天元系列活动|Asymptotic Distributions of Largest Pearson Correlation Coefficients under Dependent Structures
报 告 人: 姜铁锋 教授
所在单位: University of Minnesota and Chinese University of Hong Kong at Shenzhen
报告地点: 数学楼第二报告厅
报告时间: 2023-11-20 10:00:00
报告简介:

Given a random sample from a multivariate normal distribution whose covariance matrix is a Toeplitz matrix, we study the largest off-diagonal entry of the sample correlation matrix. Assuming the multivariate normal distribution has the covariance structure of an auto-regressive sequence, we establish a phase transition in the limiting distribution of the largest off-diagonal entry. We show that the limiting distributions are of Gumbel-type (with different parameters) depending on how large or small the parameter of the autoregressive sequence is. In the critical case, we obtain that the limiting distribution is the maximum of two independent random variables of Gumbel distributions. This phase transition establishes the exact threshold at which the auto-regressive covariance structure behaves differently than its counterpart with the covariance matrix equal to the identity. Assuming the covariance matrix is a general Toeplitz matrix, we obtain the limiting distribution of the largest entry under the ultra-high dimensional settings: it is a weighted sum of two independent random variables, one normal and the other following a Gumbel-type law. The counterpart of the non-Gaussian case is also discussed. As an application, we study a high-dimensional covariance testing problem.

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主讲人简介:
姜铁锋,美国明尼苏达大学统计系终身教授,美国NSF Career Award获得者。主要从事概率统计及其相关领域的研究工作,特别是在概率论、高维统计以及纯数学等交叉学科取得了突破性的进展。姜教授解决的“哈尔酉矩阵被独立随机变量逼近”的结果被用于量子计算的研究中。姜教授目前已发表论文40多篇,其中绝大部分发表在国际顶尖的概率统计与机器学习杂志上,包括《Ann. Probab.》、《Probab. Theor. Rel. Fields》、《Ann. Stat.》、《Ann. Appl. Probab.》、《J. Mach. Learn Res.》等。