252 research outputs found
Convergence Rate for Spectral Distribution of Addition of Random Matrices
Let and be two by deterministic Hermitian matrices and let
be an by Haar distributed unitary matrix. It is well known that the
spectral distribution of the sum converges weakly to the free
additive convolution of the spectral distributions of and , as tends
to infinity. We establish the optimal convergence rate in the
bulk of the spectrum
Universality for the largest eigenvalue of sample covariance matrices with general population
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 . Here, is
an random matrix with independent entries such that , . On
dimensionality, we assume that and as
. For a class of general deterministic positive-definite
matrices , under some additional assumptions on the
distribution of 's, we show that the limiting behavior of the largest
eigenvalue of 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\H{o}s, 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 (). Consequently, in
the standard complex case (), 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 converges weakly to the
type 2 Tracy-Widom distribution . Moreover, in the real case, we
show that when is spiked with a fixed number of subcritical spikes,
the type 1 Tracy-Widom limit holds for the normalized largest
eigenvalue of , which extends a result of F\'{e}ral and
P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of
nondiagonal and more generally distributed .Comment: 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
Universality for a global property of the eigenvectors of Wigner matrices
Let be an real (resp. complex) Wigner matrix and
be its spectral decomposition. Set
, where is a real (resp.
complex) unit vector. Under the assumption that the elements of have 4
matching moments with those of GOE (resp. GUE), we show that the process
converges weakly to the Brownian bridge for any
such that as ,
where for the real case and for the complex case. Such a
result indicates that the othorgonal (resp. unitary) matrices with columns
being the eigenvectors of Wigner matrices are asymptotically Haar distributed
on the orthorgonal (resp. unitary) group from a certain perspective.Comment: typos correcte
Local Stability of the Free Additive Convolution
We prove that the system of subordination equations, defining the free
additive convolution of two probability measures, is stable away from the edges
of the support and blow-up singularities by showing that the recent smoothness
condition of Kargin is always satisfied. As an application, we consider the
local spectral statistics of the random matrix ensemble , where is
a Haar distributed random unitary or orthogonal matrix, and and are
deterministic matrices. In the bulk regime, we prove that the empirical
spectral distribution of concentrates around the free additive
convolution of the spectral distributions of and on scales down to
.Comment: Third version: More details added to Lemma 6.3 and proof of Theorem
2.
Spectral rigidity for addition of random matrices at the regular edge
We consider the sum of two large Hermitian matrices and with a Haar
unitary conjugation bringing them into a general relative position. We prove
that the eigenvalue density on the scale slightly above the local eigenvalue
spacing is asymptotically given by the free convolution of the laws of and
as the dimension of the matrix increases.
This implies optimal rigidity of the eigenvalues and optimal rate of
convergence in Voiculescu's theorem. Our previous works [3,4] established these
results in the bulk spectrum, the current paper completely settles the problem
at the spectral edges provided they have the typical square-root behavior. The
key element of our proof is to compensate the deterioration of the stability of
the subordination equations by sharp error estimates that properly account for
the local density near the edge. Our results also hold if the Haar unitary
matrix is replaced by the Haar orthogonal matrix
Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Let the sample correlation matrix be , where with
. We assume to be a collection of independent symmetric distributed
random variables with sub-exponential tails. Moreover, for any , we assume
to be identically distributed. We assume and
with some as . In this
paper, we provide the Tracy-Widom law () for both the largest and
smallest eigenvalues of . If are i.i.d. standard normal, we can
derive the for both the largest and smallest eigenvalues of the matrix
, where with , .Comment: 35 pages, a major revisio
- …