20 research outputs found
Large Deviations for the solution of a Kac-type kinetic equation
The aim of this paper is to study large deviations for the self-similar
solution of a Kac-type kinetic equation. Under the assumption that the initial
condition belongs to the domain of normal attraction of a stable law of index
and under suitable assumptions on the collisional kernel, precise
asymptotic behavior of the large deviations probability is given
Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits
The problem of convergence in law of normed sums of exchangeable random
variables is examined. First, the problem is studied w.r.t. arrays of
exchangeable random variables, and the special role played by mixtures of
products of stable laws - as limits in law of normed sums in different rows of
the array - is emphasized. Necessary and sufficient conditions for convergence
to a specific form in the above class of measures are then given. Moreover,
sufficient conditions for convergence of sums in a single row are proved.
Finally, a potentially useful variant of the formulation of the results just
summarized is briefly sketched, a more complete study of it being deferred to a
future work
Bayesian estimation for a parametric Markov renewal model applied to seismic data.
This paper presents a complete methodology for Bayesian inference on a
semi-Markov process, from the elicitation of the prior distribution, to the
computation of posterior summaries, including a guidance for its JAGS
implementation. The holding times (conditional on the transition between two
given states) are assumed to be Weibull-distributed. We examine the elicitation
of the joint prior density of the shape and scale parameters of the Weibull
distributions, deriving a specific class of priors in a natural way, along with
a method for the determination of hyperparameters based on ``learning data''
and moment existence conditions. This framework is applied to data of
earthquakes of three types of severity (low, medium and high size) that
occurred in the central Northern Apennines in Italy and collected by the
\cite{CPTI04} catalogue. Assumptions on two types of energy accumulation and
release mechanisms are evaluated
Kinetic models with randomly perturbed binary collisions
We introduce a class of Kac-like kinetic equations on the real line, with
general random collisional rules, which include as particular cases models for
wealth redistribution in an agent-based market or models for granular gases
with a background heat bath. Conditions on these collisional rules which
guarantee both the existence and uniqueness of equilibrium profiles and their
main properties are found. We show that the characterization of these
stationary solutions is of independent interest, since the same profiles are
shown to be solutions of different evolution problems, both in the econophysics
context and in the kinetic theory of rarefied gases
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
Theoremes limites pour les chaines de Markov : application aux algorithmes stochastiques
SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc