We consider the approximate Euler scheme for Lévy-drivenstochastic differential equations.We study the rate of convergence in law of the paths.We show that when approximating the small jumps by Gaussianvariables, the convergence is much faster than when simplyneglecting them.For example, when the Lévy measure of the driving processbehaves like |z|−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the small jumps, see[S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of ordernα/(2−α), which is huge when α is close to 2. In the same spirit, we study the problem of the approximation of a Lévy-driven S.D.E.by a Brownian S.D.E. when the Lévy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlós-Major-Tsunády [J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independentrvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].

In this paper, a very useful lemma (in two versions) is proved: itsimplifies notably the essential step to establish a Lindebergcentral limit theorem for dependent processes. Then, applying thislemma to weakly dependent processes introduced in Doukhan andLouhichi (1999), a new central limit theorem is obtained forsample mean or kernel density estimator. Moreover, by using thesubsampling, extensions under weaker assumptions of these centrallimit theorems are provided. All the usual causal or non causaltime series: Gaussian, associated, linear, ARCH(∞),bilinear, Volterra processes, ..., enter this frame.

In this paper we solve the basic fractionalanalogue of the classical linear-quadratic Gaussianregulator problem in continuous time. For a completelyobservable controlled linear system driven by a fractionalBrownian motion, we describe explicitely the optimal controlpolicy which minimizes a quadratic performance criterion.

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 study the LRT statistic for testing a single population i.i.d. model against a mixture of two populations with Markov regime.We prove that the LRT statistic converges to infinity in probability as the number of observations tends to infinity. This is a consequence of a convergence result of the LRT statistic for a subproblem where the parameters are restricted to a subset of the whole parameter set.

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.

We derive necessary and sufficient conditions for a sum of i.i.d.random variables $\sum_{i=1}^n X_i/b_n$ –where $\frac {b_n} n \downarrow 0$ ,but $\frac {b_n} {\sqrt n} \uparrow \infty$ – to satisfy a moderate deviationsprinciple. Moreover we show that this equivalence is a typical moderatedeviations phenomenon. It is not true in a large deviations regime.

In this paper, we study the problem of non parametric estimationof the stationary marginal density f of an α or aβ-mixing process, observed either in continuous time or indiscrete time. We present an unified framework allowing to dealwith many different cases. We consider a collection of finitedimensional linear regular spaces. We estimate f using aprojection estimator built on a data driven selected linear spaceamong the collection. This data driven choice is performed via theminimization of a penalized contrast. We state non asymptotic riskbounds, regarding to the integrated quadratic risk, for ourestimators, in both cases of mixing. We show that they areadaptive in the minimax sense over a large class of Besov balls.In discrete time, we also provide a result for model selectionamong an exponentially large collection of models (non regularcase).

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.

We investigate the role of the initialization for the stability of the қ-means clusteringalgorithm. As opposed to other papers, we consider the actual қ-means algorithm (also knownas Lloyd algorithm). In particular we leverage on the property that this algorithm can getstuck in local optima of the қ-means objective function. We are interested in the actualclustering, not only in the costs of the solution. We analyze when differentinitializations lead to the same local optimum, and when they lead to different localoptima. This enables us to prove that it is reasonable to select the number of clustersbased on stability scores.

We consider a dynamical system in $\mathbb{R}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tailnature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

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.

Many statistical applications require establishingcentral limit theorems for sums/integrals $S_T(h)=\int_{t \in I_T} h (X_t) {\rm d}t$ or for quadratic forms $Q_T(h)=\int_{t,s \in I_T} \hat{b}(t-s) h (X_t, X_s) {\rm d}s {\rm d}t$ , where X t is a stationaryprocess. A particularly important case is that of Appellpolynomials h(X t ) = P m (X t ), h(X t ,X s ) = P m , n (X t ,X s ), since the “Appell expansion rank" determines typically thetype of central limit theorem satisfied by the functionalsS T (h), Q T (h). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu, Lect. Notes Statist.187 (2006) 259–286], a functionalanalysis approach to this problem proposed by [Avram and Brown, Proc. Amer.Math. Soc.107 (1989) 687–695] based on the method of cumulants and on integrabilityassumptions in the spectral domain; several applications arepresented as well.

We give limit theorems specifying weak andstrong rates of convergence associated to a quadratic extension of themartingale almost-sure central limit theorem. Some typical examples arediscussed to illustrate how to make use of them in statistic.

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/∞ queue. They describe in particular the exponential dissipation of Φ-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/∞ queues.Proofs are elementary and rely essentially on the development of a “Φ-calculus”.

We extend Hoggar's theorem that the sum of two independentdiscrete-valued log-concave random variables is itself log-concave. Weintroduce conditions under which the result still holds for dependentvariables. We argue that these conditions are natural by giving someapplications. Firstly, we use our main theorem to give simple proofsof the log-concavity of the Stirling numbers of the second kind and ofthe Eulerian numbers.Secondly, we prove results concerning the log-concavityof the sum of independent (not necessarily log-concave) randomvariables.

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chainwhich is irreducible and reversible outside the absorbing point.This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues.In the discrete time setting, if the underlying Dirichlet eigenvalues(namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point)are nonnegative, we show that T is distributed as a mixture ofsums of independent geometric laws whose parameters are successiveDirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law.This result leads to a probabilistic interpretation of the spectrum,in terms of strong random times and local equilibria through a simple intertwining relation.Next this study is extended to the continuous time framework,wheregeometric laws have to be replaced by exponential distributions having the (opposite)Dirichlet eigenvalues of the generator as parameters.Returning to the discrete time setting we consider the influence ofnegative eigenvalues which are given another probabilistic meaning.These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

We study the probability distribution of the location of a particleperforming a cyclic random motion in $\mathbb{R}^d$ . The particle can taken possible directions with different velocities and the changes ofdirection occur at random times. The speed-vectors as well as thesupport of the distribution form a polyhedron (the first one havingconstant sides and the other expanding with time t). Thedistribution of the location of the particle is made up of twocomponents: a singular component (corresponding to the beginning ofthe travel of the particle) and an absolutely continuous component.
We completely describe the singular component and exhibit anintegral representation for the absolutely continuous one. Thedistribution is obtained by using a suitable expression of thelocation of the particle as well as some probability calculustogether with some linear algebra. The particular case of theminimal cyclic motion (n=d+1) with Erlangian switching times isalso investigated and the related distribution can be expressed interms of hyper-Bessel functions with several arguments.

In this paper we consider BSDEs with Lipschitzcoefficient reflected on two discontinuous (RCLL) barriers. In thiscase, we prove first the existence and uniqueness of the solution,then we also prove the convergence of the solutions of the penalizedequations to the solution of the RBSDE. Since the method used in thecase of continuous barriers (see Cvitanic and Karatzas, Ann. Probab.24 (1996) 2024–2056 and Lepeltier and San Martín, J. Appl. Probab.41 (2004) 162–175) does notwork, we develop a new method, by considering the solutions of thepenalized equations as the solutions of special RBSDEs and usingsome results of Peng and Xu in Annales of I.H.P.41 (2005) 605–630.

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.