2016/03/28 by Leonetti, Paolo
#44A60 #60G09 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1603.08442
Given a partition \I1,…,Ik\ of \1,…,n\, let (X1,…,Xn) be random vector with each Xi taking values in an arbitrary measurable space (S,\mathscrS) such that their joint law is invariant under finite permutations of the indexes within each class Ij. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class Ij. 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 (X1,…,Xn) 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.