Universality for the largest eigenvalue of sample covariance matrices with general population

Abstract

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 ${\mathcal{W}}_N={\Sigma }^{1/2}X{X}^{\ast }{\Sigma }^{1/2}$. Here, $X=(x_{ij}{)}_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij},1\le i\le M,1\le j\le N$ such that $\mathbb{E}{x}_{ij}=0$, $\mathbb{E}|x_{ij}{|}^2=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\to d\in (0,\infty )$ as $N\to \infty$. 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 $X$.

Published in at http://dx.doi.org/10.1214/14-AOS1281 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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