Geodesic
Distribution of eigenvalues of sample covariance matrices with tensor product samples
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 1, pp. 82-98 
D. Tieplova. Distribution of eigenvalues of sample covariance matrices with tensor product samples. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 1, pp. 82-98. http://geodesic.mathdoc.fr/item/JMAG_2017_13_1_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the $n^2\times n^2$ real symmetric and hermitian matrices $M_n$, which are equal to the sum $m_n$ of tensor products of the vectors $X^\mu=B(Y^\mu\otimes Y^\mu)$, $\mu=1,\dots,m_n$, where $Y^\mu$ are i.i.d. random vectors from $\mathbb{R}^n(\mathbb{C}^n)$ with zero mean and unit variance of components, and $B$ is an $n^2\times n^2$ positive definite non-random matrix. We prove that if $m_n/n^2\to c\in[0,+\infty)$ and the Normalized Counting Measure of eigenvalues of $BJB$, where $J$ is defined below in (2.6), converges weakly, then the Normalized Counting Measure of eigenvalues of $M_n$ converges weakly in probability to a non-random limit, and its Stieltjes transform can be found from a certain functional equation.

[1] N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, New York, 1993 | MR | Zbl

[2] G. Akemann, J. Baik, P. Di Francesco, The Oxford Handbook of Random Matrix Theory, Oxford Univ. Press, Oxford, 2011 | MR | Zbl

[3] Z. D. Bai, J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, Springer, New York, 2010 | MR | Zbl

[4] P. J. Forrester, Log-Gases and Random Matrices, Princeton Univ. Press, Princeton, New York, 2010 | MR | Zbl

[5] J. S. Geronimo, T. P. Hill, “Necessary and Suffcient Condition that the Limit of Stieltjes Transforms is the Stieltjes Transform”, J. Approx. Theory, 2003 | MR

[6] V.L. Girko, Theory of Stochastic Canonical Equations, v. I, II, Kluwer, Dordrecht, 2001 | MR

[7] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976 | MR | Zbl

[8] A. Lytova, L. Pastur, “Central Limit Theorem for Linear Eigenvalue Statistics of Random Matrices with Independent Entries”, Annals of Probability, 37:5 (2009), 1778–1840 | DOI | MR | Zbl

[9] V. Marchenko, L. Pastur, “The Eigenvalue of Distribution in Some Ensembles of Random Matrices”, Math. USSR Sbornik, 1 (1967) | MR

[10] L. Pastur, M. Shcherbina, Eigenvalue Distribution of Large Random Matrices, Mathematical Survives and Monographs, 171, AMS, Providence, RI, 2011 | DOI | MR | Zbl

[11] M. Shcherbina, “Central Limit Theorem for Linear Eigenvalue Statistics of the Wigner and Sample Covariance Random Matrices”, J. Math. Phys., Anal., Geom., 7:2 (2011), 176–192 | MR | Zbl