We study the estimation of the mean function of a continuous-time stochastic process andits derivatives. The covariance function of the process is assumed to be nonparametric andto satisfy mild smoothness conditions. Assuming that n independent realizationsof the process are observed at a sampling design of size N generated by a positivedensity, we derive the asymptotic bias and variance of the local polynomial estimator asn,Nincrease to infinity. We deduce optimal sampling densities, optimal bandwidths, andpropose a new plug-in bandwidth selection method. We establish the asymptotic performanceof the plug-in bandwidth estimator and we compare, in a simulation study, its performancefor finite sizes n,N to the cross-validation and the optimalbandwidths. A software implementation of the plug-in method is available in the Renvironment.

This paper gives an approximation of the solution of the Boltzmann equation by stochastic interacting particle systems in a case of cut-off collision operator and small initial data. In this case,following the ideas of Mischler and Perthame, we prove the existence and uniqueness of the solution of this equation and also the existence and uniqueness of the solution of the associated nonlinear martingale problem. 
Then, we first delocalize the interaction by considering a mollified Boltzmann equation in which the interaction is averaged on cells of fixed size which cover the space. In this situation, Grahamand Méléard have obtained an approximation of the mollified solution by some stochastic interacting particle systems. Then we consider systems in which the size of the cells depends on the size of the system. We show that the associated empirical measures converge in law to a deterministic probability measure whose density flow is the solution of the full Boltzmann equation. That suggests an algorithm based on the Poisson interpretation of the integral term for the simulation of this solution.

We give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain numberof charged particles moves on the segment [1,L] according to a Markovian law. One unitary charge, positive or negative, jumps from a site k to another site k'=k+1 or k'=k-1 at a rate which depends on the charge at site k and at site k'. The total charge of the system is preserved by the dynamics, in this sense our dynamics is similar to the Kawasaki dynamics, but in our case there is no restriction on the maximum charge allowed per site. The model is equivalent to an interface dynamics connected with the stochastic Ising model at very low temperature: the “unrestrictedsolid on solid model”. Thus the results we obtain may be read as results for this model. We give necessary and sufficient conditions to ensure that the spectral gap tends towards zero as theinverse of the square of L, independently of the total charge. We follow the method outlined in some papers by Yau (Lu, Yau (1993), Yau (1994)) where a similar spectral gap is proved for theoriginal Kawasaki dynamics.

In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surelyfinite random terminal time. More precisely, we are going to show that if (Wn ) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if $(\tau^n)$ is a sequence of stopping times that converges to a stopping time τ, then the solution of the BSDE driven by Wn with random terminal time $\tau^n$ converges to the solution of the BSDE driven by W with random terminal time τ.

In this paper, we extend a result of Campanino and Pétritis [Markov Process. Relat. Fields 9 (2003) 391–412]. We study a random walk in ${\mathbb Z}^2$ with random orientations.We suppose that the orientation of the kth flooris given by $\xi_k$ , where $(\xi_k)_{k\in\mathbb Z}$ isa stationary sequence of random variables.Once the environment fixed, the random walk can goeither up or down or can stay in the present floor (but moving with respect to its orientation).This model was introduced by Campanino and Pétritisin [Markov Process. Relat. Fields 9 (2003) 391–412] whenthe $(\xi_k)_{k\in\mathbb Z}$ is a sequence ofindependent identically distributed random variables. In [Theory Probab. Appl. 52 (2007) 815–826], Guillotin-Plantard and Le Ny extend thisresult to a situation where the orientations of the floors are independentbut chosen with stationary probabilities (not equal to 0and to 1). In the present paper, we generalize the result of [Markov Process. Relat. Fields 9 (2003) 391–412]to some cases when $(\xi_k)_k$ is stationary. Moreover we extend slightlya result of [Theory Probab. Appl.52 (2007) 815–826].

Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and Meziou (2006) on the max-plus decomposition of supermartingales, and they provide a singular analogue to the non-linear Riesz representation in El Karoui and Föllmer (2005).

In this paper, we investigate a nonparametric approach to provide a recursive estimatorof the transition density of a piecewise-deterministic Markov process, from only oneobservation of the path within a long time. In this framework, we do not observe a Markovchain with transition kernel of interest. Fortunately, one may write the transitiondensity of interest as the ratio of the invariant distributions of two embedded chains ofthe process. Our method consists in estimating these invariant measures. We state a resultof consistency and a central limit theorem under some general assumptions about the mainfeatures of the process. A simulation study illustrates the well asymptotic behavior ofour estimator.

