This paper studies the almost sure location of the eigenvalues of matrices WNWN∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$\end{document}, where WN=(WN(1)T,…,WN(M)T)T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{W}}_N = ({\mathbf{W}}_N^{(1)T}, \ldots , {\mathbf{W}}_N^{(M)T})^{T}$$\end{document} is a ML×N\documentclass[12pt]{minimal}
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\begin{document}$${\textit{ML}} \times N$$\end{document} block-line matrix whose block-lines (WN(m))m=1,…,M\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbf{W}}_N^{(m)})_{m=1, \ldots , M}$$\end{document} are independent identically distributed L×N\documentclass[12pt]{minimal}
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\begin{document}$$L \times N$$\end{document} Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if M→+∞\documentclass[12pt]{minimal}
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\begin{document}$$M \rightarrow +\infty $$\end{document} and MLN→c∗(c∗∈(0,∞))\documentclass[12pt]{minimal}
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\begin{document}$$\frac{{\textit{ML}}}{N} \rightarrow c_* (c_* \in (0, \infty ))$$\end{document}, then the empirical eigenvalue distribution of WNWN∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$\end{document} converges almost surely towards the Marcenko–Pastur distribution. More importantly, it is established using the Haagerup–Schultz–Thorbjornsen ideas that if L=O(Nα)\documentclass[12pt]{minimal}
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\begin{document}$$L = O(N^{\alpha })$$\end{document} with α<2/3\documentclass[12pt]{minimal}
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\begin{document}$$\alpha < 2/3$$\end{document}, then, almost surely, for N\documentclass[12pt]{minimal}
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\begin{document}$$N$$\end{document} large enough, the eigenvalues of WNWN∗\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf{W}}_N {\mathbf{W}}_N^{*}$$\end{document} are located in the neighbourhood of the Marcenko–Pastur distribution. It is conjectured that the condition α<2/3\documentclass[12pt]{minimal}
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\begin{document}$$\alpha < 2/3$$\end{document} is optimal.
