High dimensional correlation matrices: the central limit theorem and its applications

Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, the paper establishes a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable. This result is of independent interest in large dimensional random-matrix theory. We also further investigate the sample correlation matrices of a high dimensional vector whose elements have a special correlated structure and the corresponding central limit theorem is developed. Meanwhile, we apply the linear spectral statistic to an independence test for p random variables, and then an equivalence test for p factor loadings and n factors in a factor model. The finite sample performance of the test proposed shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from various cities in China is also conducted.