Determinant of sample correlation matrix with application

Abstract

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