CHAOTICITY FOR MULTICLASS SYSTEMS AND EXCHANGEABILITY WITHIN CLASSES

Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass 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 independent and identically distributed system implies asymptotic independence between different classes. A result implying the Hewitt-Savage 0-1 law is also extended.