Linear eigenvalue statistics of XX′ matrices

Abstract

This article focuses on the fluctuations of linear eigenvalue statistics of Tn×pTn×p′, where Tn×p is an n × p Toeplitz matrix with real, complex, or time-dependent entries. We show that as n → ∞ with p/n → λ ∈ (0, ∞), the linear eigenvalue statistics of these matrices for polynomial test functions converge in distribution to Gaussian random variables. We also discuss the linear eigenvalue statistics of Hn×pHn×p′, when Hn×p is an n × p Hankel matrix. As a result of our studies, we derive in-probability limit and a central limit theorem type result for the Schettan norm of rectangular Toeplitz matrices. To establish the results, we use the method of moments.