We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate f n drawn from a sample X1,...,Xn . The estimate θn is defined as any x in {X1,...,Xn } such that $f_n(x)=\max_{i=1, \hdots,n} f_n(X_i)$ . It is shown that θn behaves asymptotically as any maximizer ${\hat{\theta}}_n$ of f n . More precisely, we prove that for any sequence $(r_n)_{n\geq 1}$ of positive real numbers such that $r_n\to\infty$ and $r_n^d\log n/n\to 0$ , one has $r_n\,\|\theta_n-{\hat{\theta}}_n\| \to 0$ in probability. The asymptotic normality of θn follows without further work.

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.

Let Y be a Ornstein–Uhlenbeck diffusion governed by astationary and ergodic process X : dYt = a(Xt)Yt dt + σ(Xt)dWt,Y0 = y0 . We establish that under the condition α = Eµ(a(X0)) < 0 with μ the stationary distribution of the regime process X, the diffusion Y is ergodic. We also consider conditions for the existence of moments for theinvariant law of Y when X is a Markov jump process having a finite number of states.Using results on random difference equationson one hand and the fact that conditionally toX, Y is Gaussian on the other hand, we give such a condition for the existence of the moment of order s ≥ 0. Actually we recover in this case a result that Basak et al. [J. Math. Anal. Appl. 202 (1996) 604–622] have established using the theory of stochastic control of linear systems.


Using probabilistic tools, this work states a pointwise convergence offunction solutions of the 2-dimensional Boltzmann equation to the functionsolution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results ofFournier (2000) on the Malliavin calculus for the Boltzmannequation. Moreover, using the particle system introduced by Guérin andMéléard (2003), some simulations of the solution of the Landau equation will be given. This result isoriginal and has not been obtained for the moment by analytical methods.

We consider the continuous time, one-dimensional random walk in random environmentin Sinai's regime. We show that the probability for theparticle to be, at time t and in a typical environment,at a distance larger than ta (0<a<1)from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.

A sample of i.i.d. continuous time Markov chains beingdefined, the sum over each component of a real function of thestate is considered. For this functional, a central limit theorem for the first hitting time of a prescribed level is proved. The result extends the classical central limit theorem for order statistics. Various reliability models are presented as examples of applications.

We provide an extension of topological methods applied to acertain class of Non Feller Models which we call Quasi-Feller. We give conditions to ensure the existence of a stationarydistribution. Finally, we strengthen the conditions to obtain apositive Harris recurrence, which in turn implies the existenceof a strong law of large numbers.

We present several functional inequalitiesfor finite difference gradients, such asa Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities,associated deviation estimates,and an exponential integrability property.In the particular case of the geometric distribution on ${\mathbb{N}}$ we use an integration by parts formula to computethe optimal isoperimetric and Poincaré constants,and to obtain an improvement of ourgeneral logarithmic Sobolev inequality.By a limiting procedure we recover the correspondinginequalities for the exponential distribution.These results have applications to interacting spin systems undera geometric reference measure.

The LISDLG process denoted by J(t) is defined in Iglói and Terdik [ESAIM: PS7 (2003) 23–86] by afunctional limit theorem as the limit of ISDLG processes. This paper gives amore general limit representation of J(t). It is shown that process J(t)has its own renormalization group and that J(t) can be represented as thelimit process of the renormalization operator flow applied to the elements ofsome set of stochastic processes. The latter set consists of IGSDLG processeswhich are generalizations of the ISDLG process.

The stochastic approximation version of EM (SAEM) proposed by Delyon et al. (1999) isa powerful alternative to EM when the E-step is intractable. Convergence ofSAEM toward a maximum of the observed likelihood is established whenthe unobserved data are simulated at each iteration under the conditionaldistribution. We show that this very restrictive assumption can be weakened. Indeed, the results of Benveniste et al. for stochastic approximationwith Markovian perturbations are used to establish the convergenceof SAEM when it is coupled with a Markov chain Monte-Carloprocedure. This result is very useful for many practical applications. Applications to the convolution model and the change-points model are presented to illustrate the proposed method.

We consider a diffusion process X t smoothed with (small)sampling parameter ε. As in Berzin, León and Ortega(2001), we consider a kernel estimate $\widehat{\alpha}_{\varepsilon}$ with window h(ε) of afunction α of its variance. In order to exhibit globaltests of hypothesis, we derive here central limit theorems forthe Lp deviations such as \[ \frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p\right).\]

We present a method for estimating the edge of a two-dimensionalbounded set, given a finite random set of points drawn from the interior.The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditionsfor various kinds of convergence and asymptotic normality.We propose a method of reducing the negative bias and edge effects,illustrated by some simulations.

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $\mathbb{Z}^d$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster.As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation.We also prove a flat edge result in the case of dimension 2. Various examples are also given.

We study the large deviation principle for stochastic processes of the form $\{\sum_{k=1}^{\infty}x_{k}(t)\xi_{k}:t\in T\}$ , where $\{\xi_{k}\}_{k=1}^{\infty}$ is a sequence of i.i.d.r.v.'s with mean zero and $x_{k}(t)\in \mathbb{R}$ . We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.