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February 2015 Universality for the largest eigenvalue of sample covariance matrices with general population
Zhigang Bao, Guangming Pan, Wang Zhou
Ann. Statist. 43(1): 382-421 (February 2015). DOI: 10.1214/14-AOS1281


This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_{N}=\Sigma^{1/2}XX^{*}\Sigma^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij}$, $1\leq i\leq M$, $1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^{2}=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$’s, we show that the limiting behavior of the largest eigenvalue of $\mathcal{W}_{N}$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341–407, Adv. Math. 229 (2012) 1435–1515] by Erdős, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935–1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^{2}=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663–714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470–490] (singular case), we show that after an appropriate normalization the largest eigenvalue of $\mathcal{W}_{N}$ converges weakly to the type 2 Tracy–Widom distribution $\mathrm{TW}_{2}$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy–Widom limit $\mathrm{TW}_{1}$ holds for the normalized largest eigenvalue of $\mathcal{W}_{N}$, which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$. In summary, we establish the Tracy–Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on $\Sigma$. Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed.


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Zhigang Bao. Guangming Pan. Wang Zhou. "Universality for the largest eigenvalue of sample covariance matrices with general population." Ann. Statist. 43 (1) 382 - 421, February 2015.


Published: February 2015
First available in Project Euclid: 6 February 2015

zbMATH: 06420692
MathSciNet: MR3311864
Digital Object Identifier: 10.1214/14-AOS1281

Primary: 60B20
Secondary: 15B52 , 62H10 , 62H25

Keywords: edge universality , Sample covariance matrices , Tracy–Widom law

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 1 • February 2015
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