Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws.

Given a partition {I₁, …, I_k} of {1, …, n}, let (X₁, …, X_n) be random vector with each X_i taking values in an arbitrary measurable space (S,S)<inline-graphic></inline-graphic> 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. 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 particular, given a finite exchangeable sequence of Bernoulli random variables, the directing measure can be chosen nonnegative if and only if two effectively computable matrices are positive semi-definite. [ABSTRACT FROM AUTHOR]