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

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

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