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arXiv · 2410.15160

Largest Eigenvalues of Principal Minors of Deformed Gaussian Orthogonal Ensembles and Wishart Matrices

Tiefeng Jiang·Yongcheng Qi
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Abstract

Consider a high-dimensional Wishart matrix $\bd{W}=\bd{X}^T\bd{X}$ where the entries of $\bd{X}$ 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 $\bd{W}$. Under certain restrictions on the sample size and the population dimension of $\bd{W}$, 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 Hájek projection. This method may also be applicable to other extreme-value problems. Some open questions are posed.

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Tiefeng Jiang, Yongcheng Qi. 2024-10-19. Largest Eigenvalues of Principal Minors of Deformed Gaussian Orthogonal Ensembles and Wishart Matrices. https://arxiv.org/abs/2410.15160

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