## The Annals of Applied Probability

### Determinant of sample correlation matrix with application

Tiefeng Jiang

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1356-1397.

Dates
Received: October 2016
Revised: August 2017
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566833

Digital Object Identifier
doi:10.1214/17-AAP1362

Mathematical Reviews number (MathSciNet)
MR3914547

Zentralblatt MATH identifier
07057457

#### Citation

Jiang, Tiefeng. Determinant of sample correlation matrix with application. Ann. Appl. Probab. 29 (2019), no. 3, 1356--1397. doi:10.1214/17-AAP1362. https://projecteuclid.org/euclid.aoap/1550566833