## Seminars

## Seminars

## Asymptotic Distributions of Largest Pearson Correlation Coefficients under Dependent Structures

**Holder：** Tiefeng Jiang (University of Minnesota)

**Time：**2023-10-30 14:00-15:00

**Location：**Room 1114, Sciences Building No. 1

**Abstract: **
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.

**About the Speaker: **

Jiang Tiefeng is a tenured professor in the Department of Statistics at the University of Minnesota and a recipient of the NSF Career Award. He is mainly engaged in research in probability, statistics and related fields, especially in interdisciplinary subjects such as probability theory, high-dimensional statistics and pure mathematics. He has made breakthrough progress. The result of Professor Jiang's solution of "Haar unitary matrix is approximated by independent random variables" has been used in the research of quantum computing. Professor Jiang has published more than 40 papers, most of which have been published in top international probability, statistics and machine learning magazines, including "Ann. Probab.", "Probab. Theor. Rel. Fields", "Ann. Stat. ", "Ann. Appl. Probab.", "J. Mach. Learn Res.", etc.

PERSONAL WEBSITE:

http://users.stat.umn.edu/~jiang040/

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