Weighted p-radial distributions on Euclidean and matrix p-balls with applications to large deviations

Abstract

A probabilistic representation for a class of weighted p-radial distributions, based on mixtures of a weighted cone probability measure and a weighted uniform distribution on the Euclidean ℓpn-ball, is derived. Large deviation principles for the empirical measure of the coordinates of random vectors on the ℓpn-ball with distribution from this weighted measure class are discussed. The class of p-radial distributions is extended to p-balls in classical matrix spaces, both for self-adjoint and non-self-adjoint matrices. The eigenvalue distribution of a self-adjoint random matrix, chosen in the matrix p-ball according to such a distribution, is determined. Similarly, the singular value distribution is identified in the non-self-adjoint case. Again, large deviation principles for the empirical spectral measures for the eigenvalues and the singular values are presented as an application.