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Abstract In this paper, we study the largest eigenvalues of sample covariance matrices with elliptically distributed data. We consider the sample covariance matrix $$Q=YY^{*},$$ where the data matrix $$Y \in \mathbb{R}^{p \times n}$$ contains i.i.d. $$p$$-dimensional observations $$\textbf{y}_{i}=\xi _{i}T\textbf{u}_{i},\;i=1,\dots ,n.$$ Here $$\textbf{u}_{i}$$ is distributed on the unit sphere, $$\xi _{i} \sim \xi $$ is some random variable that is independent of $$\textbf{u}_{i}$$ and $$T^{*}T=\varSigma $$ is some deterministic positive definite matrix. Under some mild regularity assumptions on $$\varSigma ,$$ assuming $$\xi ^{2}$$ has bounded support and certain decay behaviour near its edge so that the limiting spectral distribution of $$Q$$ has a square root decay behaviour near the spectral edge, we prove that the Tracy–Widom law holds for the largest eigenvalues of $$Q$$ when $$p$$ and $$n$$ are comparably large. Based on our results, we further construct some useful statistics to detect the signals when they are corrupted by high dimensional elliptically distributed noise.
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Block perturbation of symplectic matrices in Williamson’s theorem
Williamson's theorem states that for any 2n×2n real positive definite matrix A, there exists a 2n×2n real symplectic matrix S such that STAS=D⊕D, where D is an n×n diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of A. Let H be any 2n×2n real symmetric matrix such that the perturbed matrix A+H is also positive definite. In this paper, we show that any symplectic matrix S̃ diagonalizing A+H in Williamson's theorem is of the form S̃ =SQ+(‖H‖), where Q is a 2n×2n real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that S̃ and S can be chosen so that ‖S̃ −S‖=(‖H‖). Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45-58, 2017].
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- Award ID(s):
- 2304816
- PAR ID:
- 10534593
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Canadian Mathematical Bulletin
- ISSN:
- 0008-4395
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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