## Annals of Probability

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

#### 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 $\rho >0$ 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 3321-3374.

Dates
Revised: January 2019
First available in Project Euclid: 22 October 2019

https://projecteuclid.org/euclid.aop/1571731453

Digital Object Identifier
doi:10.1214/19-AOP1341

Mathematical Reviews number (MathSciNet)
MR4021253

Zentralblatt MATH identifier
07145319

#### Citation

Fan, Jianqing; Jiang, Tiefeng. Largest entries of sample correlation matrices from equi-correlated normal populations. Ann. Probab. 47 (2019), no. 5, 3321--3374. doi:10.1214/19-AOP1341. https://projecteuclid.org/euclid.aop/1571731453

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