## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1356-1397 |

Number of pages | 42 |

Journal | Annals of Applied Probability |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2019 |

### Bibliographical note

Funding 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.

Funding Information:

1Supported in part by NSF Grants DMS-12-09166 and DMS-14-06279. MSC2010 subject classifications. 60B20, 60F05.

## Keywords

- Central limit theorem
- Moment generating function
- Multivariate normal distribution
- Sample correlation matrix
- Smallest eigenvalue