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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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    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.
    4. 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.
    5. Balaji, Srinivasan & Mahmoud, Hosam M. & Watanabe, Osamu, 2006. "Distributions in the Ehrenfest process," Statistics & Probability Letters, Elsevier, vol. 76(7), pages 666-674, April.
    6. 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.
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