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In this paper, under the assumption that the dimension is much larger thanthe 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 entrieswhose variances are $(pn)^{-1/2}$, and $\Sigma$ is the deterministic populationcovariance matrix. We establish two classes of central limit theorems (CLTs)for the linear spectral statistics (LSS) for $Q,$ the global CLTs on themacroscopic scales and the local CLTs on the mesoscopic scales. We prove thatthe LSS converge to some Gaussian processes whose mean and covariance functionsdepending on $\Sigma$, the ratio $p/n$ and the test functions, can beidentified explicitly on both macroscopic and mesoscopic scales. We also showthat even though the global CLTs depend on the fourth cumulant of $x_{ij},$ thelocal CLTs do not. Based on these results, we propose two classes of statisticsfor testing the structures of $\Sigma,$ the global statistics and the localstatistics, and analyze their superior power under general local alternatives.To our best knowledge, the local LSS testing statistics which do not rely onthe fourth moment of $x_{ij},$ is used for the first time in hypothesis testingwhile the literature mostly uses the global statistics and requires the priorknowledge of the fourth cumulant. Numerical simulations also confirm theaccuracy and powerfulness of our proposed statistics and illustrate betterperformance compared to the existing methods in the literature.

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