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 $\mathcal{W}_N=\Sigma^{1/2}XX^*\Sigma ^{1/2}$. 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 Erd\H{o}s, 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 F\'{e}ral and P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$.