IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v25y2023i2d10.1007_s11009-023-10034-1.html
   My bibliography  Save this article

Exact Covariances and Refined Asymptotics in Dichromatic Tenable Balanced Pólya Urn Schemes

Author

Listed:
  • Soumaya Idriss

    (Monastir University)

  • Hosam Mahmoud

    (The George Washington University)

Abstract

In the present paper, we provide exact expressions and sharp asymptotics for the covariance matrix of a dichromatic time-dependent Pólyaurn. We follow a purely combinatorial approach. Although we take interest in the large-index case with a fixed replacement matrix (as it was left open in Mahmoud (2022)), the combinatorial approach we pursue also produces exact expressions and sharp asymptotics for both small- and critical-index urn schemes with fixed replacement matrices, as well as schemes with dynamic (time-dependent matrices). We provide examples elucidating each case, including two dynamic schemes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:2:d:10.1007_s11009-023-10034-1
    DOI: 10.1007/s11009-023-10034-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-023-10034-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-023-10034-1?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. Smythe, R. T., 1996. "Central limit theorems for urn models," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 115-137, December.
    2. Zhang, Panpan & Chen, Chen & Mahmoud, Hosam, 2015. "Explicit characterization of moments of balanced triangular Pólya urns by an elementary approach," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 149-153.
    3. 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.
    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. 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.
    2. Dasgupta, Amites, 2024. "Azuma-Hoeffding bounds for a class of urn models," Statistics & Probability Letters, Elsevier, vol. 204(C).
    3. Aletti, Giacomo & Ghiglietti, Andrea, 2017. "Interacting generalized Friedman’s urn systems," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2650-2678.
    4. 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.
    5. Crimaldi, Irene & Dai Pra, Paolo & Louis, Pierre-Yves & Minelli, Ida G., 2019. "Synchronization and functional central limit theorems for interacting reinforced random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 70-101.
    6. Feng, Yarong & Mahmoud, Hosam M., 2021. "Dynamic Pólya–Eggenberger urns," Statistics & Probability Letters, Elsevier, vol. 174(C).
    7. Berti, Patrizia & Crimaldi, Irene & Pratelli, Luca & Rigo, Pietro, 2010. "Central limit theorems for multicolor urns with dominated colors," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1473-1491, August.
    8. Michael D Nicholson & Tibor Antal, 2019. "Competing evolutionary paths in growing populations with applications to multidrug resistance," PLOS Computational Biology, Public Library of Science, vol. 15(4), pages 1-25, April.
    9. José Moler & Fernando Plo & Henar Urmeneta, 2013. "A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 46-61, March.
    10. 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.
    11. Mailler, Cécile & Marckert, Jean-François, 2022. "Parameterised branching processes: A functional version of Kesten & Stigum theorem," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 339-377.
    12. Yang, Li & Hu, Jiang & Bai, Zhidong, 2024. "Revisit of a Diaconis urn model," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    13. Kortchemski, Igor, 2015. "A predator–prey SIR type dynamics on large complete graphs with three phase transitions," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 886-917.
    14. Yuan, Ao & Chai, Gen Xiang, 2008. "Optimal adaptive generalized Polya urn design for multi-arm clinical trials," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 1-24, January.
    15. Dimitris Cheliotis & Dimitra Kouloumpou, 2022. "Functional Limit Theorems for the Pólya Urn," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2038-2051, September.
    16. Kaj, Ingemar & Tahir, Daniah, 2019. "Stochastic equations and limit results for some two-type branching models," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 35-46.
    17. Kaur, Gursharn & Choi, Kwok Pui & Wu, Taoyang, 2023. "Distributions of cherries and pitchforks for the Ford model," Theoretical Population Biology, Elsevier, vol. 149(C), pages 27-38.
    18. Patrizia Berti & Irene Crimaldi & Luca Pratelli & Pietro Rigo, 2009. "Central Limit Theorems For Multicolor Urns With Dominated Colors," Quaderni di Dipartimento 106, University of Pavia, Department of Economics and Quantitative Methods.
    19. Berbeglia, Franco & Berbeglia, Gerardo & Van Hentenryck, Pascal, 2021. "Market segmentation in online platforms," European Journal of Operational Research, Elsevier, vol. 295(3), pages 1025-1041.
    20. Chen Chen & Hosam Mahmoud, 2018. "The continuous-time triangular Pólya process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(2), pages 303-321, April.

    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:spr:metcap:v:25:y:2023:i:2:d:10.1007_s11009-023-10034-1. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.