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 $${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$WNWN∗, where $${\mathbf{W}}_N = ({\mathbf{W}}_N^{(1)T}, \ldots , {\mathbf{W}}_N^{(M)T})^{T}$$WN=(WN(1)T,…,WN(M)T)T is a $${\textit{ML}} \times N$$ML×N block-line matrix whose block-lines $$({\mathbf{W}}_N^{(m)})_{m=1, \ldots , M}$$(WN(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 $$\frac{{\textit{ML}}}{N} \rightarrow c_* (c_* \in (0, \infty ))$$MLN→c∗(c∗∈(0,∞)), then the empirical eigenvalue distribution of $${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$WNWN∗ 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(Nα) with $$\alpha < 2/3$$α<2/3, then, almost surely, for $$N$$N large enough, the eigenvalues of $${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$WNWN∗ are located in the neighbourhood of the Marcenko–Pastur distribution. It is conjectured that the condition $$\alpha < 2/3$$α<2/3 is optimal.