Largest Eigenvalues of Principal Minors of Deformed Gaussian Orthogonal Ensembles and Wishart Matrices
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| Format: | Preprint |
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2024
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| _version_ | 1866913556565327872 |
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| author | Jiang, Tiefeng Qi, Yongcheng |
| author_facet | Jiang, Tiefeng Qi, Yongcheng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Largest Eigenvalues of Principal Minors of Deformed Gaussian Orthogonal Ensembles and Wishart Matrices Jiang, Tiefeng Qi, Yongcheng Probability 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. |
| title | Largest Eigenvalues of Principal Minors of Deformed Gaussian Orthogonal Ensembles and Wishart Matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2410.15160 |