Poisson convergence for the largest eigenvalues of heavy tailed random matrices
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, p. 589-610

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.

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.

DOI : https://doi.org/10.1214/08-AIHP188
Classification:  15A52,  62G32,  60G55
Keywords: largest eigenvalues statistics, extreme values, random matrices, heavy tails
@article{AIHPB_2009__45_3_589_0,
     author = {Auffinger, Antonio and Ben Arous, G\'erard and P\'ech\'e, Sandrine},
     title = {Poisson convergence for the largest eigenvalues of heavy tailed random matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     pages = {589-610},
     doi = {10.1214/08-AIHP188},
     zbl = {1177.15037},
     mrnumber = {2548495},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_3_589_0}
}
Auffinger, Antonio; Ben Arous, Gérard; Péché, Sandrine. Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, pp. 589-610. doi : 10.1214/08-AIHP188. http://www.numdam.org/item/AIHPB_2009__45_3_589_0/

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