Central Limit Theory for Linear Spectral Statistics of Normalized Separable Sample Covariance Matrix

This paper focuses on the separable covariance matrix when the dimension $p$ and the sample size $n$ grow to infinity but the ratio $p/n$ tends to zero. The separable sample covariance matrix can be written as $n^{-1}A^{1/2}XBX^\top A^{1/2}$, where $A$ and $B$ correspond to the cross-row and cross-column correlations, respectively. We normalize the separable sample covariance matrix and prove the central limit theorem for corresponding linear spectral statistics, with explicit formulas for the mean and covariance function. We apply the results to testing the correlations of a large number of variables with two common examples, related to spatial-temporal model and matrix-variate model, which are beyond the scenarios considered in existing studies. The asymptotic sizes and powers are studied under mild conditions. The computations of the asymptotic mean and variance are involved under the null hypothesis where $A$ is the identity matrix, with simplified expressions which facilitate to practical usefulness. Extensive numerical studies show that the proposed testing statistic performs sufficiently reliably under both the null and alternative hypothesis, while a conventional competitor fails to control empirical sizes.