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Universality for the largest eigenvalue of sample covariance matrices with general population. (English) Zbl 1408.60006

Summary: 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 \(N/M\to d\in(0,\infty)\) 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 [L. Erdős et al., Probab. Theory Relat. Fields 154, No. 1–2, 341–407 (2012; Zbl 1277.15026); Adv. Math. 229, No. 3, 1435–1515 (2012; Zbl 1238.15017)] by Erdős, Yau and Yin for Wigner matrices and extended by N. S. Pillai and J. Yin [Ann. Appl. Probab. 24, No. 3, 935–1001 (2014; Zbl 1296.15021)] 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 N. El Karoui in [Ann. Probab. 35, No. 2, 663–714 (2007; Zbl 1117.60020)] (nonsingular case) and A. Onatski in [Ann. Appl. Probab. 18, No. 2, 470–490 (2008; Zbl 1141.60009)] (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 D. Féral and S. Péché [J. Math. Phys. 50, No. 7, 073302, 33 p. (2009; Zbl 1342.62100)] 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.

MSC:

60B20 Random matrices (probabilistic aspects)
62H10 Multivariate distribution of statistics
15B52 Random matrices (algebraic aspects)
62H25 Factor analysis and principal components; correspondence analysis
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References:

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