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

2013/04/30 by Zhigang Bao, Guangming Pan, Wang Zhou · 1 citation
Mathematics · #math.PR #math.ST #stat.TH

paper · pdf · doi:10.1214/14-aos1281

published as Annals of Statistics 2015, Vol. 43, No. 1, 382-421 · 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)

arxiv created 2015/03/05 · arxiv updated 2015/03/06

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 W_N=Σ^1/2XX^*Σ^1/2. Here, X=(x_ij)_M,N is an M× N random matrix with independent entries x_ij,1≤ i≤ M,1≤ j≤ N such that 𝔼x_ij=0, 𝔼|x_ij|²=1/N. On dimensionality, we assume that M=M(N) and N/M→ d∈(0,∞) as N→∞. For a class of general deterministic positive-definite M× M matrices Σ, under some additional assumptions on the distribution of x_ij's, we show that the limiting behavior of the largest eigenvalue of 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 (Σ=I). Consequently, in the standard complex case (𝔼x_ij²=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 W_N converges weakly to the type 2 Tracy-Widom distribution TW₂. Moreover, in the real case, we show that when Σ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW₁ 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 Σ and more generally distributed X.

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