  Statistical inferences for sample correlation matrices are important in high
dimensional data analysis. Motivated by this, this paper establishes a new
central limit theorem (CLT) for a linear spectral statistic (LSS) 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. 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 proposed test shows its applicability and
effectiveness in practice. An empirical application to test the independence of
household incomes from different cities in China is also conducted.
