报告题目：Largest Entries of Sample Correlation Matrices from Equi-correlated Normal Populations

报告人：Tiefeng Jiang   University of Minnesota

报告时间：2019年1月10日上午10：00-11：00

报告地点：数学楼第二报告厅

报告摘要： We obtain 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. 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. It turns out that the thresholds of such a regime depend on n and converge to zero. If ρ is less than the threshold, larger than the threshold 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. The results are then applied to evaluating the power of a high-dimensional testing problem of identity correlation matrix.