## Journal of the Mathematical Society of Japan

### Riesz measures and Wishart laws associated to quadratic maps

#### 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

https://projecteuclid.org/euclid.jmsj/1390600847

Digital Object Identifier
doi:10.2969/jmsj/06610317

Mathematical Reviews number (MathSciNet)
MR3161403

Zentralblatt MATH identifier
1284.62314

#### 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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