Chaoticity for multi-class systems and exchangeability within classes

TL;DR

This paper extends classical exchangeability results to multi-class systems, showing how their conditional laws relate to independent uniform orderings and establishing convergence criteria and asymptotic independence between classes.

Contribution

It introduces a natural partial exchangeability framework for multi-class systems and characterizes their convergence and independence properties, extending Hewitt-Savage law implications.

Findings

Conditional law corresponds to independent uniform orderings within classes

Convergence of systems is equivalent to convergence of empirical measures

Asymptotic independence between classes when each converges to i.i.d. systems

Abstract

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.