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 $${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$ W N W N ∗ , where $${\mathbf{W}}_N = ({\mathbf{W}}_N^{(1)T}, \ldots , {\mathbf{W}}_N^{(M)T})^{T}$$ W N = ( W N ( 1 ) T , … , W N ( M ) T ) T is a $${\textit{ML}} \times N$$ ML × N block-line matrix whose block-lines $$({\mathbf{W}}_N^{(m)})_{m=1, \ldots , M}$$ ( W N ( m ) ) m = 1 , … , M are independent identically distributed $$L \times N$$ L × N Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if $$M \rightarrow +\infty $$ M → + ∞ and $$\frac{{\textit{ML}}}{N} \rightarrow c_* (c_* \in (0, \infty ))$$ ML N → c ∗ ( c ∗ ∈ ( 0 , ∞ ) ) , then the empirical eigenvalue distribution of $${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$ W N W N ∗ converges almost surely towards the Marcenko–Pastur distribution. More importantly, it is established using the Haagerup–Schultz–Thorbjornsen ideas that if $$L = O(N^{\alpha })$$ L = O ( N α ) with $$\alpha