Abstract: Consider a high-dimensional Wishart matrix W “ XT X where the entries of X are i.i.d. random variables with mean zero, variance one, and a finite fourth moment η. Motivated by problems in signal processing and high-dimensional statistics, we study the maximum of the largest eigenvalues of any two-by-two principal minors of W. Under certain restrictions on the sample size and the population dimension of W, we obtain the limiting distribution of the maximum, which follows the Gumbel distribution when η is between 0 and 3, and a new distribution when η exceeds 3. To derive this result, we first address a simpler problem on a new object named a deformed Gaussian orthogonal ensemble (GOE). The Wishart case is then resolved using results from the deformed GOE and a high-dimensional central limit theorem. Our proof strategy combines the Stein-Poisson approximation method, conditioning, U-statistics, and the Hajek projection. This
method may also be applicable to other extreme-value problems. Some open questions are posed.
Bio: Dr. Yongcheng Qi is professor at Department of Mathematics and Statistics, University of Minnesota Duluth. He received a BS degree in Mathematics from Peking University in 1987 and PhD in Probability and Statistics in 1992. From 1994 to 1997, Dr Qi was an associate professor at Department of Probability and Statistics, Peking University. He obtained his PhD degree in Statistics from University of Georgia in 2001. His research interests include limit theorems in Probability and Statistics, empirical likelihood methods, high-dimensional statistics and theory of random matrices. Some of his papers were published in Annals of Statistics, Transactions of the American Mathematical Society, Journal of Theoretical Probability, Statistica Sinica, Scandinavian Journal of Statistics, Journal of Multivariate Analysis, Journal of Mathematical Physics, Insurance: Mathematics and Economics, Stochastic Processes and their Applications, Journal of Applied Probability, and Journal of Applied Statistics.
