---
_id: '1505'
abstract:
- lang: eng
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.
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"
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
citation:
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
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.
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.
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.
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.
short: Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.
date_created: 2018-12-11T11:52:25Z
date_published: 2015-02-01T00:00:00Z
date_updated: 2021-01-12T06:51:14Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/14-AOS1281
intvolume: ' 43'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1304.5690
month: '02'
oa: 1
oa_version: Preprint
page: 382 - 421
publication: Annals of Statistics
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5672'
quality_controlled: '1'
status: public
title: Universality for the largest eigenvalue of sample covariance matrices with
general population
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2015'
...