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Central limit theorems for a hypergeometric randomly reinforced urn. (English) Zbl 1351.60024

Summary: We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

MSC:

60F05 Central limit and other weak theorems
60G57 Random measures
60B10 Convergence of probability measures
60G42 Martingales with discrete parameter