Finite Partially Exchangeable Laws are Signed Mixtures of Product Laws

Abstract: Given a partition ${I_1,\ldots,I_k}$ of ${1,\ldots,n}$, let $(X_1,\ldots,X_n)$ be random vector with each $X_i$ taking values in an arbitrary measurable space $(S,\mathscr{S})$ such that their joint law is invariant under finite permutations of the indexes within each class $I_j$. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class $I_j$. The representation is unique if and only if the set of these signed measures is weakly compact. We provide a necessary condition for the existence of a nonnegative directing measure. This is related to the notions of infinite extendibility and reinforcement. In the special case where $(X_1,\ldots,X_n)$ is an exchangeable sequence of ${0,1}$-valued random variables, the directing measure can be chosen nonnegative if and only if two effectively computable matrices are positive semi-definite.