513 research outputs found
A Strong Szego Theorem for Jacobi Matrices
We use a classical result of Gollinski and Ibragimov to prove an analog of
the strong Szego theorem for Jacobi matrices on . In particular, we
consider the class of Jacobi matrices with conditionally summable parameter
sequences and find necessary and sufficient conditions on the spectral measure
such that and lie in
, the linearly-weighted space.Comment: 26 page
Mod-Gaussian convergence and its applications for models of statistical mechanics
In this paper we complete our understanding of the role played by the
limiting (or residue) function in the context of mod-Gaussian convergence. The
question about the probabilistic interpretation of such functions was initially
raised by Marc Yor. After recalling our recent result which interprets the
limiting function as a measure of "breaking of symmetry" in the Gaussian
approximation in the framework of general central limit theorems type results,
we introduce the framework of -mod-Gaussian convergence in which the
residue function is obtained as (up to a normalizing factor) the probability
density of some sequences of random variables converging in law after a change
of probability measure. In particular we recover some celebrated results due to
Ellis and Newman on the convergence in law of dependent random variables
arising in statistical mechanics. We complete our results by giving an
alternative approach to the Stein method to obtain the rate of convergence in
the Ellis-Newman convergence theorem and by proving a new local limit theorem.
More generally we illustrate our results with simple models from statistical
mechanics.Comment: 49 pages, 21 figure
Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
This paper deals with a one--dimensional model for granular materials, which
boils down to an inelastic version of the Kac kinetic equation, with
inelasticity parameter . In particular, the paper provides bounds for
certain distances -- such as specific weighted --distances and the
Kolmogorov distance -- between the solution of that equation and the limit. It
is assumed that the even part of the initial datum (which determines the
asymptotic properties of the solution) belongs to the domain of normal
attraction of a symmetric stable distribution with characteristic exponent
\a=2/(1+p). With such initial data, it turns out that the limit exists and is
just the aforementioned stable distribution. A necessary condition for the
relaxation to equilibrium is also proved. Some bounds are obtained without
introducing any extra--condition. Sharper bounds, of an exponential type, are
exhibited in the presence of additional assumptions concerning either the
behaviour, near to the origin, of the initial characteristic function, or the
behaviour, at infinity, of the initial probability distribution function
General model selection estimation of a periodic regression with a Gaussian noise
This paper considers the problem of estimating a periodic function in a
continuous time regression model with an additive stationary gaussian noise
having unknown correlation function. A general model selection procedure on the
basis of arbitrary projective estimates, which does not need the knowledge of
the noise correlation function, is proposed. A non-asymptotic upper bound for
quadratic risk (oracle inequality) has been derived under mild conditions on
the noise. For the Ornstein-Uhlenbeck noise the risk upper bound is shown to be
uniform in the nuisance parameter. In the case of gaussian white noise the
constructed procedure has some advantages as compared with the procedure based
on the least squares estimates (LSE). The asymptotic minimaxity of the
estimates has been proved. The proposed model selection scheme is extended also
to the estimation problem based on the discrete data applicably to the
situation when high frequency sampling can not be provided
AR and MA representation of partial autocorrelation functions, with applications
We prove a representation of the partial autocorrelation function (PACF), or
the Verblunsky coefficients, of a stationary process in terms of the AR and MA
coefficients. We apply it to show the asymptotic behaviour of the PACF. We also
propose a new definition of short and long memory in terms of the PACF.Comment: Published in Probability Theory and Related Field
Generalized Central Limit Theorem and Renormalization Group
We introduce a simple instance of the renormalization group transformation in
the Banach space of probability densities. By changing the scaling of the
renormalized variables we obtain, as fixed points of the transformation, the
L\'evy strictly stable laws. We also investigate the behavior of the
transformation around these fixed points and the domain of attraction for
different values of the scaling parameter. The physical interest of a
renormalization group approach to the generalized central limit theorem is
discussed.Comment: 16 pages, to appear in J. Stat. Phy
Complete characterization of convergence to equilibrium for an inelastic Kac model
Pulvirenti and Toscani introduced an equation which extends the Kac
caricature of a Maxwellian gas to inelastic particles. We show that the
probability distribution, solution of the relative Cauchy problem, converges
weakly to a probability distribution if and only if the symmetrized initial
distribution belongs to the standard domain of attraction of a symmetric stable
law, whose index is determined by the so-called degree of
inelasticity, , of the particles: . This result is
then used: (1) To state that the class of all stationary solutions coincides
with that of all symmetric stable laws with index . (2) To determine
the solution of a well-known stochastic functional equation in the absence of
extra-conditions usually adopted
Entanglement in the quantum Ising model
We study the asymptotic scaling of the entanglement of a block of spins for
the ground state of the one-dimensional quantum Ising model with transverse
field. When the field is sufficiently strong, the entanglement grows at most
logarithmically in the number of spins. The proof utilises a transformation to
a model of classical probability called the continuum random-cluster model, and
is based on a property of the latter model termed ratio weak-mixing. Our proof
applies equally to a large class of disordered interactions
Boundary driven zero-range processes in random media
The stationary states of boundary driven zero-range processes in random media
with quenched disorder are examined, and the motion of a tagged particle is
analyzed. For symmetric transition rates, also known as the random barrier
model, the stationary state is found to be trivial in absence of boundary
drive. Out of equilibrium, two further cases are distinguished according to the
tail of the disorder distribution. For strong disorder, the fugacity profiles
are found to be governed by the paths of normalized -stable
subordinators. The expectations of integrated functions of the tagged particle
position are calculated for three types of routes.Comment: 23 page
Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble
The paper studies the spectral properties of large Wigner, band and sample
covariance random matrices with heavy tails of the marginal distributions of
matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at
Giens, France on September 13-17, 200
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