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Central limit theorems for a hypergeometric randomly reinforced urn

Published online by Cambridge University Press:  24 October 2016

Irene Crimaldi*
Affiliation:
IMT School for Advanced Studies Lucca
*
* Postal address: IMT School for Advanced Studies Lucca, Piazza San Ponziano 6, 55100 Lucca, Italy. Email address: irene.crimaldi@imtlucca.it

Abstract

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2016 

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