报告题目 (Title)：分块相关系数矩阵的极限律

报告人 (Speaker)：鲍志刚 教授（香港科技大学）

报告时间 (Time)：2022年12月20日 (周二) 10:00-11：30

报告地点 (Place)：腾讯会议（会议号：555-280-309 无密码）

邀请人(Inviter)：张阳春

主办部门：理学院数学系

报告摘要：Independence test for the components of a random vector is a classical problem. When the variances of the components are unknown, various statistics constructed from the sample correlation matrices are often used. In the literature, the limiting distributions of these statistic have been well-studied in both low and high dimensional cases. In this talk, we will discuss a rather general extension of this independence test problem, in high dimensional case. We consider the independence test for k subvectors of a random vector with dimension p, where the dimension of the subvector p_i’s can vary from 1 to order p. When the population covariance matrices of the subvectors are unknown, we construct a random matrix model called (sample) block correlation matrix, based on n samples. It turns out that the spectral statistics of the block correlation matrix do not depend on the unknown population covariance. Further, under the null hypothesis, the limiting behavior of the spectral statistics can be described with the aid of the free probability theory. Specifically, under three different settings of possibly n-dependent k and p_i’s, we show that the empirical spectral distribution of the block correlation matrix converges to the free Poisson binomial distribution, free Poisson distribution (Marchenko-Pastur law) and free Gaussian distribution (semicircle law), respectively. We then further derive the CLT for the linear spectral statistics of the block correlation matrix. Our results are established under general distribution assumption on the random vector. It turns out that the CLT is universal and does not depend on the 4-th cumulants of the vector components, due to a self-normalizing effect of the correlation type matrices.