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

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.

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.

DOI : 10.1214/08-AIHP188
Classification : 15A52, 62G32, 60G55
Mots-clés : 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},
     pages = {589--610},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP188},
     mrnumber = {2548495},
     zbl = {1177.15037},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/08-AIHP188/}
}
TY  - JOUR
AU  - Auffinger, Antonio
AU  - Ben Arous, Gérard
AU  - Péché, Sandrine
TI  - Poisson convergence for the largest eigenvalues of heavy tailed random matrices
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 589
EP  - 610
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/08-AIHP188/
DO  - 10.1214/08-AIHP188
LA  - en
ID  - AIHPB_2009__45_3_589_0
ER  - 
%0 Journal Article
%A Auffinger, Antonio
%A Ben Arous, Gérard
%A Péché, Sandrine
%T Poisson convergence for the largest eigenvalues of heavy tailed random matrices
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 589-610
%V 45
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/08-AIHP188/
%R 10.1214/08-AIHP188
%G en
%F 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, Tome 45 (2009) no. 3, pp. 589-610. doi : 10.1214/08-AIHP188. http://archive.numdam.org/articles/10.1214/08-AIHP188/

[1] Z. D. Bai, P. R. Krishnaiah and Y. Q. Yin. On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Related Fields 78 (1988) 509-521. | MR | Zbl

[2] Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16 (1988) 1729-1741. | MR | Zbl

[3] S. Belinschi, A. Dembo and A. Guionnet. Spectral measure of heavy tailed band and covariance random matrices. Commun. Math. Phys. (2009). To appear. | MR

[4] G. Ben Arous and A. Guionnet. The spectrum of heavy tailed random matrices. Comm. Math. Phys. 278 (2008) 715-751. | MR | Zbl

[5] R. Bhatia. Matrix Analysis. Springer, New York, 1996. | MR | Zbl

[6] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl

[7] G. Biroli, J. P. Bouchaud and M. Potters. On the top eigenvalue of heavy-tailed random matrices. Europhys. Lett. 78 (2007) 10001. | MR

[8] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York, 1966. | MR | Zbl

[9] O. Kallenberg. Foundations of Modern Probability. Springer, New York, 2001. | MR | Zbl

[10] V. Marchenko and L. Pastur. The distribution of eigenvalues in certain sets of random matrices. Mat. Sb. 72 (1967) 507-536. | MR | Zbl

[11] S. Péché and A. Soshnikov. Wigner random matrices with non-symmetrically distributed entries. J. Stat. Phys. 129 (2007) 857-884. | MR | Zbl

[12] R. Resnick. Extreme Values, Regular Variation and Point Processes 4. Springer, New York, 1987. | MR | Zbl

[13] A. Ruzmaikina. Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Comm. Math. Phys. 261 (2006) 277-296. | MR | Zbl

[14] A. Soshnikov. A note on universality of the distribution of largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108 (2002) 1033-1056. | MR | Zbl

[15] A. Soshnikov. Poisson statistics for the largest eigenvalues in random matrix ensembles. In Mathematical Physics of Quantum Mechanics 351-364. Lecture Notes in Phys. 690. Springer, Berlin, 2006. | MR | Zbl

[16] A. Soshnikov. Poisson statistics for the largest eigenvalue of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 (2004) 82-91. | MR | Zbl

[17] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697-733. | MR | Zbl

Cité par Sources :