Preprints du L.P.S. C.N.R.S. Toulouse (Toulouse Mathematical Institute) (1995-2004)
Del Moral, P., Doucet A., G.W. Peters
Asymptotic and Increasing Propagation of Chaos Expansions for Genealogical
Particle models Publications du L.S.P., [32p], (2004). This article is concerned with the propagation-of-chaos properties of genetic
type particle models. This class of models arises in a variety of scientific
disciplines including theoretical physics, macromolecular biology, engineering
sciences, and more particularly in advanced signal processing. From the pure
mathematical point of view, these interacting particle systems can be regarded
as a mean field particle interpretation of a class of Feynman-Kac measures on
path spaces.
In the present article, we design an original integration theory of
propagation-of-chaos based on the fluctuation analysis of a class of
interacting particle random fields. We provide analytic functional
representations of the asymptotic distributions of finite particle blocks,
yielding what seems to be the first result of this kind for this class of
interacting particle systems. These asymptotic expansions are expressed in
terms of the limiting Feynman-Kac semigroups and a class of interacting jump
operators. These results provide both sharp estimates of the negligible bias
introduced by the interaction mechanisms, and central limit theorems for
nondegenerate $U$-statistics and von Mises statistics associated with genealogical tree
models. Applications to nonlinear filtering problems and interacting Markov
chain Monte Carlo algorithms are discussed.
Del Moral, P., Garnier J.
Genealogical Particle Analysis of Rare Events Publications du L.S.P., [32p], (2004). In this paper an original interacting particle system
approach is developed for studying Markov chains in rare event regimes.
The proposed particle system is theoretically studied through a genealogical
tree interpretation of Keynman-Kac path measures.
The algorithmic implementation of the particle system is presented.
An efficient estimator for the probability of ocurrence of a rare event
is proposed and its variance is computed.
Applications and numerical implementations are discussed.
First, we apply the particle system technique
to a toy model (a Gaussian random walk),
which permits to illustrate the theoretical predictions.
Second, we address a physically relevant problem consisting in
the estimation of the outage probability
due to polarization-mode dispersion in optical fibers.
Del Moral, P., Tindel S.
A Berry-Esseen theorem
for Feynman-Kac and interacting particle models. Publications du L.S.P., [30p], (2003). In this paper we investigate the speed of convergence
of the fluctuations of a general class
of Feynman-Kac particle approximation models.
We design an original approach based
on new Berry-Esseen type estimates for
abstract martingale sequences combined with
original exponential concentration estimates of
interacting processes. These
results extend the corresponding
statements in the classical theory and apply to
a class of branching and genealogical path-particle
models arising in non linear
filtering literature as well as in statistical physics and biology.
Del Moral P., Miclo L., Viens F. Precise Propagations of Chaos Estimates
for Feynman-Kac and Genealogical Particle Models Technical Report, Purdue University, [29p], (2003). Strong propagations of chaos estimates for interacting particle
and Feynman-Kac approximating models are studied.
We use as
a tool a tensor product Feynman-Kac semi-group approach with
respect to time horizons and particle block sizes.
Propagations of chaos estimates for Boltzmann-Gibbs
measures are derived
from a precise moment analysis of empirical measures
and from an
original transport equation relating symmetric
statistic type and
tensor product empirical measures.
The analysis
presented in this article apply to study
the asymptotic behavior
of genetic historical processes and their complete
genealogical tree evolution
yielding what seems to be the first
precise propagations of chaos estimates for this
type of path-particle models. Incidently this can be also be considered
as an extension of the traditional
asymptotic theory of q-symmetric statistics to
interacting random sequences.
Del Moral P. and Miclo L.
Annealed Feynman-Kac Models. Publications du Laboratoire de Statistiques et Probabilites,
Toulouse III (2002).
We analyze the concentration properties of an annealed
Feynman-Kac model in distribution space. We characterize the
concentration regions in terms of a variational problem involving
a competition between the potential function and the mutation kernel.
When the temperature parameter is evanescent with time and under appropriate hypotheses, the probability mass
tends to concentrate on regions with minimal potential values. We give
a precise description of these areas using non-linear semi-group
contractions and large deviations techniques. We illustrate this
annealed model with two physical interpretations related respectively
to Markov motions in absorbing media and
interacting measure valued processes.
Del Moral P. and Miclo L.
On the convergence of chains with time empirical self-interactions. Publications du Laboratoire de Statistiques et Probabilites,
Toulouse III (2002).
