CLT for linear spectral statistics of a rescaled sample precision matrix

Abstract

Central limit theorems (CLTs) of linear spectral statistics (LSS) of general Fisher matrices [Formula: see text] are widely used in multivariate statistical analysis where [Formula: see text] with a deterministic complex matrix [Formula: see text] and two sample covariance matrices [Formula: see text] and [Formula: see text] from two independent samples with sample sizes [Formula: see text] and [Formula: see text]. As the first step to obtain the CLT, it is necessary to establish the CLT for LSS of the random matrix [Formula: see text], or equivalently that of [Formula: see text], that is a sample precision matrix rescaled by a general non-negative definite Hermitian matrix [Formula: see text]. Because the scaling matrix [Formula: see text] 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 [Formula: see text] where the inverse of [Formula: see text] may not exist, thus extending Bai and Silverstein’s CLT. As a further innovation of the paper, general populations for the sample covariance matrix [Formula: see text] are covered requiring the existence a fourth-order moment of arbitrary value, that is not necessarily matching the values of the Gaussian case.