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 \(W_n\): the number of white balls after n draws. Under the self-reinforcement rule, we prove that, after suitable normalization, the number of white balls \(W_n\) converges almost surely to a random variable \(W_\infty \) which has an absolutely continuous distribution.
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Acknowledgements
The first author is grateful to the King Saud University, Deanship of Scientific Research, College of Science Research Center. The authors are grateful to anonymous referees for their valuable comments and corrections that improved the presentation of this paper.
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Rafik, A., Nabil, L. & Olfa, S. A generalized urn with multiple drawing and random addition. Ann Inst Stat Math 71, 389–408 (2019). https://doi.org/10.1007/s10463-018-0651-3
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DOI: https://doi.org/10.1007/s10463-018-0651-3