We consider stochastic chains on abstract measurable spaces
whose evolution at any given time depend on the present
position and on the occupation measure created by the path up to this instant. This generalization of reinforced random walks enables us to impose conditions insuring $Lp$ or
a.s. convergence of the empirical measures toward some fixed point of a probability-valued dynamical system. We present two sets of hypotheses based on weak contraction properties, leading to two different proofs, but in both situations the rates of convergence are optimal in the examined level of generality.
Del Moral P. and Miclo L.
Markov chains with self-interactions. Publications du Laboratoire de Statistiques et Probabilites½,
Toulouse III (2002).
In this article we study a class of time self-interacting ``Markov''
chain models. We propose a novel theoretical basis based on measure valued
processes and semigroup technics to analyze their
asymptotic behavior as the time parameter tends to infinity. We
exhibit different types of decays to equilibrium depending on the
level of interaction.
We illustrate these results in a variety of
examples including Gaussian or Poisson self-interacting models. We
analyze the long time behavior of a new
class of evolutionary self-interacting chain models. These
genetic type algorithms can also be regarded as
reinforced stochastic explorations of an environment
with obstacles related to a
potential function.
Cerou F., Del Moral P., Le Gland F. and Lezaud P.
Genealogical Models in Rare Event Analysis. Publications du Laboratoire de Statistiques et Probabilites,
Toulouse III, (2002).
We present in this article a genetic type
interacting particle systems
algorithm and a genealogical model for
estimating a class of rare events arising in physics and network
analysis. We represent the distribution of a Markov process
hitting a rare target in terms of a
Feynman-Kac model in path space. We show how these
branching
particle models described in previous works can be used to estimate the
probability of the corresponding rare events as well as the
distribution
of the process in this regime.
Del Moral P. and Zajic T.
A note on Laplace-Varadhan's integral lemma. Publications du Laboratoire de Statistiques et Probabilites,
Toulouse III, (2001).
We propose a complement to
an integral lemma of Laplace-Varadhan arising in large deviations
literature. We examine a situation in which the state space may
depend on the rate of deviations.
In the final part of this work we use this framework
to discuss large deviations principles for a class of mean field
interacting jump processes and a class of mean field interacting
non linear diffusions.
Del Moral P., Miclo L.
Particle approximations of Lyapunov exponents
connected to Schr\"odinger operators and Feynman-Kac semigroups. Publications du Laboratoire de Statistiques et Probabilites,
Toulouse III, (2001).
We present an interacting particle system
methodology for the numerical solving of the Lyapunov exponent
of Feynman-Kac semigroups and for estimating the principal
eigenvalue of Schr\"odinger generators. The continuous or discrete time
models studied in this work
consists of N interacting particles evolving in an environment
with soft obstacles related to a potential function $V$. These
models are related to genetic algorithms and Moran type particle
schemes. Their choice
is not unique. We will examine a class of models
extending the hard obstacle model
of K. Burdzy, R. Holyst and P. March
and including the Moran type scheme presented by the authors in a
previous work.
We provide precise
uniform estimates with respect to the time parameter and we
analyse
the fluctuations of continuous time particle models.
Del Moral P., Ledoux M. and Miclo L.
About supercontraction properties of Markov kernels. Publications du Laboratoire de Statistiques et Probabilites, Toulouse III, No 01-01 (2001).
We study Lipschitz's contraction properties of general Markovian kernels seen as operators on spaces of probabilities endowed with entropy-like ``distances". Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's coefficient and strong contraction properties in Orlicz's spaces for relative densities are proved under restrictive mixing assumptions. Next we obtain contraction estimates in the entropy sense around an arbitrary probability by introducing a particular Dirichlet form and the corresponding modified logarithmic Sobolev inequalities. The interest of these bounds will be illustred by inhomogeneous Gaussian examples, emphasizing the irrelevence of the existence of an invariant measure assumption.
Del Moral P., Miclo L.
About the strong propagation of chaos for interacting
particle approximations of Feynman-Kac formulae. Publications du Laboratoire de Statistiques et Probabilites, Toulouse III, No 08-00 (2000).
Recently we have introduced Moran type interacting particle systems in order to solve numerically
normalized continuous time Feynman-Kac formulae. These schemes can also be seen as approximating procedures
for certain generalized spatially homogeneous Boltzman equations, so strong propagation of chaos is known to hold for them. We will give a new proof of this result by studying the evolution of tensorized empirical
measures and then applying two straightforward coupling arguments. The only difficulty is in the first step
to find nice martingales, and this will be done via the introduction of another family
of Moran semigroups. This work also procures us the opportuneness to present an appropriate abstract
setting, in particular without any topological assumption on the state space, and to apply
a genealogical algorithm for the smoothing problem in non linear filtering context.
