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

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.