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Revisit of a Diaconis urn model

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  • Yang, Li
  • Hu, Jiang
  • Bai, Zhidong

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 (2005) 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.

Suggested Citation

  • Yang, Li & Hu, Jiang & Bai, Zhidong, 2024. "Revisit of a Diaconis urn model," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    • Handle: RePEc:eee:spapps:v:172:y:2024:i:c:s0304414924000589
    DOI: 10.1016/j.spa.2024.104352
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    References listed on IDEAS

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    1. Bai, Z. D. & Hu, Feifang & Shen, Liang, 2002. "An Adaptive Design for Multi-Arm Clinical Trials," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 1-18, April.
    2. Smythe, R. T., 1996. "Central limit theorems for urn models," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 115-137, December.
    3. A. Abrams & H. Landau & Z. Landau & J. Pommersheim & E. Zaslow, 2007. "Random Multiplication Approaches Uniform Measure in Finite Groups," Journal of Theoretical Probability, Springer, vol. 20(1), pages 107-118, March.
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