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

Philippe Loubaton ()
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Philippe Loubaton: Université Paris-Est

Journal of Theoretical Probability, 2016, vol. 29, issue 4, 1339-1443

Abstract: Abstract 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

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; 60B20; 15B52 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-015-0614-z

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