The Matsumoto–Yor Property and Its Converse on Symmetric Cones

Bartosz Kołodziejek ()
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Bartosz Kołodziejek: Warsaw University of Technology

Journal of Theoretical Probability, 2017, vol. 30, issue 2, 624-638

Abstract: Abstract The Matsumoto–Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As was observed in Letac and Wesołowski (Ann Probab 28:1371–1383, 2000), the natural framework for the multivariate MY property is symmetric cones; however, they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.

Keywords: Matsumoto–Yor property; Generalized inverse Gaussian distributions; Wishart distributions; Symmetric cones; Hua’s identity; Functional equation; Primary 62E10; Secondary 60E05 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0648-2

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