Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws

Paolo Leonetti ()
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Paolo Leonetti: Università “Luigi Bocconi” – via Roentgen 1

Sankhya A: The Indian Journal of Statistics, 2018, vol. 80, issue 2, No 1, 195-214

Abstract: 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 ) $(S,\mathcal {S})$ 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.

Keywords: Finite partial exchangeability; Signed measure; De Finetti representation; True mixture; Reduced Hausdorff moment problem.; Primary 44A60; 60G09; Secondary 15A24; 46A55; 62E99. (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s13171-017-0123-5

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