LARGEST ENTRIES OF SAMPLE CORRELATION MATRICES FROM EQUI-CORRELATED NORMAL POPULATIONS

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(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. 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.