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 ${\mathbf{W}}_N{\mathbf{W}}_N^{\ast }$ where ${\mathbf{W}}_N=({\mathbf{W}}_N^{(1)T},...,{\mathbf{W}}_N^{(M)T}{)}^T$ is a $ML\times N$ block-line matrix whose block-lines $({\mathbf{W}}_N^{(m)}{)}_{m=1,...,M}$ are independent identically distributed $L\times N$ Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if $M\to +\infty$ and $\frac{ML}{N}\to c_{\ast }$ ($c_{\ast }\in (0,\infty )$), then the empirical eigenvalue distribution of ${\mathbf{W}}_N{\mathbf{W}}_N^{\ast }$ 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 ${\mathbf{W}}_N{\mathbf{W}}_N^{\ast }$ are located in the neighbourhood of the Marcenko-Pastur distribution.

67 pages. Revised version, to appear in Journal of Theoretical Probability, published on line at http://link.springer.com/article/10.1007/s10959-015-0614-z