Journal of Theoretical Probability

, Volume 29, Issue 4, pp 1339–1443 | Cite as

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

Article

Abstract

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

Keywords

Singular value limit distribution of random complex Gaussian large block-Hankel matrices Almost sure location of the singular values Marcenko–Pastur distribution Poincaré–Nash inequality Integration by parts formula 

Mathematics Subject Classification (2010)

60B20 15B52 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratoire d’Informatique Gaspard Monge, UMR CNRS 8049Université Paris-EstMarne la Vallée Cedex 2France

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