UNIVERSALITY FOR THE LARGEST EIGENVALUE OF SAMPLE COVARIANCE MATRICES WITH GENERAL POPULATION

• 成果类型: Article
• 署名作者: Bao, Zhigang; Pan, Guangming; Zhou, Wang
• 署名单位: Zhejiang University; Nanyang Technological University; National University of Singapore
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/14-AOS1281
• 发表日期: 2015
• 页码: 382-421
• 关键词: limiting spectral distribution; tracy-widom limit; edge universality; local statistics; semicircle law; delocalization; signals; tests

摘要

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 = Sigma(XX)-X-1/2*E-1/2. Here, X = (xij)(M,N) is an M x N random matrix with independent entries x(ij), 1 <= i <= M, 1 <= j <= N such that Ex(ij) = 0, E vertical bar x(ij)vertical bar(2) = 1/N. On dimensionality, we assume that M = M(N) and N/M -> d is an element of(0, infinity) as N -> infinity. For a class of general deterministic positive-definite M x 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 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 Erdos, 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 (Ex(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 WN converges weakly to the type 2 Tracy-Widom distribution TW2. 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 TW1 holds for the normalized largest eigenvalue of W-N, which extends a result of Feral and Peche 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.