A generalized urn with multiple drawing and random addition

Aguech Rafik (), Lasmar Nabil () and Selmi Olfa ()
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Aguech Rafik: King Saoud University, Riyadh
Lasmar Nabil: Institut Préparatoire aux Études d’ingénieurs de Monastir
Selmi Olfa: Département de Mathématiques, Faculté des Sciences de Monastir

Annals of the Institute of Statistical Mathematics, 2019, vol. 71, issue 2, No 6, 389-408

Abstract: 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.

Keywords: Unbalanced urn; Stochastic approximation; Martingale; Maximal inequality (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10463-018-0651-3

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