Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices

Suppose that X-n = (x(jk)) is N x n whose elements are independent complex variables with mean zero, variance 1. The separable sample covariance matrix is defined as Bn = 1/N T-2n(1/2) XnT1nXn* T-2n(1/2) where T-1n is a Hermitian matrix and T-2n(1/2) is a Hermitian square root of the nonnegative definite Hermitian matrix T-2n. Its linear spectral statistics (LSS) are shown to have Gaussian limits when n/N approaches a positive constant under some conditions.