LIMIT DISTRIBUTIONS FOR LARGE POLYA URNS

• 成果类型: Article
• 署名作者: Chauvin, Brigitte; Pouyanne, Nicolas; Sahnoun, Reda
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP696
• 发表日期: 2011
• 页码: 1-32
• 关键词: branching-processes; THEOREMS

摘要

We consider a two-color Polya urn in the case when a fixed number S of balls is added at each step. Assume it is a large urn that is, the second eigenvalue m of the replacement matrix satisfies 1/2 < m/S <= 1. After n drawings, the composition vector has asymptotically a first deterministic term of order n and a second random term of order n(m/S). The object of interest is the limit distribution of this random term. The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree m. The limit laws appear to constitute a new family of probability densities supported by the whole real line.