Central limit theorems for a hypergeometric randomly reinforced urn

TL;DR

This paper establishes advanced central limit theorems for a hypergeometric reinforced urn model, enabling more accurate asymptotic analysis of the urn's composition and extending to multiple urns with shared random influences.

Contribution

It introduces new CLTs for hypergeometric reinforced urns with simultaneous ball draws and additions, including multiple urns under common random factors, strengthening the theoretical understanding.

Findings

Proved CLTs in the sense of stable and almost sure conditional convergence.

Derived asymptotic confidence intervals for the unknown limit proportion.

Extended results to multiple urns with shared random influences.

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.