We consider the nonparametric regression estimation problem of recovering an unknownresponse function f on the basis of spatially inhomogeneous data when thedesign points follow a known density g with a finite number ofwell-separated zeros. In particular, we consider two different cases: wheng has zeros of a polynomial order and when g has zerosof an exponential order. These two cases correspond to moderate and severe data losses,respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds forthe L2-risk when f is assumed to belong to aBesov ball, and construct adaptive wavelet thresholding estimators of fthat are asymptotically optimal (in the minimax sense) or near-optimal within alogarithmic factor (in the case of a zero of a polynomial order), over a wide range ofBesov balls. The spatially inhomogeneous ill-posed problem that we investigate isinherently more difficult than spatially homogeneous ill-posed problems like,e.g., deconvolution. In particular, due to spatial irregularity,assessment of asymptotic minimax global convergence rates is a much harder task than thederivation of asymptotic minimax local convergence rates studied recently in theliterature. Furthermore, the resulting estimators exhibit very different behavior andasymptotic minimax global convergence rates in comparison with the solution of spatiallyhomogeneous ill-posed problems. For example, unlike in the deconvolution problem, theasymptotic minimax global convergence rates are greatly influenced not only by the extentof data loss but also by the degree of spatial homogeneity of f.Specifically, even if 1/g is non-integrable, one can recoverf as well as in the case of an equispaced design (in terms ofasymptotic minimax global convergence rates) when it is homogeneous enough since theestimator is “borrowing strength” in the areas where f is adequatelysampled.

We consider stationary processes with long memory which are non-Gaussian and representedas Hermite polynomials of a Gaussian process. We focus on the corresponding waveletcoefficients and study the asymptotic behavior of the sum of their squares since this sumis often used for estimating the long–memory parameter. We show that the limit is notGaussian but can be expressed using the non-Gaussian Rosenblatt process defined as aWiener–Itô integral of order 2. This happens even if the original process is definedthrough a Hermite polynomial of order higher than 2.

Let Y ∈ ℝn be a random vector with means and covariance matrix σ2PntPn wherePn is some knownn × n-matrix. We construct a statistical procedure toestimate s as well as under moment condition on Y orGaussian hypothesis. Both cases are developed for known or unknownσ2. Our approach is free from any prior assumption ons and is based on non-asymptotic model selection methods. Given somelinear spaces collection {Sm, m ∈ ℳ}, we consider, for any m ∈ ℳ, the least-squaresestimator ŝm of s inSm. Considering a penalty function that isnot linear in the dimensions of the Sm’s, weselect some m̂ ∈ ℳ in order to get an estimatorŝm̂ with a quadratic risk as close aspossible to the minimal one among the risks of theŝm’s. Non-asymptotic oracle-typeinequalities and minimax convergence rates are proved forŝm̂. A special attention is given to theestimation of a non-parametric component in additive models. Finally, we carry out asimulation study in order to illustrate the performances of our estimators inpractice.

We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with nd−1, or with nd−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φn decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.

In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (K. Dziedziul, M. Kucharska, B. Wolnik, Estimation of the smoothness parameter). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes possible for some classes of the density functions.

Given an autoregressive process X of order p(i.e.Xn = a1Xn−1 + ··· + apXn−p + Ynwhere the random variables Y1, Y2,... are i.i.d.), we study the asymptotic behaviour of the probability that the processdoes not exceed a constant barrier up to time N (survival or persistenceprobability). Depending on the coefficients a1,...,ap and the distribution ofY1, we state conditions under which the survival probabilitydecays polynomially, faster than polynomially or converges to a positive constant. Specialemphasis is put on AR(2) processes.

We pursue the study of a random coloring first passage percolation model introduced byFontes and Newman. We prove that the asymptotic shape of this first passage percolationmodel continuously depends on the law of the coloring. The proof uses several couplings,particularly with greedy lattice animals.

Let M be acomplete Riemannian manifold, M ∈ ℕ andp ≥ 1. Weprove that almost everywhere on x = (x1,...,xN) ∈ MNfor Lebesgue measure in MN, the measure \hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$}$\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\mathit{\delta }}_{{\mathit{x}}_{\mathit{k}}}$ has a unique p–mean ep(x).As a consequence, if X = (X1,...,XN)is a MN-valued randomvariable with absolutely continuous law, then almost surely μ(X(ω)) has aunique p–mean. In particular if (Xn)n ≥ 1is an independent sample of an absolutely continuous law in M, then the processep,n(ω) = ep(X1(ω),...,Xn(ω))is well-defined. Assume M is compact and consider a probability measureν inM. Usingpartial simulated annealing, we define a continuous semimartingale which converges inprobability to the set of minimizers of the integral of distance at power p with respect toν. When theset is a singleton, it converges to the p–mean.

We consider representations of a joint distribution law of a family of categorical randomvariables (i.e., a multivariate categorical variable) as a mixture ofindependent distribution laws (i.e. distribution laws according to whichrandom variables are mutually independent). For infinite families of random variables, wedescribe a class of mixtures with identifiable mixing measure. This class is interestingfrom a practical point of view as well, as its structure clarifies principles of selectinga “good” finite family of random variables to be used in applied research. For finitefamilies of random variables, the mixing measure is never identifiable; however, it alwayspossesses a number of identifiable invariants, which provide substantial informationregarding the distribution under consideration.

