Characterization of the law of a finite exchangeable sequence through the finite-dimensional distributions of the empirical measure

A finite exchangeable sequence (ξ1,…,ξN) need not satisfy de Finetti's conditional representation, but there is a one-to-one relationship between its law and the law of its empirical measure, i.e. 1N∑i=1Nδξi. The aim of this paper is to identify the law of a finite exchangeable sequence through the finite-dimensional distributions of its empirical measure. The problem will be approached by singling out conditions that are necessary and sufficient so that a family of finite-dimensional distributions provides a complete characterization of the law of the empirical measure. This result is applied to construct laws of finite exchangeable sequences.