A generalized urn with multiple drawing and random addition

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 Wn: the number of white balls after n draws. Under the self-reinforcement rule, we prove that, after suitable normalization, the number of white balls Wn converges almost surely to a random variable W which has an absolutely continuous distribution.