Spectral properties of high-dimensional rescaled sample correlation matrices

High-dimensional sample correlation matrices are a crucial class of random matrices in multivariate statistical analysis. The central limit theorem (CLT) provides a theoretical foundation for statistical inference. In this paper, assuming that the data dimension increases proportionally with the sample size, we derive the limiting spectral distribution of the matrix [Formula: see text] and establish the CLT for the linear spectral statistics (LSS) of [Formula: see text] under the linear independent component structure. In contrast to the existing literature, our proposed spectral properties do not require [Formula: see text] to be an identity matrix. Moreover, we also derive the joint limiting distribution of LSS of [Formula: see text] and propose a numerical method for computing the asymptotic covariance of CLT of the LSS for rescaled sample correlation matrices. As an illustration, an application is given for the CLT.