Linear Eigenvalue Statistics of $X{X}^{\prime }$ matrices

Abstract

This article focuses on the fluctuations of linear eigenvalue statistics of $T_{n\times p}{T}_{n\times p}^{\prime }$, where $T_{n\times p}$ is an $n\times p$ Toeplitz matrix with real, complex or time-dependent entries. We show that as $n\to \infty$ and $p/n\to \lambda \in (0,\infty )$, 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 $H_{n\times p}{H}_{n\times p}^{\prime }$, when $H_{n\times p}$ is an $n\times p$ Hankel matrix. As a result of our studies, we also derive in-probability limit and a central limit theorem type result for Schettan norm of rectangular Toeplitz matrices. To establish the results, we use method of moments.

arXiv comment30 pages

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