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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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.

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

First available in Project Euclid: 24 January 2014

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Zentralblatt MATH identifier

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

convex cones homogeneous cones Riesz measures Wishart laws


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.

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