Weighted $p$-radial Distributions on Euclidean and Matrix $p$-balls with Applications to Large Deviations

TL;DR

This paper introduces a probabilistic framework for weighted p-radial distributions on Euclidean and matrix p-balls, deriving large deviation principles and analyzing eigenvalue and singular value distributions.

Contribution

It extends p-radial distributions to matrix spaces and provides new large deviation results for spectral measures of random matrices.

Findings

Derived probabilistic representations for weighted p-radial distributions.

Established large deviation principles for empirical spectral measures.

Determined eigenvalue and singular value distributions in matrix p-balls.

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 $\ell_p^n$-ball, is derived. Large deviation principles for the empirical measure of the coordinates of random vectors on the $\ell_p^n$-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.