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 \((S,\mathcal {S})\) 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.
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Acknowledgments
The author is supported by a PhD scholarship from Università Bocconi. He thanks Nate Eldredge (Northern Colorado, US) and Pierpaolo Battigalli, Sandra Fortini, Fabio Maccheroni, Pietro Muliere, and Sonia Petrone (Università Bocconi, IT) for useful comments. He is also grateful to an anonymous referee for a careful proofreading that allowed a substantial improvement of the presentation.
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Leonetti, P. Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws. Sankhya A 80, 195–214 (2018). https://doi.org/10.1007/s13171-017-0123-5
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DOI: https://doi.org/10.1007/s13171-017-0123-5
Keywords and phrases.
- Finite partial exchangeability
- Signed measure
- De Finetti representation
- True mixture
- Reduced Hausdorff moment problem.