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In this paper, we consider the separable covariance model, which plays animportant role in wireless communications and spatio-temporal statistics anddescribes a process where the time correlation does not depend on the spatiallocation and the spatial correlation does not depend on time. We established acentral limit theorem for linear spectral statistics of general separablesample covariance matrices in the form of $\mathbf S_n=\frac1n\mathbfT_{1n}\mathbf X_n\mathbf T_{2n}\mathbf X_n^*\mathbf T_{1n}^*$ where $\mathbfX_n=(x_{jk})$ 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 complexvariables with zero means and unit variances, $\mathbf T_{1n}$ is a $p\timesm_1 $ complex matrix and $\mathbf T_{2n}$ is an $m_2\times m_2$ Hermitianmatrix. We then apply this general central limit theorem to the problem oftesting white noise in time series.

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