Central limit theorem for linear spectral statistics of general separable sample covariance matrices with applications

Abstract In this paper, we consider the separable covariance model, which plays an important role in wireless communications and spatio-temporal statistics and describes a process where the time correlation does not depend on the spatial location and the spatial correlation does not depend on time. We established a central limit theorem for linear spectral statistics of general separable sample covariance matrices in the form of S n = 1 n T 1 n X n T 2 n X n ∗ T 1 n ∗ where X n = ( x j k ) is of m 1 × m 2 dimension, the entries { x j k , j = 1 , … , m 1 , k = 1 , … , m 2 } are independent and identically distributed complex variables with zero means and unit variances, T 1 n is a p × m 1 complex matrix and T 2 n is an m 2 × m 2 Hermitian matrix. We then apply this general central limit theorem to the problem of testing white noise in time series.