On the universality of the distribution of the generalized eigenvalues of a pencil of Hankel random matrices

TL;DR

This paper proves that the distribution of generalized eigenvalues of Hankel random matrix pencils exhibits universality under certain conditions, with implications for exponential interpolation in stationary processes.

Contribution

It establishes universality properties for eigenvalue distributions of Hankel matrix pencils and derives an integral representation of their density, independent of specific process distributions.

Findings

Distribution depends only on stationarity asymptotically

Integral representation of eigenvalue density derived

Universality holds under multivariate spherical distribution assumption

Abstract

Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.