Spherical ensembles

Abstract

Four classes of matrix variate spherical distributions are defined for real normed division algebras, after which some of the properties of these classes of distributions are studied. In addition, matrix variate elliptical distributions are defined and their corresponding densities found. In an analogous way to the procedure used for the Hermite ensemble, we construct vector-spherical ensembles and spherical ensembles for real normed division algebras, which contain as particular cases the Hermite and Fourier ensembles, among many others. In addition, we construct the vector-spherical-Laguerre and spherical-Laguerre ensembles for real normed division algebras, which in turn contain as particular cases the classical Laguerre and Jacobi ensembles, among many others. For each of these ensembles, the joint element density and the joint density of the eigenvalues are obtained. Finally, two classes of tridiagonal matrix models are constructed for the general (β>0) β-vector-spherical and β-vector-spherical-Laguerre ensembles, extending some of the results in Dumitriu (2002) [14]. Observe that, potentially, our results contain an infinite number of ensembles, analogous to the Hermite and classical Laguerre ensembles for real normed division algebras.