Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Antonio Auffinger, Gérard Ben Arous, and Sandrine Péché

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Abstract

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in (Electron. Commun. Probab. 9 (2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

Résumé

On étudie la loi des plus grandes valeurs propres de matrices aléatoires symétriques réelles et de covariance empirique quand les coefficients des matrices sont à queue lourde. On étend le résultat obtenu par Soshnikov dans (Electron. Commun. Probab. 9 (2004) 82–91) et on montre que le comportement asymptotique des plus grandes valeurs propres est déterminé par les plus grandes entrées de la matrice.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 3 (2009), 589-610.

Dates
First available: 4 August 2009

Permanent link to this document
http://projecteuclid.org/euclid.aihp/1249391376

Digital Object Identifier
doi:10.1214/08-AIHP188

Zentralblatt MATH identifier
05611480

Mathematical Reviews number (MathSciNet)
MR2548495

Subjects
Primary: 15A52 62G32: Statistics of extreme values; tail inference 60G55: Point processes

Keywords
Largest eigenvalues statistics Extreme values Random matrices Heavy tails

Citation

Auffinger, Antonio; Ben Arous, Gérard; Péché, Sandrine. Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 45 (2009), no. 3, 589--610. doi:10.1214/08-AIHP188. http://projecteuclid.org/euclid.aihp/1249391376.


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