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

报告人：Tiefeng Jiang Universityof 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-diagonalentry of the sample correlation matrices of high-dimensional Gaussianpopulations with equi-correlation structure. Assume the entries of thepopulation distribution have a common correlation coefficient ρ > 0 and boththe 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 correlationmatrix converges to a Gaussian distribution, and the same is true for the samplecovariance matrix as 0 < ρ < 1/2. This differs substantially from awell-known result for the independent case where ρ = 0, in which the abovelimiting distribution is an extreme-value distribution. We then study the phasetransition between these two limiting distributions and identify the regime ofρ where the transition occurs. It turns out that the thresholds of such aregime 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 correspondinglimiting distribution is the extreme-value distribution, the Gaussiandistribution and a convolution of the two distributions, respectively. Theproofs rely on a subtle use of the Chen-Stein Poisson approximation method,conditioning, a coupling to create independence and a special property ofsample correlation matrices. The results are then applied to evaluating thepower of a high-dimensional testing problem of identity correlation matrix.