Comptes Rendus
Functional Analysis/Probability Theory
Sharp bounds on the rate of convergence of the empirical covariance matrix
[Ordre asymptotique des valeurs singulières extrêmes de la matrice de covariance empirique]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 195-200.

Soient X1,,XNRn des vecteurs aléatoires indépendants centrés, de matrice de covariance l'identité et à densité log-concave. On démontre qu'avec une grande probabilité, on a

supxSn1|1Ni=1N(|Xi,x|2E|Xi,x|2)|CnN,
C>0 est une constante numérique. Ce résultat reste vrai dans le cadre beaucoup plus général où les formes linéaires (Xi,x)iN,xSn1 et les normes euclidiennes (|Xi|/n)iN vérifient des inégalités de type sous-exponentiel. Il en résulte que si A désigne la matrice dont les colonnes sont (Xi), alors avec grande probabilité, les valeurs singulières extrêmes λmin et λmax de AA vérifient 1CnNλminNλmaxN1+CnN, ce qui est une version quantitative du théorème de Bai–Yin (Z.D. Bai, Y.Q. Yin, 1993 [4]) bien connu pour les matrices aléatoires à coefficients i.i.d.

Let X1,,XNRn be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has

supxSn1|1Ni=1N(|Xi,x|2E|Xi,x|2)|CnN,
where C is an absolute positive constant. This result is valid in a more general framework when the linear forms (Xi,x)iN,xSn1 and the Euclidean norms (|Xi|/n)iN exhibit uniformly a sub-exponential decay. As a consequence, if A denotes the random matrix with columns (Xi), then with overwhelming probability, the extremal singular values λmin and λmax of AA satisfy the inequalities 1CnNλminNλmaxN1+CnN which is a quantitative version of Bai–Yin theorem (Z.D. Bai, Y.Q. Yin, 1993 [4]) known for random matrices with i.i.d. entries.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.12.014

Radosław Adamczak 1 ; Alexander E. Litvak 2 ; Alain Pajor 3 ; Nicole Tomczak-Jaegermann 2

1 Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
3 Équipe d'analyse et mathématiques appliquées, université Paris Est, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee cedex 2, France
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     title = {Sharp bounds on the rate of convergence of the empirical covariance matrix},
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Radosław Adamczak; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann. Sharp bounds on the rate of convergence of the empirical covariance matrix. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 195-200. doi : 10.1016/j.crma.2010.12.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.12.014/

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[4] Z.D. Bai; Y.Q. Yin Limit of the smallest eigenvalue of a large dimensional sample covariance matrix, Ann. Probab., Volume 21 (1993), pp. 1275-1294

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Cité par Sources :

The research was conducted while the authors participated in the Thematic Program on Asymptotic Geometric Analysis at the Fields Institute in Toronto in Fall 2010.

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