Determinant of sample correlation matrix with application

Let x1,.,xn be independent random vectors of a common p-dimensional normal distribution with population correlation matrix Rn. The sample correlation matrix R.n=(r.ij)p.p is generated from x1,.,xn such that r.ij is the Pearson correlation coefficient between the i th column and the j th column of the data matrix (x1,.,xn).. The matrix R.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 R.n for a big class of Rn. 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 Rn is larger than 1/2. Besides, a formula of the moments of |R.n| and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.