Geodesic
Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) (2012).

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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. 
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     author = {Morcrette, Basile and Mahmoud, Hosam M.},
     title = {Exactly {Solvable} {Balanced} {Tenable} {Urns} with {Random} {Entries} via the {Analytic} {Methodology}},
     journal = {Discrete mathematics & theoretical computer science},
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     volume = {DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)},
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Morcrette, Basile; Mahmoud, Hosam M. Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) (2012). doi : 10.46298/dmtcs.2996. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2996/

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