### 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}

Original language | English (US) |
---|---|

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 |

### Fingerprint

### Keywords

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

### Cite this

**Determinant of sample correlation matrix with application.** / Jiang, Tiefeng.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 29, no. 3, pp. 1356-1397. https://doi.org/10.1214/17-AAP1362

}

TY - JOUR

T1 - Determinant of sample correlation matrix with application

AU - Jiang, Tiefeng

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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

AB - 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

KW - Central limit theorem

KW - Moment generating function

KW - Multivariate normal distribution

KW - Sample correlation matrix

KW - Smallest eigenvalue

UR - http://www.scopus.com/inward/record.url?scp=85063327141&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063327141&partnerID=8YFLogxK

U2 - 10.1214/17-AAP1362

DO - 10.1214/17-AAP1362

M3 - Article

AN - SCOPUS:85063327141

VL - 29

SP - 1356

EP - 1397

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 3

ER -