On the Almost Sure Location of the Singular Values of Certain Gaussian Block-Hankel Large Random Matrices

This paper studies the almost sure location of the eigenvalues of matrices , where is a block-line matrix whose block-lines are independent identically distributed Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if and , then the empirical eigenvalue distribution of converges almost surely towards the Marcenko-Pastur distribution. More importantly, it is established using the Haagerup-Schultz-Thorbjornsen ideas that if with , then, almost surely, for large enough, the eigenvalues of are located in the neighbourhood of the Marcenko-Pastur distribution. It is conjectured that the condition is optimal.