A generalized urn with multiple drawing and random addition
Annals of the Institute of Statistical Mathematics(2018)
摘要
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_∞ which has an absolutely continuous distribution.
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关键词
Unbalanced urn, Stochastic approximation, Martingale, Maximal inequality
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