Basile Morcrette ; Hosam M. Mahmoud

Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology
dmtcs:2996 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2012,
DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)

https://doi.org/10.46298/dmtcs.2996
Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic MethodologyArticle
Authors: Basile Morcrette ^{1,}^{2}; Hosam M. Mahmoud ^{3}
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Basile Morcrette;Hosam M. Mahmoud
1 Algorithms
2 Algorithmes, Programmes et Résolution
3 Department of Statistics [Washington]
This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.
Volume: DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)
Kiyoshi Inoue, 2023, Exactly solvable urn models under random replacement schemes and their applications, Journal of Applied Probability, 60, 3, pp. 835854, 10.1017/jpr.2022.103.
Léon Brenig, Fundamental theories of physics, Simulating Deterministic Dynamics by Drawing Coloured Balls at Random in Urns, pp. 117134, 2022, 10.1007/9783031044588_6.
Cyril Banderier;Philippe Marchal;Michael Wallner, 2020, Periodic Pólya urns, the density method and asymptotics of Young tableaux, Annals of Probability, 48, 4, 10.1214/19aop1411, https://doi.org/10.1214/19aop1411.