# Largest entries of sample correlation matrices from equi-correlated normal populations

Jianqing Fan, Tiefeng Jiang

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

## Abstract

The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient ρ >0 and both the population dimension p and the sample size n tend to infinity with logp = o(n1/3). As 0<ρ <1, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as 0<ρ <1/2. This differs substantially from a well-known result for the independent case where ρ = 0, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of ρ where the transition occurs. If ρ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen-Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.

Original language English (US) 3321-3374 54 Annals of Probability 47 5 https://doi.org/10.1214/19-AOP1341 Published - Sep 1 2019

### Bibliographical note

Funding Information:
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3322 2. Main results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3324 2.1. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3324 2.2. Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3326 2.3. An application to a high-dimensional test . . . . . . . . . . . . . . . . . . . . . . . . . 3328 3. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3329 3.1. Some technical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3329 3.2. Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3337 3.3. The proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3349 3.4. The proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3358 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3373 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3373 Received September 2017; revised January 2019. 1Supported by NSF Grants DMS-1406266 and DMS-1712591. 2Supported by NSF Grant DMS-1406279. MSC2010 subject classifications. Primary 62H10, 62E20; secondary 60F05. Key words and phrases. Maximum sample correlation, phase transition, multivariate normal distribution, Gumbel distribution, Chen–Stein Poisson approximation.

© Institute of Mathematical Statistics, 2019.

## Keywords

• Chen-stein poisson approximation
• Gumbel distribution
• Maximum sample correlation
• Multivariate normal distribution
• Phase transition

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