Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Antonio Auffinger; Gérard Ben Arous; Sandrine Péché

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 589-610
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Auffinger, Antonio, Ben Arous, Gérard, and Péché, Sandrine. "Poisson convergence for the largest eigenvalues of heavy tailed random matrices." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 589-610. <http://eudml.org/doc/78035>.

@article{Auffinger2009,
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.},
author = {Auffinger, Antonio, Ben Arous, Gérard, Péché, Sandrine},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {largest eigenvalues statistics; extreme values; random matrices; heavy tails; largest value statistics; largest eigenvalues; random covariance matrices; random symmetric matrices},
language = {eng},
number = {3},
pages = {589-610},
publisher = {Gauthier-Villars},
title = {Poisson convergence for the largest eigenvalues of heavy tailed random matrices},
url = {http://eudml.org/doc/78035},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 589
EP - 610
AB - 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.
LA - eng
KW - largest eigenvalues statistics; extreme values; random matrices; heavy tails; largest value statistics; largest eigenvalues; random covariance matrices; random symmetric matrices
UR - http://eudml.org/doc/78035
ER -

References

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  13. [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. Zbl1130.82313MR2191882
  14. [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. Zbl1018.62042MR1933444
  15. [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. Zbl1169.15302MR2234922
  16. [16] A. Soshnikov. Poisson statistics for the largest eigenvalue of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 (2004) 82–91. Zbl1060.60013MR2081462
  17. [17] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697–733. Zbl1062.82502MR1727234

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