IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v172y2024ics0304414924000589.html
   My bibliography  Save this article

Revisit of a Diaconis urn model

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414924000589
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2024.104352?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li-Xin, Zhang, 2006. "Asymptotic results on a class of adaptive multi-treatment designs," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 586-605, March.
    2. Bai, Z. D. & Hu, Feifang, 1999. "Asymptotic theorems for urn models with nonhomogeneous generating matrices," Stochastic Processes and their Applications, Elsevier, vol. 80(1), pages 87-101, March.
    3. Feng, Yarong & Mahmoud, Hosam M., 2021. "Dynamic Pólya–Eggenberger urns," Statistics & Probability Letters, Elsevier, vol. 174(C).
    4. Soumaya Idriss, 2022. "Nonlinear Unbalanced Urn Models via Stochastic Approximation," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 413-430, March.
    5. Matthews, Peter C. & Rosenberger, William F., 1997. "Variance in randomized play-the-winner clinical trials," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 233-240, October.
    6. Moler, José A. & Plo, Fernando & San Miguel, Miguel, 2006. "An adaptive design for clinical trials with non-dichotomous response and prognostic factors," Statistics & Probability Letters, Elsevier, vol. 76(17), pages 1940-1946, November.
    7. Bélisle, Claude & Melfi, Vince, 2008. "Independence after adaptive allocation," Statistics & Probability Letters, Elsevier, vol. 78(3), pages 214-224, February.
    8. Janson, Svante, 2004. "Functional limit theorems for multitype branching processes and generalized Pólya urns," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 177-245, April.
    9. 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.
    10. Aletti, Giacomo & Ghiglietti, Andrea, 2017. "Interacting generalized Friedman’s urn systems," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2650-2678.
    11. Soumaya Idriss & Hosam Mahmoud, 2023. "Exact Covariances and Refined Asymptotics in Dichromatic Tenable Balanced Pólya Urn Schemes," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-16, June.
    12. Gopal K. Basak & Amites Dasgupta, 2006. "Central and Functional Central Limit Theorems for a Class of Urn Models," Journal of Theoretical Probability, Springer, vol. 19(3), pages 741-756, December.
    13. Dasgupta, Amites, 2024. "Azuma-Hoeffding bounds for a class of urn models," Statistics & Probability Letters, Elsevier, vol. 204(C).
    14. Yi, Yanqing & Wang, Xikui, 2007. "Goodness-of-fit test for response adaptive clinical trials," Statistics & Probability Letters, Elsevier, vol. 77(10), pages 1014-1020, June.
    15. Kotz, Samuel & Mahmoud, Hosam & Robert, Philippe, 2000. "On generalized Pólya urn models," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 163-173, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:172:y:2024:i:c:s0304414924000589. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.