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, S) 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. 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.

Finite Partially Exchangeable Laws are Signed Mixtures of Product Laws

Leonetti P
2018-01-01

Abstract

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, S) 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. 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.
2018
2018
https://link.springer.com/article/10.1007/s13171-017-0123-5
46A55; 60G09; 62E99; De Finetti representation; Finite partial exchangeability; Primary 44A60; Reduced Hausdorff moment problem; Secondary 15A24; Signed measure; True mixture
Leonetti, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2142095
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