Chaoticity for multi-class systems and echangeability within classes

Under the natural partial exchangeability assumption for multi-class interacting particle systems, we prove that these converge to an independent system with innite i.i.d. classes if and only if the empirical measure of each class satises a weak law of large numbers. This extension of a classical result for exchangeable systems (related to the de Finetti Theorem) is somewhat surprising, since then convergence of each class to innite i.i.d. particles implies asymptotic independence of particles of dierent classes.