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 ${\widehat{\mathbf{R}}}_n\mathbf{M}$ and establish the CLTs for the linear spectral statistics (LSS) of ${\widehat{\mathbf{R}}}_n\mathbf{M}$ in two structures: linear independent component structure and elliptical structure. In contrast to existing literature, our proposed spectral properties do not require $\mathbf{M}$ to be an identity matrix. Moreover, we also derive the joint limiting distribution of LSSs of ${\widehat{\mathbf{R}}}_n{\mathbf{M}}_1,\dots ,{\widehat{\mathbf{R}}}_n{\mathbf{M}}_K$. As an illustration, an application is given for the CLT.