CLT for linear spectral statistics of a rescaled sample precision matrix

Central limit theorems (CLTs) of linear spectral statistics (LSS) of general Fisher matrices F are widely used in multivariate statistical analysis where F = SyMSx-1M* with a deterministic complex matrix M and two sample covariance matrices S-x and S-y from two independent samples with sample sizes m and n. As the first step to obtain the CLT, it is necessary to establish the CLT for LSS of the random matrix MSx-1M*,or equivalently that of Sx-1T, that is a sample precision matrix rescaled by a general non-negative definite Hermitian matrix T = M*M. Because the scaling matrix T in many large-dimensional problems may not be invertible, the result does not simply follow from the celebrated CLT by Bai and Silverstein (2004). Thus, we have to alternatively derive the CLT of LSS of Sx-1T where the inverse of T may not exist, thus extending Bai and Silverstein's CLT. As a further innovation of the paper, general populations for the sample covariance matrix S-x are covered requiring the existence a fourth-order moment of arbitrary value, that is not necessarily matching the values of the Gaussian case.