Limit distributions for large Pólya urns.

Brigitte Chauvin (1, 2) , Nicolas Pouyanne (2) , Réda Sahnoun (2)

1 ALGORITHMS - Algorithms
2 LMV - Laboratoire de Mathématiques de Versailles

We consider a two colors Pólya urn with balance $S$. Assume it is a \emph{large} urn \emph{i.e.} the second eigenvalue $m$ of the replacement matrix satisfies $1/2 /l m/S \leq 1$. 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 di erential equations satis ed by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitely related to Abelian integrals on the Fermat curve of degree $m$.

Domaines: Probabilités [math.PR]

Brigitte Chauvin, Nicolas Pouyanne, Réda Sahnoun. Limit distributions for large Pólya urns.. The Annals of Applied Probability, 2011, 21 (1), pp.1-32. ⟨10.1214/10-AAP696⟩. ⟨hal-00403100⟩