The subject of this paper is to estimate adaptively the common probabilitydensity of n independent, identically distributed random variables. Theestimation is done at a fixed point $x_{0}\in \mathbb R$ , over the densityfunctions that belong to the Sobolev class Wn(β,L). We consider theadaptive problem setup, where the regularity parameter β is unknownand varies in a given set B n . A sharp adaptive estimator is obtained,and the explicit asymptotical constant, associated to its rate ofconvergence is found.

In this paper, we study the problem of non parametric estimationof an unknown regression function from dependent data withsub-Gaussian errors. As a particular case, we handle theautoregressive framework. For this purpose, we consider acollection of finite dimensional linear spaces (e.g. linear spacesspanned by wavelets or piecewise polynomials on a possiblyirregular grid) and we estimate the regression function by aleast-squares estimator built on a data driven selected linearspace among the collection. This data driven choice is performedvia the minimization of a penalized criterion akin to the Mallows'C p . We state non asymptotic risk bounds for our estimator insome ${\mathbb{L}}_2$ -norm and we show that it is adaptive in the minimaxsense over a large class of Besov balls of the form Bα,p,∞(R) with p ≥ 1.

In this paper we prove a Central Limit Theorem forstandard kernel estimates of the invariant density of one-dimensionaldynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergencefor the variance of the estimator and a variation on the Lindeberg–Riomethod. We also give an extension in the case of weaklydependent sequences in a sense introduced by Doukhan and Louhichi.

We establish new exponential inequalities for partial sums of random fields. Next, using classicalchaining arguments, we give sufficient conditions for partial sum processes indexed by large classes ofsets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, thecondition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixingrandom fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.

We consider a generalization of the so-called divide andcolor model recently introduced by Häggström. We investigate thebehavior of the magnetization in large boxes of the lattice $\mathbb{Z}^d$ and its fluctuations. Thus, Laws of Large Numbers and CentralLimit Theorems are proved, both quenched and annealed. We showthat the properties of the underlying percolation process deeplyinfluence the behavior of the coloring model. In the subcriticalcase, the limit magnetization is deterministic and the CentralLimit Theorem admits a Gaussian limit. Conversely, the limitmagnetization is not deterministic in the supercritical case andthe limit of the Central Limit Theorem is not Gaussian, except inthe particular model with exactly two colors which are equallyprobable. We also prove a Central Limit Theorem for the size of the intersection of the infinite cluster with large boxes in supercritical bond percolation.

We consider the problem of hypothesis testing within a monotoneregression model. We propose a new test of the hypothesisH 0: “ƒ = ƒ0 ” against the composite alternative H a : “ƒ ≠ ƒ0 ” under the assumption that the true regression functionf is decreasing. The test statistic is based on the ${\mathbb L}_{1}$ -distance between the isotonic estimator of f and thefunction f 0, since it is known that a properly centered and normalized version of this distance is asymptotically standardnormally distributed under H 0. We study the asymptotic powerof the test under alternatives that converge to the nullhypothesis.

The aim of this paper is to take an in-depth look at the longtime behaviour of some continuous time Markovian dynamical systems and at itsnumerical analysis. We first propose a short overview of the main ergodicity propertiesof time continuous homogeneous Markov processes (stability, positive recurrence). The basictool is a Lyapunov function. Then, we investigate if these properties still hold forthe time discretization of these processes, either with constant or decreasing step (ODE method in stochasticapproximation, Euler scheme for diffusions). We point out severaladvantages of the weighted empirical random measures associated to these procedures, especially withdecreasing step, in terms of convergence and of rate of convergence. Several simulations illustrate theseresults.


Many protein sequences present non trivial periodicities, such ascysteine signatures and leucine heptads. These known periodicitiesprobably represent a small percentage of the total number of sequencesperiodic structures, and it is useful to have general tools todetect such sequences and their period in large databases ofsequences. We compare three statistics adapted from those used in timeseries analysis: a generalisation of the simple autocovariance basedon a similarity score and two statistics intending to increase thepower of the method. Theoretical behaviour of these statistics arederived, and the corresponding tests are then described. In this paperwe also present an application of these tests to a protein known tohave sequence periodicity.

In this paper, we develop bounds on the distribution function of the empiricalmean for general ergodic Markov processes having a spectral gap. Our approach isbased on the perturbation theory for linear operators, following the techniqueintroduced by Gillman.

The Brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N x N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic inter-particles repulsion. The aim of this paper is to define the finite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to infinity (through the empirical measure process). We prove that a limiting measure-valued process exists and is the unique solution of a deterministic second-order PDE. The uniform law on [-π;π] is the only limiting distribution of µt when t goes to infinity and µt has an analytical density.

We consider a diffusion process X which is observed at times i/nfor i = 0,1,...,n, each observation being subject to a measurementerror. All errors are independent and centered Gaussian with knownvariance pn . There is an unknown parameter within the diffusioncoefficient, to be estimated. In this first paper thecase when X is indeed a Gaussian martingale is examined: we can provethat the LAN property holds under quite weak smoothness assumptions,with an explicit limiting Fisher information. What is perhaps the mostinteresting is the rate at which this convergence takes place:it is $1/\sqrt{n}$ (as when there is no measurement error) when pn goes fastenough to 0, namely npn is bounded. Otherwise, and provided thesequence pn itself is bounded, the rate is (pn / n) 1/4. Inparticular if pn = p does not depend on n, we get a rate n -1/4.

We consider a diffusion process X which is observed at times i/nfor i = 0,1,...,n, each observation being subject to a measurementerror. All errors are independent and centered Gaussian with knownvariance pn . There is an unknown parameter to estimate within thediffusion coefficient. In this second paper weconstruct estimators which are asymptotically optimal when theprocess X is a Gaussian martingale, and we conjecture that they arealso optimal in the general case.

This paper is concerned with the problem of simulation of (Xt)0≤t≤T , thesolution of a stochastic differential equation constrained by some boundary conditions in a smooth domainD: namely, we consider the case where the boundary ∂D is killing, or where it is instantaneouslyreflecting in an oblique direction. Given N discretization times equally spaced on the interval [0,T],we propose new discretization schemes: they are fully implementable and provide a weak error of orderN -1 under some conditions. The construction of these schemes is based on a natural principle of localapproximation of the domain into a half space, for which efficient simulations are available.