Chaoticity for multi-class systems and exchangeability within classes

Carl Graham (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

Classical results for exchangeable systems of random variables are extended to multi-class systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multi-class system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within each class, and that a family of such systems converges in law if and only if the corresponding empirical measure vectors converge in law. As a corollary, convergence within each class to an infinite i.i.d. system implies asymptotic independence between different classes. A result implying the Hewitt-Savage 0-1 Law is also extended.

Mots clés: Interacting particle systems; multi-class; multi-type; multi-species; mixtures; partial exchangeability; chaoticity; convergence of empirical measures; de Finetti Theorem; directing measures; Hewitt-Savage 0-1 Law

Domaines: Probabilités [math.PR]

Carl Graham. Chaoticity for multi-class systems and exchangeability within classes. Journal of Applied Probability, 2008, 45 (4), pp.1196-1203. ⟨10.1239/jap/1231340243⟩. ⟨hal-00171668v4⟩