Journal of the Mathematical Society of Japan

Riesz measures and Wishart laws associated to quadratic maps

Piotr GRACZYK and Hideyuki ISHI

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Abstract

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset$ R$^n$ is a regular open convex cone. In this context we prove new general formulas for moments of the Wishart laws on non-symmetric cones. For homogeneous cases, all the quadratic maps are characterized and the associated Riesz measure and Wishart law with its moments are described explicitly. We apply the theory of relatively invariant distributions and a matrix realization of homogeneous cones obtained recently by the second author.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 1 (2014), 317-348.

Dates
First available in Project Euclid: 24 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1390600847

Digital Object Identifier
doi:10.2969/jmsj/06610317

Mathematical Reviews number (MathSciNet)
MR3161403

Zentralblatt MATH identifier
1284.62314

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices 43A35: Positive definite functions on groups, semigroups, etc.

Keywords
convex cones homogeneous cones Riesz measures Wishart laws

Citation

GRACZYK, Piotr; ISHI, Hideyuki. Riesz measures and Wishart laws associated to quadratic maps. J. Math. Soc. Japan 66 (2014), no. 1, 317--348. doi:10.2969/jmsj/06610317. https://projecteuclid.org/euclid.jmsj/1390600847


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