Title: 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 $\mathbf S_n=\frac1n\mathbf
T_{1n}\mathbf X_n\mathbf T_{2n}\mathbf X_n^*\mathbf T_{1n}^*$ where $\mathbf
X_n=(x_{jk})$ is of $m_1\times m_2$ dimension, the entries $\{x_{jk},
j=1,...,m_1, k=1,...,m_2\}$ are independent and identically distributed complex
variables with zero means and unit variances, $\mathbf T_{1n}$ is a $p\times
m_1 $ complex matrix and $\mathbf T_{2n}$ is an $m_2\times m_2$ Hermitian
matrix. We then apply this general central limit theorem to the problem of
testing white noise in time series.