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

Abstract

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. © Association des Publications de l'Institut Henri Poincaré, 2012.