This paper is dedicated to the analysis of backward stochastic differential equations(BSDEs) with jumps, subject to an additional global constraint involving all thecomponents of the solution. We study the existence and uniqueness of a minimal solutionfor these so-called constrained BSDEs with jumps via a penalizationprocedure. This new type of BSDE offers a nice and practical unifying framework to thenotions of constrained BSDEs presented in [S. Peng and M. Xu, Preprint.(2007)] and BSDEs with constrained jumps introduced in [I. Kharroubi, J. Ma, H.Pham and J. Zhang, Ann. Probab. 38 (2008) 794–840]. Moreremarkably, the solution of a multidimensional Brownian reflected BSDE studied in [Y. Huand S. Tang, Probab. Theory Relat. Fields 147 (2010) 89–121]and [S. Hamadène and J. Zhang, Stoch. Proc. Appl. 120 (2010)403–426] can also be represented via a well chosen one-dimensionalconstrained BSDE with jumps. This last result is very promising from a numerical point ofview for the resolution of high dimensional optimal switching problems and more generallyfor systems of coupled variational inequalities.

We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (U.R.E.) method, we build an estimator of the risk which allows to select an estimator in a collection of models. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology.

This paper deals with uniform consistency and uniform confidence bands for the quantile function and its derivatives. We describe a kernel local polynomial estimator of quantile function and give uniform consistency. Furthermore, we derive its maximal deviation limit distribution using an approximation in the spirit of Bickel and Rosenblatt [P.J. Bickel and M. Rosenblatt, Ann. Statist. 1 (1973) 1071–1095].

Consider an autoregressive model with measurement error: we observe Zi =Xi +εi, where theunobserved Xi is a stationarysolution of the autoregressive equation Xi =gθ0(Xi− 1) + ξi. Theregression function gθ0 isknown up to a finite dimensional parameter θ0 to be estimated. The distributions ofξ1 and X0 are unknownand gθ belongs to a largeclass of parametric regression functions. The distribution of ε0 is completelyknown. We propose an estimation procedure with a new criterion computed as the Fouriertransform of a weighted least square contrast. This procedure provides an asymptoticallynormal estimator \hbox{$\hat \theta$}θ̂ of θ0, for a large class of regressionfunctions and various noise distributions.

The purpose of this paper is to investigate moderate deviations for the Durbin–Watsonstatistic associated with the stable first-order autoregressive process where the drivennoise is also given by a first-order autoregressive process. We first establish a moderatedeviation principle for both the least squares estimator of the unknown parameter of theautoregressive process as well as for the serial correlation estimator associated with thedriven noise. It enables us to provide a moderate deviation principle for theDurbin–Watson statistic in the case where the driven noise is normally distributed and inthe more general case where the driven noise satisfies a less restrictive Chen–Ledoux typecondition.

We establish posterior consistency for non-parametric Bayesian estimation of thedispersion coefficient of a time-inhomogeneous Brownian motion.

Many mathematical models involve input parameters, which are not precisely known. Globalsensitivity analysis aims to identify the parameters whose uncertainty has the largestimpact on the variability of a quantity of interest (output of the model). One of thestatistical tools used to quantify the influence of each input variable on the output isthe Sobol sensitivity index. We consider the statistical estimation of this index from afinite sample of model outputs: we present two estimators and state a central limittheorem for each. We show that one of these estimators has an optimal asymptotic variance.We also generalize our results to the case where the true output is not observable, and isreplaced by a noisy version.

This paper presents a new model of asymmetric bifurcating autoregressive process withrandom coefficients. We couple this model with a Galton−Watson tree to take into account possiblymissing observations. We propose least-squares estimators for the various parameters ofthe model and prove their consistency, with a convergence rate, and asymptotic normality.We use both the bifurcating Markov chain and martingale approaches and derive new resultsin both these frameworks.

In this paper we are interested in the estimation of a density − defined on a compact interval ofℝ− from n independent andidentically distributed observations. In order to avoid boundary effect, beta kernelestimators are used and we propose a procedure (inspired by Lepski’s method) in order toselect the bandwidth. Our procedure is proved to be adaptive in an asymptotically minimaxframework. Our estimator is compared with both the cross-validation algorithm and theoracle estimator using simulated data.

Fine regularity of stochastic processes is usually measured in a local way by localHölder exponents and in a global way by fractal dimensions. In the case of multiparameterGaussian random fields, Adler proved that these two concepts are connected under theassumption of increment stationarity property. The aim of this paper is to consider thecase of Gaussian fields without any stationarity condition. More precisely, we prove thatalmost surely the Hausdorff dimensions of the range and the graph in any ballB(t0,ρ)are bounded from above using the local Hölder exponent at t0. We definethe deterministic local sub-exponent of Gaussian processes, which allows to obtain analmost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of thesample path on an open interval are controlled almost surely by the minimum of the localexponents. Then, we apply these generic results to the cases of the set-indexed fractionalBrownian motion on RN, the multifractionalBrownian motion whose regularity function H is irregular and the generalized Weierstrassfunction, whose Hausdorff dimensions were unknown so far.

We consider a strong Markov process with killing and prove an approximation method forthe distribution of the process conditioned not to be killed when it is observed. Themethod is based on a Fleming−Viot type particle system with rebirths, whose particles evolve asindependent copies of the original strong Markov process and jump onto each others insteadof being killed. Our only assumption is that the number of rebirths of theFleming−Viot type systemdoesn’t explode in finite time almost surely and that the survival probability of theoriginal process remains positive in finite time. The approximation method generalizesprevious results and comes with a speed of convergence. A criterion for the non-explosionof the number of rebirths is also provided for general systems of time and environmentdependent diffusion particles. This includes, but is not limited to, the case of theFleming−Viot type system ofthe approximation method. The proof of the non-explosion criterion uses an originalnon-attainability of (0,0)result for pair of non-negative semi-martingales with positive jumps.