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A generalized urn with multiple drawing and random addition

Author

Listed:
  • Aguech Rafik

    (King Saoud University, Riyadh
    Département de Mathématiques, Faculté des Sciences de Monastir)

  • Lasmar Nabil

    (Institut Préparatoire aux Études d’ingénieurs de Monastir)

  • Selmi Olfa

    (Département de Mathématiques, Faculté des Sciences de Monastir)

Abstract

In this paper, we consider an unbalanced urn model with multiple drawing. At each discrete time step n, we draw m balls at random from an urn containing white and blue balls. The replacement of the balls follows either opposite or self-reinforcement rule. Under the opposite reinforcement rule, we use the stochastic approximation algorithm to obtain a strong law of large numbers and a central limit theorem for $$W_n$$ W n : the number of white balls after n draws. Under the self-reinforcement rule, we prove that, after suitable normalization, the number of white balls $$W_n$$ W n converges almost surely to a random variable $$W_\infty $$ W ∞ which has an absolutely continuous distribution.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:2:d:10.1007_s10463-018-0651-3
    DOI: 10.1007/s10463-018-0651-3
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    References listed on IDEAS

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    1. 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.
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