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Characterization and Enumeration of Certain Classes of Tenable Pólya Urns Grown by Drawing Multisets of Balls

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  • Scott R. Konzem

    (The George Washington University
    Board of Governors of the Federal Reserve System)

  • Hosam M. Mahmoud

    (The George Washington University)

Abstract

We study necessary and sufficient conditions for the strong tenability of Pólya urn schemes under the sampling of multisets of balls. We also investigate sufficient conditions for the tenability (not necessarily in the strong sense) of Pólya urn schemes under the sampling of multisets of balls. We enumerate certain balanced classes and give algorithmic constructions for the replacement matrices for members in the class. We probabilistically analyze the zero-balanced tenable class, and find the asymptotic average proportion of each color, when the starting number of balls is large. We also give an algorithm to determine tenability and construct the Markov chain underlying the scheme, when it is tenable.

Suggested Citation

  • Scott R. Konzem & Hosam M. Mahmoud, 2016. "Characterization and Enumeration of Certain Classes of Tenable Pólya Urns Grown by Drawing Multisets of Balls," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 359-375, June.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:2:d:10.1007_s11009-014-9421-8
    DOI: 10.1007/s11009-014-9421-8
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Aguech Rafik & Lasmar Nabil & Selmi Olfa, 2019. "A generalized urn with multiple drawing and random addition," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 389-408, April.

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