On the almost sure location of the singular values of certain Gaussian block-Hankel large random matrices

TL;DR

This paper proves that the eigenvalues of certain large Gaussian block-Hankel matrices almost surely follow the Marcenko-Pastur distribution under specific growth conditions, extending understanding of their spectral behavior.

Contribution

It establishes the almost sure convergence of eigenvalue distributions of Gaussian block-Hankel matrices to the Marcenko-Pastur law and characterizes eigenvalue localization for specific matrix dimensions.

Findings

Eigenvalue distribution converges to Marcenko-Pastur law as matrix size grows.

Eigenvalues are localized near the Marcenko-Pastur distribution under certain growth conditions.

Results hold for matrices with block sizes growing slower than N^{2/3}.

Abstract

This paper studies the almost sure location of the eigenvalues of matrices ${\bf W}_N {\bf W}_N^{*}$ where ${\bf W}_N = ({\bf W}_N^{(1)T}, ..., {\bf W}_N^{(M)T})^{T}$ is a $ML \times N$ block-line matrix whose block-lines $({\bf W}_N^{(m)})_{m=1, ..., M}$ are independent identically distributed $L \times N$ Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if $M \rightarrow +\infty$ and $\frac{ML}{N} \rightarrow c_*$ ($c_* \in (0, \infty)$), then the empirical eigenvalue distribution of ${\bf W}_N {\bf W}_N^{*}$ converges almost surely towards the Marcenko-Pastur distribution. More importantly, it is established that if $L = \mathcal{O}(N^{\alpha})$ with $\alpha < 2/3$, then, almost surely, for $N$ large enough, the eigenvalues of ${\bf W}_N {\bf W}_N^{*}$ are located in the neighbourhood of the Marcenko-Pastur distribution.