Comparison between two types of large sample covariance matrices
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, p. 655-677

Let {X ij }, i,j=, be a double array of independent and identically distributed (i.i.d.) real random variables with EX 11 =μ, E|X 11 -μ| 2 =1 and E|X 11 | 4 <. Consider sample covariance matrices (with/without empirical centering) 𝒮=1 n j=1 n (𝐬 j -𝐬 ¯)(𝐬 j -𝐬 ¯) T and 𝐒=1 n j=1 n 𝐬 j 𝐬 j T , where 𝐬 ¯=1 n j=1 n 𝐬 j and 𝐬 j =𝐓 n 1/2 (X 1j ,...,X pj ) T with (𝐓 n 1/2 ) 2 =𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n with p/n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.

Soit {X ij }, i,j=1,2,..., un tableau à double entrées, les X ij étant des variables aléatoires réelles indépendantes et identiquement distribuées (i.i.d.) et où 𝐄X 11 =μ, 𝐄|X 11 -μ| 2 =1 et 𝐄|X 11 | 4 <. Considérons les matrices de covariances empiriques suivantes (avec/sans centrage empirique): 𝒮=1 n j=1 n (𝐬 j -𝐬 ¯)(𝐬 j -𝐬 ¯) T et 𝐒=1 n j=1 n 𝐬 j 𝐬 j T , avec 𝐬 ¯=1 n j=1 n 𝐬 j et 𝐬 j =𝐓 n 1/2 (X 1j ,...,X pj ) T , où (𝐓 n 1/2 ) 2 =𝐓 n est une matrice déterministe définie positive. Nous démontrons que, sous le régime asymptotique n et p/n converge vers une constante positive, le théorème central limite pour la statistique 𝒮 est différent de celui concernant la statistique 𝐒. En outre, nous montrons que cette différence de comportement n’est pas observée pour le comportement moyen des vecteurs propres.

DOI : https://doi.org/10.1214/12-AIHP506
Classification:  15A52,  60F15,  62E20,  60F17
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},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {655-677},
     doi = {10.1214/12-AIHP506},
     zbl = {1295.15023},
     mrnumber = {3189088},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIHPB_2014__50_2_655_0/

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