Riesz measures and Wishart laws associated to quadratic maps

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset \real^n$ is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone $\Omega$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones

Characterization of Wishart–Laplace distributions via Jordan algebra homomorphisms

AbstractFor a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart–Laplace distribution (the distribution of the difference of two independent central Wishart Wp(mi,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Qk(Y)} is independent Laplace–Wishart if and only if in addition to the aforementioned conditions, the images ρk(Σ+) of the Moore–Penrose inverse Σ+ of Σ are mutually orthogonal: ρk(Σ+)ρℓ(Σ+)=0 for k≠ℓ

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

AbstractLet Y be an n×p multivariate normal random matrix with general covariance ΣY and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed

Wishart laws and variance function on homogeneous cones

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.Comment: 24 pages, Probab. Math. Statist (2019