IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v110y2004i2p177-245.html
   My bibliography  Save this article

Functional limit theorems for multitype branching processes and generalized Pólya urns

Author

Listed:
  • Janson, Svante

Abstract

A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the Athreya-Karlin embedding, these results yield asymptotic results for generalized Pólya urns. We investigate such results in detail and obtain explicit formulas for the asymptotic variances and covariances. The general formulas involve integrals of matrix functions; we show how they can be evaluated and simplified in important special cases. We also consider the numbers of drawn balls of different types and functional limit theorems for the urns. We illustrate our results by some examples, including several applications to random trees where our theorems and variance formulas give simple proofs of some known results; we also give some new results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:110:y:2004:i:2:p:177-245
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(03)00179-0
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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, 1999. "Asymptotic theorems for urn models with nonhomogeneous generating matrices," Stochastic Processes and their Applications, Elsevier, vol. 80(1), pages 87-101, March.
    2. Kotz, Samuel & Mahmoud, Hosam & Robert, Philippe, 2000. "On generalized Pólya urn models," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 163-173, August.
    3. Smythe, R. T., 1996. "Central limit theorems for urn models," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 115-137, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Brigitte Chauvin & Cécile Mailler & Nicolas Pouyanne, 2015. "Smoothing Equations for Large Pólya Urns," Journal of Theoretical Probability, Springer, vol. 28(3), pages 923-957, September.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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).
    19. Dasgupta, Amites, 2024. "Azuma-Hoeffding bounds for a class of urn models," Statistics & Probability Letters, Elsevier, vol. 204(C).
    20. Dimitris Cheliotis & Dimitra Kouloumpou, 2022. "Functional Limit Theorems for the Pólya Urn," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2038-2051, September.

    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. 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. Aletti, Giacomo & Ghiglietti, Andrea, 2017. "Interacting generalized Friedman’s urn systems," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2650-2678.
    3. 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.
    4. Davidson, Allison & D. Ward, Mark, 2018. "The characterization of tenable Pólya urns," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 38-43.
    5. 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.
    6. Franchini, Simone, 2017. "Large deviations for generalized Polya urns with arbitrary urn function," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3372-3411.
    7. Feng, Yarong & Mahmoud, Hosam M., 2021. "Dynamic Pólya–Eggenberger urns," Statistics & Probability Letters, Elsevier, vol. 174(C).
    8. Soumaya Idriss, 2022. "Nonlinear Unbalanced Urn Models via Stochastic Approximation," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 413-430, March.
    9. 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.
    10. 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.
    11. Wioletta M. Ruszel & Debleena Thacker, 2024. "Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation," Journal of Theoretical Probability, Springer, vol. 37(4), pages 2859-2885, November.
    12. Yanqing Yi & Yuan Yuan, 2013. "An optimal allocation for response-adaptive designs," Journal of Applied Statistics, Taylor & Francis Journals, vol. 40(9), pages 1996-2008, September.
    13. Yang, Li & Hu, Jiang & Bai, Zhidong, 2024. "Revisit of a Diaconis urn model," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    14. 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.
    15. 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.
    16. Dasgupta, Amites, 2024. "Azuma-Hoeffding bounds for a class of urn models," Statistics & Probability Letters, Elsevier, vol. 204(C).
    17. 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.
    18. 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:110:y:2004:i:2:p:177-245. 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.