CLT for Generalized Linear Spectral Statistics of High-dimensional Sample Covariance Matrices and Applications

In this paper, we introduce the Generalized Linear Spectral Statistics (GLSS) of a high-dimensional sample covariance matrix Sₙ, denoted as trf (Sₙ) Bₙ, which effectively captures distinct spectral properties of Sₙ by involving an ancillary matrix Bₙ and a test function f. The joint asymptotic normality of GLSS associated with different test functions is established under weak assumptions on Bₙ and the underlying distribution, when the dimension n and sample size N are comparable. Specifically, we allow the rank of Bₙ to diverge with n. The convergence rate of GLSS is determined by N/{rank (Bₙ) }. As a natural application, we propose a novel approach based on GLSS for hypothesis testing on eigenspaces of spiked covariance matrices. The theoretical accuracy of the results established for GLSS and the advantages of the newly suggested testing procedure are demonstrated through various numerical studies.