An algebraic approach to Pólya processes

Abstract

Pólya processes are natural generalizations of Pólya- Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small when 1 is a simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤ 1/2; otherwise, it is called large). © Association des Publications de l'Institut Henri Poincaré, 2008.