{"title":"Universality for the largest eigenvalue of sample covariance matrices with general population","date_created":"2018-12-11T11:52:25Z","oa_version":"Preprint","publication":"Annals of Statistics","acknowledgement":"B.Z. was supported in part by NSFC Grant 11071213, ZJNSF Grant R6090034 and SRFDP Grant 20100101110001. P.G. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11. Z.W. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11, and by a Grant R-155-000-131-112 at the National University of Singapore\r\n","department":[{"_id":"LaEr"}],"citation":{"chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” *Annals of Statistics*. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/14-AOS1281.","ista":"Bao Z, Pan G, Zhou W. 2015. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 43(1), 382–421.","ieee":"Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample covariance matrices with general population,” *Annals of Statistics*, vol. 43, no. 1. Institute of Mathematical Statistics, pp. 382–421, 2015.","short":"Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.","mla":"Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” *Annals of Statistics*, vol. 43, no. 1, Institute of Mathematical Statistics, 2015, pp. 382–421, doi:10.1214/14-AOS1281.","ama":"Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. *Annals of Statistics*. 2015;43(1):382-421. doi:10.1214/14-AOS1281","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. *Annals of Statistics*. Institute of Mathematical Statistics. https://doi.org/10.1214/14-AOS1281"},"issue":"1","status":"public","oa":1,"doi":"10.1214/14-AOS1281","day":"01","abstract":[{"text":"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 = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij '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 Erd″os, 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 (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 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 whenΣ 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 Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ 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.","lang":"eng"}],"date_published":"2015-02-01T00:00:00Z","year":"2015","quality_controlled":"1","date_updated":"2021-01-12T06:51:14Z","publist_id":"5672","main_file_link":[{"url":"https://arxiv.org/abs/1304.5690","open_access":"1"}],"intvolume":" 43","type":"journal_article","page":"382 - 421","volume":43,"publisher":"Institute of Mathematical Statistics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"02","_id":"1505","language":[{"iso":"eng"}],"publication_status":"published","author":[{"first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao"},{"last_name":"Pan","first_name":"Guangming","full_name":"Pan, Guangming"},{"last_name":"Zhou","first_name":"Wang","full_name":"Zhou, Wang"}]}