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 WNW∗N , where WN=(W(1)TN,…,W(M)TN)T is a ML×N block-line matrix whose block-lines (W(m)N)m=1,…,M are independent identically distributed L×N Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if M→+∞ and MLN→c∗(c∗∈(0,∞)) , then the empirical eigenvalue distribution of WNW∗N converges almost surely towards the Marcenko–Pastur distribution. More importantly, it is established using the Haagerup–Schultz–Thorbjornsen ideas that if L=O(Nα) with α<2/3 , then, almost surely, for N large enough, the eigenvalues of WNW∗N are located in the neighbourhood of the Marcenko–Pastur distribution. It is conjectured that the condition α<2/3 is optimal.
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Philippe Loubaton. On the almost sure location of the singular values of certain Gaussian block-Hankel large random matrices. Journal of Theoretical Probability, Springer, 2016, 29 (4), pp.1339-1443. ⟨10.1007/s10959-015-0614-z⟩. ⟨hal-01579037⟩

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