Global and local CLTs for linear spectral statistics of general sample covariance matrices when the dimension is much larger than the sample size with applications

In this paper, under the assumption that the dimension is much larger than the sample size, i.e., $p \asymp n^{\alpha}, \alpha>1,$ we consider the (unnormalized) sample covariance matrices $Q = \Sigma^{1/2} XX^*\Sigma^{1/2}$, where $X=(x_{ij})$ is a $p \times n$ random matrix with centered i.i.d entries whose variances are $(pn)^{-1/2}$, and $\Sigma$ is the deterministic population covariance matrix. We establish two classes of central limit theorems (CLTs) for the linear spectral statistics (LSS) for $Q,$ the global CLTs on the macroscopic scales and the local CLTs on the mesoscopic scales. We prove that the LSS converge to some Gaussian processes whose mean and covariance functions depending on $\Sigma$, the ratio $p/n$ and the test functions, can be identified explicitly on both macroscopic and mesoscopic scales. We also show that even though the global CLTs depend on the fourth cumulant of $x_{ij},$ the local CLTs do not. Based on these results, we propose two classes of statistics for testing the structures of $\Sigma,$ the global statistics and the local statistics, and analyze their superior power under general local alternatives. To our best knowledge, the local LSS testing statistics which do not rely on the fourth moment of $x_{ij},$ is used for the first time in hypothesis testing while the literature mostly uses the global statistics and requires the prior knowledge of the fourth cumulant. Numerical simulations also confirm the accuracy and powerfulness of our proposed statistics and illustrate better performance compared to the existing methods in the literature.