no. 3
Probability theory
A short proof of the Marchenko–Pastur theorem
[Une courte démonstration du théorème de Marchenko–Pastur]

Présenté par : Jean-François le Gall

Yaskov, Pavel1, 2
1 Steklov Mathematical Institute of RAS, Russia
2 National University of Science and Technology MISIS, Russia
Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 319-322.

Nous prouvons le théorème de Marchenko–Pastur pour les matrices aléatoires avec des colonnes i.i.d. et une structure de dépendance générale à l'intérieur des colonnes par une simple modification de la méthode standard résolvante de Cauchy–Stieltjes standard.

We prove the Marchenko–Pastur theorem for random matrices with i.i.d. columns and a general dependence structure within the columns by a simple modification of the standard Cauchy–Stieltjes resolvent method.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.12.008
Yaskov, Pavel 1, 2

1 Steklov Mathematical Institute of RAS, Russia
2 National University of Science and Technology MISIS, Russia 
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Yaskov, Pavel. A short proof of the Marchenko–Pastur theorem. Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 319-322. doi : 10.1016/j.crma.2015.12.008. http://www.numdam.org/articles/10.1016/j.crma.2015.12.008/

[1] Adamczak, R. On the Marchenko–Pastur and circular laws for some classes of random matrices with dependent entries, Electron. J. Probab., Volume 16 (2011), pp. 1065-1095

[2] Adamczak, R. Some remarks on the Dozier–Silverstein theorem for random matrices with dependent entries, Random Matrices: Theory Appl., Volume 2 (2013) (46pp)

[3] Bai, Z.; Zhou, W. Large sample covariance matrices without independence structures in columns, Stat. Sin., Volume 18 (2008), pp. 425-442

[4] Bai, Z.; Silverstein, J.W. On the empirical distribution of eigenvalues of a class of large dimensional random matrices, J. Multivar. Anal., Volume 54 (1995), pp. 175-192

[5] Grenander, U.; Silverstein, J.W. Spectral analysis of networks with random topologies, SIAM J. Appl. Math., Volume 32 (1977) no. 2, pp. 499-519

[6] Hall, P. On the Lp convergence of sums of independent random variables, Math. Proc. Camb. Philos. Soc., Volume 82 (1977), pp. 439-446

[7] Jonsson, D. Some limit theorems for the eigenvalues of a sample covariance matrix, J. Multivar. Anal., Volume 12 (1982), pp. 1-38

[8] Marcenko, V.A.; Pastur, L.A. Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. (N. S.), Volume 72 (1967), pp. 507-536

[9] Pajor, A.; Pastur, L. On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution, Stud. Math., Volume 195 (2009), pp. 11-29

[10] Pastur, L. On the spectrum of random matrices, Teor. Mat. Fiz., Volume 10 (1972), pp. 102-112

[11] Pastur, L.; Shcherbina, M. Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, vol. 171, American Mathematical Society, Providence, RI, USA, 2011

[12] Tao, T. Topics in Random Matrix Theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, USA, 2012

[13] Wachter, K.W. The strong limits of random matrix spectra for sample matrices of independent elements, Ann. Probab., Volume 6 (1978) no. 1, pp. 1-18

[14] Yin, Y.Q.; Krishnaiah, P.R. Limit theorem for the eigenvalues of the sample covariance matrix when the underlying distribution is isotropic, Teor. Veroâtn. Primen., Volume 30 (1985) no. 4, pp. 810-816

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