Central limit theorems for linear spectral statistics of large dimensional $F$-matrices

Zheng, Shurong

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 444-476.

Dans beaucoup d'applications, on cherche à effectuer une inférence statistique sur des paramètres définis à partir de la mesure spectrale d'une F-matrice, matrice obtenue comme le produit d'une matrice de covariance du tableau de variables indépendantes (Xjk)p×n1 et de l'inverse d'une autre matrice de covariance (Yjk)p×n2. Les variables sont soient toutes réelles soient complexes. Il est donc utile d'étudier les distributions asymptotiques des estimateurs de ces paramètres associés à la F-matrice. Dans cet article, nous établissons des théorèmes centraux limites pour les statistiques linéaires du spectre de la F-matrice dans la limite où p, n1, n2 tendent vers l'infini en restant de même ordre, et donnons des formules exactes pour leurs moyennes et covariances. De plus, l'hypothèse que les variables (Xjk)p×n1 et (Yjk)p×n2 sont i.i.d. et la restriction que le quatrième moment est égal à 2 ou 3 comme dans Bai et Silverstein (Ann. Probab. 32 (2004) 553-605) sont affaiblies de la manière suivante; les coefficients (Xjk)p×n1 et (Yjk)p×n2 sont indépendants mais non nécessairement équidistribués, pourvu qu'ils aient le même quatrième moment dans chaque tableau. Par conséquent, nous obtenons le théorème de la limite centrale pour les statistiques linéaires de la matrice beta qui est de la forme (I + d ⋅ F matrix)-1, où d est une constante et I la matrice identité.

In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the F matrix. In this paper, we establish the central limit theorems with explicit expressions of means and covariance functions for the linear spectral statistics of the large dimensional F matrix, where the dimension p of the two samples tends to infinity proportionally to the sample sizes (n1, n2). Moreover, the assumptions of the i.i.d. structures of arrays (Xjk)p×n1, (Yjk)p×n2 and the restriction of the fourth moments equaling 2 or 3 made in Bai and Silverstein (Ann. Probab. 32 (2004) 553-605) are relaxed to that arrays (Xjk)p×n1 and (Yjk)p×n2 are independent respectively but not necessarily identically distributed except for a common fourth moment for each array. As a consequence, we obtain the central limit theorems for the linear spectral statistics of the beta matrix that is of the form (I + d ⋅ F matrix)-1, where d is a constant and I is an identity matrix.

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