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

Pencils of matrices whose elements have a joint noncentral Gaussian distribution with nonidentical 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.