We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t 1 < t 2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i ≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

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].

In this paper, we are interested in the asymptotical behaviorof the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation.The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, fortwo-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered.

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.

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.

We propose a general partition-based strategy to estimate conditional density with candidate densities that are piecewise constant with respect to the covariate. Capitalizing on a general penalized maximum likelihood model selection result, we prove, on two specific examples, that the penalty of each model can be chosen roughly proportional to its dimension. We first study a classical strategy in which the densities are chosen piecewise conditional according to the variable. We then consider Gaussian mixture models with mixing proportion that vary according to the covariate but with common mixture components. This model proves to be interesting for an unsupervised segmentation application that was our original motivation for this work.

We propose a test of a qualitative hypothesis on the mean of a n-Gaussianvector. The testing procedure is available when the variance of theobservations is unknown and does not depend on any prior information onthe alternative. The properties of the test are non-asymptotic. Fortesting positivity or monotonicity, weestablish separation rates with respect to the Euclidean distance, oversubsets of $\mathbb{R}^{n}$ which are related to Hölderian balls in functionalspaces. We provide a simulation study in order to evaluate theprocedure when the purpose is to test monotonicity in a functionalregression model and to check the robustness of the procedure tonon-Gaussian errors.

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).

Given a general critical or sub-critical branching mechanism and its associated Lévy continuum random tree, we consider a pruning procedure on this tree using a Poisson snake. It defines a fragmentation process on the tree. We compute the family of dislocation measures associated with this fragmentation. This work generalizes the work made for a Brownian tree [R. Abraham and L. Serlet, Elect. J. Probab. 7 (2002) 1–15] and for a tree without Brownian part [R. Abraham and J.-F. Delmas, Probab. Th. Rel. Fiel 141 (2008) 113–154].

We observe an infinitely dimensional Gaussian random vector x = ξ + vwhereξ is a sequence of standard Gaussian variables and v ∈ l2 is anunknownmean. We consider the hypothesis testing problem H0 : v = 0versusalternatives $H_{\varepsilon,\tau}:v\in V_{\varepsilon}$ for the sets $V_{\varepsilon}=V_{\varepsilon}(\tau,\rho_{\varepsilon})\subset l_2$ .The sets Vε are l q -ellipsoidsof semi-axes ai = i-s R/ε with l p -ellipsoidof semi-axes bi = i-r pε/ε removed orsimilar Besov bodies Bq,t;s (R/ε) with Besovbodies Bp,h;r (pε/ε) removed. Here $\tau =(\kappa,R)$ or $\tau =(\kappa,h,t,R);\ \\kappa=(p,q,r,s)$ are the parameters which define thesets Vε for given radii pε → 0,0 < p,q,h,t ≤ ∞, -∞ ≤ r,s ≤ ∞, R > 0; ε → 0 is theasymptotical parameter.We study the asymptotics of minimaxsecond kind errors $\beta_{\varepsilon}(\alpha)=\beta(\alpha, V_{\varepsilon}(\tau,\rho_{\varepsilon}))$ and construct asymptotically minimax or minimax consistent families oftests $\psi_{\alpha;\varepsilon,\tau,\rho_{\varepsilon}}$ , if it is possible.We describe the partition of the set of parameters κ into regions withdifferent types of asymptotics: classical, trivial, degenerate and Gaussian(of various types).Analogous rates have been obtained in a signal detectionproblem for continuous variant of white noise model: alternativescorrespond to Besov or Sobolev balls with Besov or Sobolev balls removed.The study is based on an extension of methods of constructions ofasymptotically least favorable priors.These methods are applicable to wide class of “convex separablesymmetrical" infinite-dimensional hypothesis testingproblems in white Gaussian noise model. Under some assumptionsthese methodsare based on the reduction of hypothesis testing problemto convex extreme problem: to minimize specially defined Hilbert normover convex sets of sequences $\bar{\pi}$ of measures πi on thereal line. The study of this extreme problem allows to obtain differenttypes of Gaussian asymptotics.If necessary assumptions do not hold, then we obtain other types ofasymptotics.

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.

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.

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.

We present the rough path theory introduced by Lyons, using the swewing lemma of Feyel and de Lapradelle.

These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.

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 dependence coefficients for stationary Markov chains. We emphasize on someequivalencies for reversible Markov chains. We improve some known results and provide anecessary condition for Markov chains based on Archimedean copulas to be exponentialρ-mixing.We analyse the example of the Mardia and Frechet copula families using small sets.

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 consider the problem of hypothesis testing within a monotoneregression model. We propose a new test of the hypothesisH 0: “ƒ = ƒ0 ” against the composite alternative H a : “ƒ ≠ ƒ0 ” under the assumption that the true regression functionf is decreasing. The test statistic is based on the ${\mathbb L}_{1}$ -distance between the isotonic estimator of f and thefunction f 0, since it is known that a properly centered and normalized version of this distance is asymptotically standardnormally distributed under H 0. We study the asymptotic powerof the test under alternatives that converge to the nullhypothesis.

Let $\tau _{D}(Z) $ be the first exit time ofiterated Brownian motion from a domain $D \subset \mathbb{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau _{D}(Z)>t]$ be itsdistribution. In this paper we establish the exact asymptotics of $P_{z}[\tau _{D}(Z)>t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116(2006) 905–916], for $z\in D$ 
 $ \displaystyle \lim_{t\to\infty}t^{-1/2}\exp\left(\frac{3}{2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}\right)P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber$ 
where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}\left(\psi(z)\int_{D}\psi(y){\rm d}y\right) ^{2}$ . Here λD is thefirst eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ inD, and ψ is the eigenfunction corresponding toλD . We also study lifetime asymptotics of Brownian-time Brownian motion, $Z^{1}_{t} = z+X(|Y(t)|)$ , where X t and Y t are independentone-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.