We consider the problem of estimating the integral of the square of a densityf from the observation of a n sample. Our method to estimate $\int_{\mathbb{R}} f^2(x){\rm d}x$ isbased on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponentialinequality for U-statistics of order 2 due to Houdré and Reynaud.

In the present paper, by using theinequality due to Talagrand's isoperimetric method, severalversions of the bounded law of iterated logarithm for a sequenceof independent Banach space valued random variables are developedand the upper limits for the non-random constant are given.

We continue the investigation started in a previous paper, onweak convergence to infinitely divisible distributions with finitevariance. In the present paper, we study this problem for someweakly dependent random variables, including in particularassociated sequences. We obtain minimal conditions expressed interms of individual random variables. As in the i.i.d. case, wedescribe the convergence to the Gaussian and the purelynon-Gaussian parts of the infinitely divisible limit. We alsodiscuss the rate of Poisson convergence and emphasize the specialcase of Bernoulli random variables. The proofs aremainly based on Lindeberg's method.

Sample path large deviationsfor the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero arepresented. The noise is a complex additive Gaussian noise. It iswhite in time and colored in space. The solutions may be global orblow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologiesanalogue to projective limit topologies. In this setting, thesupport of the law of the solution is also characterized. As aconsequence, results on the law of the blow-up time andasymptotics when the noise converges to zero are obtained. Anapplication to the transmission of solitary waves in fiber opticsis also given.

Let (Sn)n≥0 be a $\mathbb Z$ -random walk and $(\xi_{x})_{x\in \mathbb Z}$ be a sequence of independent andidentically distributed $\mathbb R$ -valued random variables,independent of the random walk. Let h be a measurable, symmetricfunction defined on $\mathbb R^2$ with values in $\mathbb R$ . We study theweak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$ , withvalues in D[0,1] the set of right continuous real-valuedfunctions with left limits, defined by \[ \sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1].\] Statistical applications are presented, in particular we prove a strong law of large numbersfor U-statistics indexed by a one-dimensional random walk using a result of [1].

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}\sum_1^n\mathbf{f}(x_i^n) \cdot Z^n_i$ where the deterministic probability measure $\frac{1}{n}\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure R and $(Z^n_i)_{i\in \mathbb{N}}$ are $\mathbb{R}^d$ -valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$ \sup_{i,n} {\mathbb E}{\rm e}^{\alpha^* |Z_i^n|}< +\infty \quad\textrm{for some}\quad 0<\alpha^*<+\infty.$$

In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend Erdös and Rényi's functional law of large numbers.

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, “the structural intensity”, that computes the “density” of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed.

We consider a differential equation with a random rapidly varying coefficient.The random coefficient is aGaussian process with slowly decaying correlations and compete with a periodic component. In theasymptotic framework corresponding to the separation of scales present in theproblem, we prove that the solution of the differential equationconverges in distribution to the solution of a stochastic differential equationdriven by a classical Brownian motion in some cases, by a fractional Brownianmotion in other cases. The proofs of these results are based on the Lyons theory ofrough paths. Finally we discuss applications in two physical situations.

In this paper we solve the basic fractionalanalogue of the classical infinite time horizon linear-quadratic Gaussianregulator problem. For a completely observable controlled linearsystem driven by a fractional Brownian motion, we describeexplicitely the optimal control policy which minimizes anasymptotic quadratic performance criterion.

In this paper we consider three measures of overlap, namely Matusia's measure ρ, Morisita's measure λ andWeitzman's measure Δ. These measures are usually used inquantitative ecology and stress-strength models of reliabilityanalysis. Herein we consider two Weibull distributions havingthe same shape parameter and different scale parameters. Thisdistribution is known to be the most flexible life distributionmodel with two parameters. Monte Carlo evaluations are used tostudy the bias and precision of some estimators of these overlapmeasures. Confidence intervals for the measures are alsoconstructed via bootstrap methods and Taylor series approximation.

In this paper we focus on the problem of estimating a boundeddensity using a finite combination of densities from a givenclass. We consider the Maximum Likelihood Estimator (MLE) and thegreedy procedure described by Li and Barron (1999)under the additional assumption of boundedness of densities. Weprove an $O(\frac{1}{\sqrt{n}})$ bound on the estimation errorwhich does not depend on the number of densities in the estimatedcombination. Under the boundedness assumption,this improves the bound of Li and Barron by removing the $\log n$ factor and also generalizes it to the base classes with convergingDudley integral.

To validate pollution data, subject-matter experts in Airpl (an organizationthat maintains a network of air pollution monitoring stations in westernFrance) daily perform visual examinations of the data and check theirconsistency. In this paper, we describe these visual examinations andpropose a formalization for this problem. The examinations consistin comparisons of so-called shorth intervals so we build a statisticaltest that compares such intervals in a nonparametric regression model.This allows to detect atypical data. A practical applicationof the test is given.

An iterative method based on a fixed-point property is proposed for finding maximum likelihoodestimators for parameters in a model of network reliability withspatial dependence. The method is shown to converge at a geometric rate under natural conditions on data.

In this article we prove new results concerning thestructure and the stability properties of the global attractor associatedwith a class of nonlinear parabolic stochastic partial differential equationsdriven by a standard multidimensional Brownian motion.We first use monotonicity methodsto prove that the random fields either stabilize exponentially rapidly withprobability one around one of the two equilibrium states, or that they set outto oscillate between them. In the first case we can also compute exactly thecorresponding Lyapunov exponents.The last case of our analysis reveals a phenomenon of exchange of stabilitybetween the two components of the global attractor. In order to prove thisasymptotic property, we show an exponential decay estimate between the randomfield and its spatial average under an additional uniform ellipticityhypothesis.

In Benaïm and Ben Arous (2003) is solved a multi-armed bandit problem arising in the theory of learning in games. We proposea short and elementary proof of this result based on a variant of the Kronecker lemma.

In this paper, we prove a conditional principle of Gibbs type forrandom weighted measures of the form ${L_n=\frac{1}{n}\sum_{i=1}^nZ_i\delta_{x_i^n}}$ , ((Zi)i being asequence of i.i.d. real random variables. Our work extends thepreceding results of Gamboa and Gassiat (1997), in allowing to consider thinconstraints. Transportation-like ideas are used in the proof.

Let F n be the empirical distribution function (df) pertainingto independent random variables with continuous df F. Weinvestigate the minimizing point $\hat\tau_n$ of the empiricalprocess Fn - F0 , where F 0 is another df which differs fromF. If F and F 0 are locally Hölder-continuous of orderα at a point τ our main result states that $n^{1/\alpha}(\hat\tau_n - \tau)$ converges in distribution. Thelimit variable is the almost sure unique minimizing point of atwo-sided time-transformed homogeneous Poisson-process with adrift. The time-transformation and the drift-function are of thetype |t|α.

The last few years have witnessed important new developments inthe theory and practice of pattern classification. We intend tosurvey some of the main new ideas that have led to theserecent results.