Determinant of sample correlation matrix with application

Let x ₁ ,...,x _n be independent random vectors of a common p-dimensional normal distribution with population correlation matrix R _n . The sample correlation matrix Ř _n = (ř _ij ) _p × _p is generated from x ₁ ,...,x _n such that ř _ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x ₁ ,...,x _n ). The matrix Ř _n is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of Ř _n for a big class of R _n . The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R _n is larger than 1/2. Besides, a formula of the moments of |Ř _n | and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.