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., <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> ≍ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup>, α > 1, we consider the sample covariance matrices <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> = Σ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup><italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XX</i>*Σ<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup>, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> = (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_ij</i>) is a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> × <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> random matrix with centered i.i.d. entries whose variances are (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pn</i>)<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1/2</sup>, and Σ is the deterministic population covariance matrix. This ultra-high dimensional setting is increasingly relevant in various applications in information theory, communications, and signal processing, such as massive MIMO, compressed sensing, and large-scale sensor networks. We establish two classes of central limit theorems (CLTs) for the linear spectral statistics (LSS) of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i>, the global CLTs on the macroscopic scales where all the eigenvalues will be used, and the local CLTs on the mesoscopic scales where only part of the eigenvalues will be utilized. We prove that the LSS converge to some Gaussian processes whose mean and covariance functions, depending on Σ, the ratio <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>/<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>, 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_ij</i>, the local CLTs do not. Leveraging these results, we propose two classes of statistics for testing the structures of Σ, 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_ij</i>, are used for the first time in hypothesis testing while the literature mostly uses the global statistics and requires prior knowledge of the fourth moment. Numerical simulations also confirm the accuracy and powerfulness of our proposed statistics and illustrate better performance compared to the existing methods in the literature.