On considère des matrices de covariance empirique complexes MN = (1/N)YY* où Y est une matrice de taille N × p dont les coefficients Yij, 1 ≤ i ≤ N, 1≤j ≤ p, sont des variables aléatoires i.i.d. de loi F. Sous certaines hypothèses de régularité et de décroissance sur F, on montre l'universalité de certaines statistiques locales de valeurs propres au milieu du spectre quand N → ∞ et limN→∞ p/N = γ pour tout réel γ ∈ (0, ∞).
We consider complex sample covariance matrices MN = (1/N)YY* where Y is a N × p random matrix with i.i.d. entries Yij, 1 ≤ i ≤ N, 1 ≤ j ≤ p, with distribution F. Under some regularity and decay assumptions on F, we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where N → ∞ and limN→∞ p/N = γ for any real number γ ∈ (0, ∞).
Mots clés : random matrix, bulk universality, sample covariance matrices
@article{AIHPB_2012__48_1_80_0, author = {P\'ech\'e, Sandrine}, title = {Universality in the bulk of the spectrum for complex sample covariance matrices}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {80--106}, publisher = {Gauthier-Villars}, volume = {48}, number = {1}, year = {2012}, doi = {10.1214/11-AIHP442}, mrnumber = {2919199}, zbl = {1238.60010}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP442/} }
TY - JOUR AU - Péché, Sandrine TI - Universality in the bulk of the spectrum for complex sample covariance matrices JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 80 EP - 106 VL - 48 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP442/ DO - 10.1214/11-AIHP442 LA - en ID - AIHPB_2012__48_1_80_0 ER -
%0 Journal Article %A Péché, Sandrine %T Universality in the bulk of the spectrum for complex sample covariance matrices %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 80-106 %V 48 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP442/ %R 10.1214/11-AIHP442 %G en %F AIHPB_2012__48_1_80_0
Péché, Sandrine. Universality in the bulk of the spectrum for complex sample covariance matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 80-106. doi : 10.1214/11-AIHP442. http://www.numdam.org/articles/10.1214/11-AIHP442/
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