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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 RₙM and establish the CLTs for the linear spectral statistics (LSS) of RₙM in two structures: linear independent component structure and elliptical structure. In contrast to existing literature, our proposed spectral properties do not require M to be an identity matrix. Moreover, we also derive the joint limiting distribution of LSSs of Rₙ M₁, , Rₙ MK. As an illustration, an application is given for the CLT.
Chen et al. (Sat,) studied this question.