High-dimensional sample correlation matrix is an important random matrix in multivariate statistical analysis. Its central limit theory is one main theoretical basis for making statistical inference on high-dimensional correlation matrix. Under the high-dimensional framework that the dimension tends to infinity proportionally with the sample size, we establish the central limit theorems (CLT) for linear spectral statistics (LSS) of sample correlation matrices under two settings: (1). The population follows an independent component structure; (2). The population follows an elliptical structure including some heavy-tailed distributions. It shows that the CLTs of LSS of sample correlation matrices are very different under the two settings.

Especially, even if the population correlation matrix is an identity matrix, the CLTs are different under the two settings. An application of our established two CLTs is given.