1,752 research outputs found
Analysis of the limiting spectral measure of large random matrices of the separable covariance type
Consider the random matrix where
and are deterministic Hermitian nonnegative matrices with
respective dimensions and , and where is a random
matrix with independent and identically distributed centered elements with
variance . Assume that the dimensions and grow to infinity at the
same pace, and that the spectral measures of and converge as
towards two probability measures. Then it is known that the
spectral measure of converges towards a probability measure
characterized by its Stieltjes Transform.
In this paper, it is shown that has a density away from zero, this
density is analytical wherever it is positive, and it behaves in most cases as
near an edge of its support. A complete characterization
of the support of is also provided. \\ Beside its mathematical interest,
this analysis finds applications in a certain class of statistical estimation
problems.Comment: Correction of the proof of Lemma 3.
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
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