## Article

#### Title: Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws

##### Author(s): Paolo Leonetti

##### Issue: Volume 80 Series A Part 2 Year 2018

###### Pages: 195 -- 214

###### 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 $(S, {\mathcal S})$ 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 $I_j$. 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.