An algebraic approach to Pólya processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 293-323.

Les processus de Pólya sont une généralisation naturelle des modèles d'urnes de Pólya-Eggenberger. Cet article présente une nouvelle approche de leur comportement asymptotique via les moments, basée sur la décomposition spectrale d'un opérateur aux différences finies sur des espaces de polynômes. En particulier, elle fournit de nouveaux résultats sur les grands processus (un processus de Pólya est dit petit lorsque 1 est valeur propre simple de sa matrice de remplacement et lorsque toutes les autres valeurs propres ont une partie réelle ≤1/2 ; sinon, on dit qu’il est grand).

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).

DOI : 10.1214/07-AIHP130
Classification : 60F15, 60F17, 60F25, 60G05, 60G42, 60J05, 68W40
Mots-clés : Pólya processes, Pólya-Eggenberger urn processes, strong asymptotics, finite difference transition operator, vector-valued martingale methods 
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Pouyanne, Nicolas. An algebraic approach to Pólya processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 293-323. doi : 10.1214/07-AIHP130. http://www.numdam.org/articles/10.1214/07-AIHP130/

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