On the condensed density of the generalized eigenvalues of pencils of Hankel Gaussian random matrices and applications

TL;DR

This paper derives a closed-form approximation for the condensed density of generalized eigenvalues of Hankel Gaussian random matrix pencils, with implications for moments problems and numerical analysis.

Contribution

It introduces a novel approximation method for the distribution of eigenvalues in Hankel Gaussian matrix pencils with nonzero mean and diverse covariance.

Findings

Provides a closed-form approximation for the eigenvalue density.

Demonstrates applications to moments problems.

Includes numerical examples validating the approach.

Abstract

Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean and not identical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided.