50 research outputs found
On Finite Rank Deformations of Wigner Matrices
We study the distribution of the outliers in the spectrum of finite rank
deformations of Wigner random matrice under the assumption that the
off-diagonal matrix entries have uniformly bounded fifth moment and the
diagonal entries have uniformly bounded third moment. Using our recent results
on the fluctuation of resolvent entries [31],[28], and ideas from [9], we
extend results by M.Capitaine, C.Donati-Martin, and D.F\'eral [12], [13].Comment: accepted for publication in Annales de l'Institut Henri Poincar\'e
(B) Probabilit\'es et Statistique
Weak Gibbs property and system of numeration
We study the selfsimilarity and the Gibbs properties of several measures
defined on the product space \Omega\_r:=\{0,1,...,\break r-1\}^{\mathbb N}.
This space can be identified with the interval by means of the
numeration in base . The last section is devoted to the Bernoulli
convolution in base , called the Erd\H os measure, and
its analogue in base , that we study by means of a
suitable system of numeration
Universal Level dynamics of Complex Systems
. We study the evolution of the distribution of eigenvalues of a
matrix subject to a random perturbation drawn from (i) a generalized Gaussian
ensemble (ii) a non-Gaussian ensemble with a measure variable under the change
of basis. It turns out that, in the case (i), a redefinition of the parameter
governing the evolution leads to a Fokker-Planck equation similar to the one
obtained when the perturbation is taken from a standard Gaussian ensemble (with
invariant measure). This equivalence can therefore help us to obtain the
correlations for various physically-significant cases modeled by generalized
Gaussian ensembles by using the already known correlations for standard
Gaussian ensembles.
For large -values, our results for both cases (i) and (ii) are similar to
those obtained for Wigner-Dyson gas as well as for the perturbation taken from
a standard Gaussian ensemble. This seems to suggest the independence of
evolution, in thermodynamic limit, from the nature of perturbation involved as
well as the initial conditions and therefore universality of dynamics of the
eigenvalues of complex systems.Comment: 11 Pages, Latex Fil
Central limit theorems for eigenvalues in a spiked population model
In a spiked population model, the population covariance matrix has all its
eigenvalues equal to units except for a few fixed eigenvalues (spikes). This
model is proposed by Johnstone to cope with empirical findings on various data
sets. The question is to quantify the effect of the perturbation caused by the
spike eigenvalues. A recent work by Baik and Silverstein establishes the almost
sure limits of the extreme sample eigenvalues associated to the spike
eigenvalues when the population and the sample sizes become large. This paper
establishes the limiting distributions of these extreme sample eigenvalues. As
another important result of the paper, we provide a central limit theorem on
random sesquilinear forms.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP118 the Annales de
l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques
(http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Performance analysis of the dominant mode rejection beamformer
In array signal processing over challenging environments, due to the non-stationarity nature of data, it is difficult to obtain enough number of data snapshots to construct an adaptive beamformer (ABF) for detecting weak signal embedded in strong interferences. One type of adaptive method targeting for such applications is the dominant mode rejection (DMR) method, which uses a reshaped eigen-decomposition of sample covariance matrix (SCM) to define a subspace containing the dominant interferers to be rejected, thereby allowing it to detect weak signal in the presence of strong interferences. The DMR weight vector takes a form similar to the adaptive minimum variance distortion-less response (MVDR), except with the SCM being replaced by the DMR-SCM.
This dissertation studies the performance of DMR-ABF by deriving the probability density functions of three important metrics: notch depth (ND), white noise gain (WNG), and signal-to-interference-and-noise ratio (SINR). The analysis contains both single interference case and multiple interference case, using subspace transformation and the random matrix theory (RMT) method for deriving and verifying the analytical results. RMT results are used to approximate the random matrice. Finally, the analytical results are compared with RMT Monte-Carlo based empirical results