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.
Bibliographical noteFunding Information:
Received October 2016; revised August 2017. 1Supported in part by NSF Grants DMS-12-09166 and DMS-14-06279. MSC2010 subject classifications. 60B20, 60F05. Key words and phrases. Central limit theorem, sample correlation matrix, smallest eigenvalue, multivariate normal distribution, moment generating function.
1Supported in part by NSF Grants DMS-12-09166 and DMS-14-06279. MSC2010 subject classifications. 60B20, 60F05.
- Central limit theorem
- Moment generating function
- Multivariate normal distribution
- Sample correlation matrix
- Smallest eigenvalue