The Annals of Applied Probability

Determinant of sample correlation matrix with application

Tiefeng Jiang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems

Central limit theorem sample correlation matrix smallest eigenvalue multivariate normal distribution moment generating function


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

Export citation


  • Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York.
  • Bai, Z. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer, New York.
  • Bao, Z., Pan, G. and Zhou, W. (2012). Tracy–Widom law for the extreme eigenvalues of sample correlation matrices. Electron. J. Probab. 17 1–32.
  • Bartlett, M. S. (1954). A note on multiplying factors for various chi-squared approximations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 16 296–298.
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Brockwell, P. J. and Davis, R. A. (2002). Introduction to Time Series and Forecasting. Springer, New York.
  • Cai, T., Fan, J. and Jiang, T. (2013). Distributions of angles in random packing on spheres. J. Mach. Learn. Res. 14 1837–1864.
  • Cai, T., Liang, T. and Zhou, H. (2015). Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for high-dimensional Gaussian distributions. J. Multivariate Anal. 137 161–172.
  • Chow, Y. S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd ed. Springer, New York.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
  • Dong, Z., Jiang, T. and Li, D. (2012). Circular law and arc law for truncation of random unitary matrix. J. Math. Phys. 53 Article ID 013301.
  • Eaton, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley, New York.
  • Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 1079–1083.
  • Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge Univ. Press, Cambridge.
  • Jiang, T. (2004a). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865–880.
  • Jiang, T. (2004b). The limiting distributions of eigenvalues of sample correlation matrices. Sankhyā 66 35–48.
  • Jiang, T. and Qi, Y. (2015). Likelihood ratio tests for high-dimensional normal distributions. Scand. J. Stat. 42 988–1009.
  • Jiang, T. and Yang, F. (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Ann. Statist. 41 2029–2074.
  • Li, D., Liu, W. and Rosalsky, A. (2010). Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Related Fields 148 5–35.
  • Li, D. and Rosalsky, A. (2006). Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16 423–447.
  • Morrison, D. F. (2004). Multivariate Statistical Methods, 4th ed. Duxbury Press, Pacific Grove, CA.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • Nguyen, H. H. and Vu, V. (2014). Random matrices: Law of the determinant. Ann. Probab. 42 146–167.
  • Rudelson, M. and Vershynin, R. (2013). Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 Article ID 82.
  • Smale, S. (2000). Mathematical problems for the next century. In Mathematics: Frontiers and Perspectives (V. Arnold, M. Atiyah, P. Lax and B. Mazur, eds.) 271–294. Amer. Math. Soc., Providence, RI.
  • Tao, T. and Vu, V. (2012). A central limit theorem for the determinant of a Wigner matrix. Adv. Math. 231 74–101.
  • Wilks, S. S. (1932). Certain generalizations in the analysis of variance. Biometrika 24 471–494.
  • Yurinskiĭ, V. V. (1976). Exponential inequalities for sums of random vectors. J. Multivariate Anal. 6 473–499.
  • Zhou, W. (2007). Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc. 359 5345–5363.