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September 2019 Largest entries of sample correlation matrices from equi-correlated normal populations
Jianqing Fan, Tiefeng Jiang
Ann. Probab. 47(5): 3321-3374 (September 2019). DOI: 10.1214/19-AOP1341


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 $\rho >0$ and both the population dimension $p$ and the sample size $n$ tend to infinity with $\log p=o(n^{\frac{1}{3}})$. As $0<\rho <1$, 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 $0<\rho <1/2$. This differs substantially from a well-known result for the independent case where $\rho =0$, 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 $\rho $ where the transition occurs. If $\rho $ 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.


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Jianqing Fan. Tiefeng Jiang. "Largest entries of sample correlation matrices from equi-correlated normal populations." Ann. Probab. 47 (5) 3321 - 3374, September 2019.


Received: 1 September 2017; Revised: 1 January 2019; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145319
MathSciNet: MR4021253
Digital Object Identifier: 10.1214/19-AOP1341

Primary: 62E20 , 62H10
Secondary: 60F05

Keywords: Chen–Stein Poisson approximation , Gumbel distribution , Maximum sample correlation , multivariate normal distribution , phase transition

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 5 • September 2019
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