Beta-Riesz distributions on symmetric cones

Some new properties of the Riesz probability distribution

“…We close this section by the following result (Hassairi et al. ) concerning the Riesz probability distribution.…”

“…This probability distribution represents an important generalization of Wishart probability distribution; it has also allowed the definition of many other probability distributions such as the beta‐Riesz and the Riesz–Dirichlet probability distributions. Many properties of the Riesz and the related probability distributions have been established . Of course, all the results concerning the Riesz probability distribution have particular forms corresponding to the particular case of the Wishart; in other words, the results may be seen as extensions to the Riesz probability distribution of the properties of the Wishart probability distribution.…”

On the projections of the Dirichlet probability distribution on symmetric cones

“…It relies on the following fundamental theorem proved by Hassairi et al [4]. Recall that the absolutely continuous Riesz distribution on a symmetric cone is defined by these authors by…”

On Riesz distribution

“…Essentially, these advances have been archived through two approaches based on the theory of Jordan algebra and the real normed division algebras. A basic source of the general theory of symmetric cones under Jordan algebras can be found in Faraut and Korányi (1994); and specifically, some works in the context of distribution theory in symmetric cones based on Jordan algebras are provided in Massam (1994), Casalis and Letac (1996), Hassairi and Lajmi (2001), and Hassairi et al (2005), and the references therein. In the field of spherical functions (Jack polynomials including James' zonal polynomials) we can mention the work of Sawyer (1997).…”