116 research outputs found
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles
We consider properties of determinants of some random symmetric matrices
issued from multivariate statistics: Wishart/Laguerre ensemble (sample
covariance matrices), Uniform Gram ensemble (sample correlation matrices) and
Jacobi ensemble (MANOVA). If is the size of the sample, the
number of variates and such a matrix, a generalization of the
Bartlett-type theorems gives a decomposition of into a product
of independent gamma or beta random variables. For fixed, we study the
evolution as grows, and then take the limit of large and with . We derive limit theorems for the sequence of {\it processes with
independent increments} for .. Since the logarithm of the determinant is a linear
statistic of the empirical spectral distribution, we connect the results for
marginals (fixed ) with those obtained by the spectral method. Actually, all
the results hold true for models, if we define the determinant as the
product of charges.Comment: 51 pages ; it replaces and extends arXiv:math/0607767 and
arXiv:math/0509021 Third version: corrected constants in Theorem 3.
Asymptotical behaviour of the presence probability in branching random walks and fragmentations
For a subcritical Galton-Watson process , it is well known that
under an condition, the quotient has a
finite positive limit. There is an analogous result for a (one-dimensional)
supercritical branching random walk: when is in the so-called subcritical
speed area, the probability of presence around in the -th generation is
asymptotically proportional to the corresponding expectation. In Rouault (1993)
this result was stated under a natural assumption on the offspring
point process and a (unnatural) condition on the offspring mean. Here we prove
that the result holds without this latter condition, in particular we allow an
infinite mean and a dimension for the state-space. As a consequence
the result holds also for homogeneous fragmentations as defined in Bertoin
(2001), using the method of discrete-time skeletons; this completes the proof
of Theorem 4 in Bertoin-Rouault (2004 see math/PR/0409545). Finally, an
application to conditioning on the presence allows to meet again the
probability tilting and the so-called additive martingale.Comment: 15 pages, companion paper of math.PR/040954
Large Deviations for Random Spectral Measures and Sum Rules
We prove a Large Deviation Principle for the random spec- tral measure
associated to the pair where is sampled in the GUE(N) and e is
a fixed unit vector (and more generally in the - extension of this
model). The rate function consists of two parts. The contribution of the
absolutely continuous part of the measure is the reversed Kullback information
with respect to the semicircle distribution and the contribution of the
singular part is connected to the rate function of the extreme eigenvalue in
the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but
in thoses cases the expression of the rate function is not so explicit
Discretization methods for homogeneous fragmentations
Homogeneous fragmentations describe the evolution of a unit mass that breaks
down randomly into pieces as time passes. They can be thought of as continuous
time analogs of a certain type of branching random walks, which suggests the
use of time-discretization to shift known results from the theory of branching
random walks to the fragmentation setting. In particular, this yields
interesting information about the asymptotic behaviour of fragmentations.
On the other hand, homogeneous fragmentations can also be investigated using
a powerful technique of discretization of space due to Kingman, namely, the
theory of exchangeable partitions of . Spatial discretization is especially
well-suited to develop directly for continuous times the conceptual method of
probability tilting of Lyons, Pemantle and Peres.Comment: 21 page
Large deviations for near-extreme eigenvalues in the beta-ensembles
For beta ensembles with convex poynomial potentials, we prove a large
deviation principle for the empirical spectral distribution seen from the
rightmost particle. This modified spectral distribution was introduced by
Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal
eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.Comment: We fixed typos and changed Remarks 2.13 and 2.1
Truncations of Haar distributed matrices, traces and bivariate Brownian bridges
Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after
centering the double index process converges in distribution to
the bivariate tied-down Brownian bridge. The proof relies on the notion of
second order freeness.Comment: Random matrices: Theory and Applications (RMTA) To appear (2012)
http://www.editorialmanager.com/rmta
Circular Jacobi Ensembles and deformed Verblunsky coefficients
Using the spectral theory of unitary operators and the theory of orthogonal
polynomials on the unit circle, we propose a simple matrix model for the
following circular analogue of the Jacobi ensemble: c_{\delta,\beta}^{(n)}
\prod_{1\leq k with . If is
a cyclic vector for a unitary matrix , the spectral measure of
the pair is well parameterized by its Verblunsky coefficients
. We introduce here a deformation of these coefficients so that the associated Hessenberg
matrix (called GGT) can be decomposed into a product of elementary reflections parameterized by these coefficients.
If are independent random variables with some
remarkable distributions, then the eigenvalues of the GGT matrix follow the
circular Jacobi distribution above.
These deformed Verblunsky coefficients also allow to prove that, in the
regime with \delta(n)/n \to \dd, the spectral measure
and the empirical spectral distribution weakly converge to an explicit
nontrivial probability measure supported by an arc of the unit circle. We also
prove the large deviations for the empirical spectral distribution.Comment: New section on large deviations for the empirical spectral
distribution, Corrected value for the limiting free energ
Large Deviations and Branching Processes
These lecture notes are devoted to present several uses of Large Deviation asymptotics in Branching Processes
Sum rules and large deviations for spectral matrix measures
A sum rule relative to a reference measure on R is a relationship between the
reversed Kullback-Leibler divergence of a positive measure on R and some
non-linear functional built on spectral elements related to this measure (see
for example Killip and Simon 2003). In this paper, using only probabilistic
tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel
and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover
the earlier result of Damanik, Killip and Simon (2010) when the reference
measure is the (matrix-valued) semicircle law and obtain a new sum rule when
the reference measure is the (matrix-valued) Marchenko-Pastur law
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