IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v70y2018i2d10.1007_s10463-016-0594-5.html
   My bibliography  Save this article

The continuous-time triangular Pólya process

Author

Listed:
  • Chen Chen

    (The George Washington University)

  • Hosam Mahmoud

    (The George Washington University)

Abstract

We study poissonized triangular (reducible) urns on two colors, which we take to be white and blue. We analyze the number of white and blue balls after a certain period of time has elapsed. We show that for balanced processes in this class, a different scaling is needed for each color to produce nontrivial limits, contrary to the distributions in the usual irreducible urns which only require the same scaling for both colors. The limit distributions (of the scaled variables) underlying triangular urns are Gamma. The technique we use couples partial differential equations with the method of moments applied in a bootstrapped manner to produce exact and asymptotic moments. For the dominant color, we get exact moments, while relaxing the balance condition. The exact moments include alternating signs and Stirling numbers of the second kind.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:2:d:10.1007_s10463-016-0594-5
    DOI: 10.1007/s10463-016-0594-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10463-016-0594-5
    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/s10463-016-0594-5?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. 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.
    2. Sparks, Joshua & Mahmoud, Hosam M., 2013. "Phases in the two-color tenable zero-balanced Pólya process," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 265-271.
    3. Balaji, Srinivasan & Mahmoud, Hosam M. & Watanabe, Osamu, 2006. "Distributions in the Ehrenfest process," Statistics & Probability Letters, Elsevier, vol. 76(7), pages 666-674, April.
    4. Srinivasan Balaji & Hosam Mahmoud, 2006. "Exact and limiting distributions in diagonal Pólya processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(4), pages 813-813, December.
    5. Zhang, Panpan & Mahmoud, Hosam M., 2016. "Distributions in a class of Poissonized urns with an application to Apollonian networks," Statistics & Probability Letters, Elsevier, vol. 115(C), pages 1-7.
    6. Srinivasan Balaji & Hosam Mahmoud, 2006. "Exact and Limiting Distributions in Diagonal Pólya Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 171-185, 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. Sparks, Joshua & Mahmoud, Hosam M., 2013. "Phases in the two-color tenable zero-balanced Pólya process," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 265-271.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Jun-ya Gotoh & Hui Jin & Ushio Sumita, 2011. "Numerical Evaluation of Dynamic Behavior of Ornstein–Uhlenbeck Processes Modified by Various Boundaries and its Application to Pricing Barrier Options," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 193-219, March.
    9. 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.
    10. 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.
    11. 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.
    12. Dimitris Cheliotis & Dimitra Kouloumpou, 2022. "Functional Limit Theorems for the Pólya Urn," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2038-2051, September.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. Mitsokapas, Evangelos & Harris, Rosemary J., 2022. "Decision-making with distorted memory: Escaping the trap of past experience," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
    18. Dasgupta, Amites, 2024. "Azuma-Hoeffding bounds for a class of urn models," Statistics & Probability Letters, Elsevier, vol. 204(C).
    19. Yuan Ao & Li Qizhai & Xiong Ming & Tan Ming T., 2016. "Adaptive Design for Staggered-Start Clinical Trial," The International Journal of Biostatistics, De Gruyter, vol. 12(2), pages 1-17, November.
    20. Kolesko, Konrad & Sava-Huss, Ecaterina, 2023. "Limit theorems for discrete multitype branching processes counted with a characteristic," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 49-75.

    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:aistmt:v:70:y:2018:i:2:d:10.1007_s10463-016-0594-5. 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.