hosam M.

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).

Suggested Citation

  • Feng, Yarong & Mahmoud, Hosam M., 2021. "Dynamic Pólya–Eggenberger urns," Statistics & Probability Letters, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:stapro:v:174:y:2021:i:c:s0167715221000511
    DOI: 10.1016/j.spl.2021.109089
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    References listed on IDEAS

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    1. Smythe, R. T., 1996. "Central limit theorems for urn models," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 115-137, December.
    2. Panpan Zhang & Hosam M. Mahmoud, 2020. "On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 625-645, June.
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