---
res:
bibo_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 = (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.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: Zhigang
foaf_name: Bao, Zhigang
foaf_surname: Bao
foaf_workInfoHomepage: http://www.librecat.org/personId=442E6A6C-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0003-3036-1475
- foaf_Person:
foaf_givenName: Guangming
foaf_name: Pan, Guangming
foaf_surname: Pan
- foaf_Person:
foaf_givenName: Wang
foaf_name: Zhou, Wang
foaf_surname: Zhou
bibo_doi: 10.1214/14-AOS1281
bibo_issue: '1'
bibo_volume: 43
dct_date: 2015^xs_gYear
dct_language: eng
dct_publisher: Institute of Mathematical Statistics@
dct_title: Universality for the largest eigenvalue of sample covariance matrices
with general population@
...