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Central limit theorems for a hypergeometric randomly reinforced urn

Part of:Stochastic processesProbability theory on algebraic and topological structuresLimit theorems

Published online by Cambridge University Press:  24 October 2016

Irene Crimaldi
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
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Abstract

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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.

Keywords

Central limit theoremPòlya urnpreferential attachmentrandom process with reinforcementrandomly reinforced urnstable convergence

MSC classification

Primary:60F05: Central limit and other weak theorems
Secondary:60G57: Random measures60B10: Convergence of probability measures60G42: Martingales with discrete parameter

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016, pp. 899 - 913
Copyright
Copyright © Applied Probability Trust 2016 

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