Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Abstract

Onétudie la loi des plus grandes valeurs propres de matrices aléatoires symétriques réelles et de covariance empirique quand les coefficients des matrices sontà queue lourde. Oń etend le résultat obtenu par A. Soshnikov dans [18] et on montre que le comportement asymptotique des plus grandes valeurs propres est déterminé par les plus grandes entrées de la matrice.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 A. Soshnikov in [18], 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.

Random matrix theory in statistics: A review

“…Soshnikov analyzed the setting when the tail index α (the rate at which tails of the probability distribution decay) of the entries is between 0 and 2. Auffinger et al (2009) extended the results by Soshnikov (2004), to prove that for α A ð0; 4Þ, and for both Wigner and Wishart-type matrices, the point process of the largest eigenvalues (properly normalized) converges to an inhomogeneous Poisson point process whose intensity function is determined by the tail index of the (common) distribution of the entries of the matrix. A version of these results was also proved by Biroli et al (2007) …”

An Introduction to Random Matrices

“…Some partial results relaxing the symmetry assumption can be found in [PeS07], [PeS08b], although at this time the universality at the edge of Wigner matrices with entries possessing non-symmetric distribution remains open. When the entries possess heavy tail, limit laws for the largest eigenvalue change, see [Sos04], [AuBP07]. Concerning the spacing in the bulk, universality was proved when the i.i.d.…”

Localization and delocalization for heavy tailed band matrices

“…α , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov in [29,30] when α < 2 and by Auffinger et al in [1] when α < 4). On the other hand, when α > 2(1 + µ −1 ), the largest eigenvalues have order N µ 2 and most eigenvectors of the matrix are delocalized, i.e.…”

“…On s'intéresse aux plus grandes valeurs propres et aux vecteurs propres associés et prouve la transition de phase suivante. D'une part, quand α < 2(1 + µ −1 ), les plus grandes valeurs propres ont pour ordre N 1+µ α , sont asymptotiquement distribuées selon un processus de Poisson et les vecteurs propres associés sont essentiellement portés par deux coordonnées (ce phénomène a déjàété remarqué pour des matrices pleines par Soshnikov dans [29,30] quand α < 2, et par Auffinger et al dans [1] quand α < 4). D'autre part, quand α > 2(1 + µ −1 ), les plus grandes valeurs propres ont pour ordre N…”

“…It is known that the limiting spectral measure of such Wigner matrices is the semi-circle distribution provided that the variance of the entries is finite (otherwise another limiting distribution has been identified by Guionnet and Ben Arous [5]). Regarding eigenvectors, it was shown by Soshnikov [29,30] and Auffinger, Ben Arous and Péché [1] that eigenvectors associated to the largest eigenvalues have a localization length of order 1 if the entries do not admit a finite fourth moment. The localization length is not so clear in the bulk but some progress has been obtained by Bordenave and Guionnet [7].…”