Let , , be a double array of independent and identically distributed (i.i.d.) real random variables with , and . Consider sample covariance matrices (with/without empirical centering) and , where and with , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of and are different as with approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.
Soit , , un tableau à double entrées, les étant des variables aléatoires réelles indépendantes et identiquement distribuées (i.i.d.) et où , et . Considérons les matrices de covariances empiriques suivantes (avec/sans centrage empirique): et , avec et , où est une matrice déterministe définie positive. Nous démontrons que, sous le régime asymptotique et converge vers une constante positive, le théorème central limite pour la statistique est différent de celui concernant la statistique . En outre, nous montrons que cette différence de comportement n’est pas observée pour le comportement moyen des vecteurs propres.
Keywords: central limit theorems, eigenvectors and eigenvalues, sample covariance matrix, Stieltjes transform, strong convergence
@article{AIHPB_2014__50_2_655_0, author = {Pan, Guangming}, title = {Comparison between two types of large sample covariance matrices}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {655--677}, publisher = {Gauthier-Villars}, volume = {50}, number = {2}, year = {2014}, doi = {10.1214/12-AIHP506}, mrnumber = {3189088}, zbl = {1295.15023}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/12-AIHP506/} }
TY - JOUR AU - Pan, Guangming TI - Comparison between two types of large sample covariance matrices JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2014 SP - 655 EP - 677 VL - 50 IS - 2 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/12-AIHP506/ DO - 10.1214/12-AIHP506 LA - en ID - AIHPB_2014__50_2_655_0 ER -
%0 Journal Article %A Pan, Guangming %T Comparison between two types of large sample covariance matrices %J Annales de l'I.H.P. Probabilités et statistiques %D 2014 %P 655-677 %V 50 %N 2 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/12-AIHP506/ %R 10.1214/12-AIHP506 %G en %F AIHPB_2014__50_2_655_0
Pan, Guangming. Comparison between two types of large sample covariance matrices. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 655-677. doi : 10.1214/12-AIHP506. http://archive.numdam.org/articles/10.1214/12-AIHP506/
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