In this paper a multi-scaled diffusion-approximation theorem is proved so as to unify various applications in wave propagation in random media: transmission of optical modes through randomplanar waveguides; time delay in scattering for the linear wave equation; decay of the transmission coefficient for large lengths with fixed output and phase difference in weakly nonlinear randommedia.

Spider walks are systems of interacting particles. The particles move independently as long as their movements do not violate some given rules describing the relative position ofthe particles; moves that violate the rules are not realized. The goal of this paper is to study qualitative properties, as recurrence, transience, ergodicity, and positive rate of escape of these Markov processes.

In a multiple testing context, we consider a semiparametric mixture model with two components where one component is known and corresponds to the distribution of p-values under the null hypothesis and the other component f is nonparametric and stands for the distribution under the alternative hypothesis. Motivated by the issue of local false discovery rate estimation, we focus here on the estimation of the nonparametric unknown component f in the mixture, relying on a preliminary estimator of the unknown proportion θ of true null hypotheses. We propose and study the asymptotic properties of two different estimators for this unknown component. The first estimator is a randomly weighted kernel estimator. We establish an upper bound for its pointwise quadratic risk, exhibiting the classical nonparametric rate of convergence over a class of Hölder densities. To our knowledge, this is the first result establishing convergence as well as corresponding rate for the estimation of the unknown component in this nonparametric mixture. The second estimator is a maximum smoothed likelihood estimator. It is computed through an iterative algorithm, for which we establish a descent property. In addition, these estimators are used in a multiple testing procedure in order to estimate the local false discovery rate. Their respective performances are then compared on synthetic 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.

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.

The P.O.T. (Peaks-Over-Threshold) approachconsists of using the Generalized ParetoDistribution (GPD)to approximate the distribution of excesses over a threshold.We use the probability-weighted momentsto estimate the parameters of the approximating distribution.We study the asymptotic behaviour ofthese estimators (in particular their asymptotic bias) and also thefunctional bias of the GPD as an estimate of thedistribution function of the excesses. We adapt penultimateapproximation results to the case where parameters are estimated.

This paper presents a practical and simple fully nonparametric multivariate smoothingprocedure that adapts to the underlying smoothness of the true regression function. Ourestimator is easily computed by successive application of existing base smoothers (withoutthe need of selecting an optimal smoothing parameter), such as thin-plate spline or kernelsmoothers. The resulting smoother has better out of sample predictive capabilities thanthe underlying base smoother, or competing structurally constrained models (MARS, GAM) forsmall dimension (3 ≤ d ≤7) and moderate sample size n ≤ 1000. Moreover our estimator is still usefulwhen d > 10and to our knowledge, no other adaptive fully nonparametric regression estimator isavailable without constrained assumption such as additivity for example. On a realexample, the Boston Housing Data, our method reduces the out of sample prediction error by20%. An R package ibr, available at CRAN, implements the proposedmultivariate nonparametric method in R.

We prove a central limit theorem for linear triangulararrays under weak dependence conditions. Our result is then appliedto dependent random variables sampled by a ${\mathbb Z}$ -valued transient random walk. This extends the resultsobtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics3 (2003) 477–497]. An applicationto parametric estimation by random sampling is also provided.

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a newfinite difference operator ∇n, defined as a linear combination ofleft and right finite differences. We show that this operator ∇n plays a keyrole in a new Poincaré (spectral gap) inequality with respect to binomial weights, withthe orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator.We briefly discuss the relationship of this operator to the problem of optimal transportof probability measures.

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.

The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible.

We prove a strong law of large numbers for moving averages of independent, identically distributed random variables with certain subexponential distributions. These random variables show a behavior that can be considered intermediate between the classical strong law and the Erdös-Rényi law. We further show that the difference from the classical behavior is due to the influence of extreme terms.

As in preceding papers inwhich we studied the limits of penalized 1-dimensional Wienermeasures with certain functionals Γt, we obtain here theexistence of the limit, as t → ∞, of d-dimensional Wienermeasures penalized by a function of the maximum up to time t ofthe Brownian winding process (for d = 2), or in {d}≥ 2dimensions for Brownian motionprevented to exit a cone before time t. Various extensions of these multidimensional penalisations arestudied, and the limit laws are described. Throughout this paper, the skew-product decomposition ofd-dimensional Brownian motion plays an important role.