Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.

In this paper, under the linear regression model with heteroscedastic and/or correlated errors when the stochastic linear restrictions on the parameter vector are assumed to be held, a generalization of the ordinary mixed estimator (GOME), ordinary ridge regression estimator (GORR) and Generalized least squares estimator (GLSE) is proposed. The performance of this new estimator against GOME, GORR, GLS and the stochastic restricted Liu estimator (SRLE) [Yang and Xu, Statist. Papers 50 (2007) 639–647] are examined in terms of matrix mean square error criterion. A numerical example is considered to illustrate the theoretical results.

Specific Gaussian mixtures are considered to solve simultaneouslyvariable selection and clustering problems. A non asymptoticpenalized criterion is proposed to choose the number of mixturecomponents and the relevant variable subset. Because of the nonlinearity of the associated Kullback-Leibler contrast on Gaussianmixtures, a general model selection theorem for maximum likelihoodestimation proposed by [Massart Concentration inequalities and model selection Springer, Berlin (2007). Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23 (2003)] is used to obtainthe penalty function form. This theorem requires to control thebracketing entropy of Gaussian mixture families. The ordered andnon-ordered variable selection cases are both addressed in thispaper.

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by twoparameters. The first parameter governs the lacunarity of the waveletcoefficients while the second one governs its intensity. In this paper,we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal (in the Le Cam sense) tests on the intensity parameter.

We prove that the large deviation principle holds for a class ofprocesses inspired by semi-Markov additive processes. For theprocesses we consider, the sojourn times in the phase process neednot be independent and identically distributed. Moreover thestate selection process need not be independent of the sojourntimes. We assume that the phase process takes values in a finite set andthat the order in which elements in the set, called states, arevisited is selected stochastically. The sojourn times determinehow long the phase process spends in a state once it has beenselected. The main tool is a representation formula for the samplepaths of the empirical laws of the phase process. Then, based onassumed joint large deviation behavior of the state selection andsojourn processes, we prove that the empirical laws of the phaseprocess satisfy a sample path large deviation principle. From thislarge deviation principle, the large deviations behavior of aclass of modulated additive processes is deduced. As an illustration of the utility of the general results, we providean alternate proof of results for modulated Lévy processes. As apractical application of the results, we calculate the large deviationrate function for a processes that arises as the InternationalTelecommunications Union's standardized stochastic model of two-wayconversational speech.

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x)){\rm d}t+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients f and gk,driven by a sequence (βk)k of i.i.d. fractional Brownianmotions of index H>1/2. Using the Malliavin calculus techniquesand a p-th moment maximal inequality for the infinite sum ofSkorohod integrals with respect to (βk)k, we prove that theequation has a unique solution (in a Banach space of summabilityexponent p ≥ 2), and this solution is Hölder continuous inboth time and space.

We give a stochastic expansion for estimates $\widehat{\theta}$that minimise the arithmetic mean of (typically independent) random functions of a known parameter θ.Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates.This is used to obtain leading cumulant coefficients of $\widehat{\theta}$needed for the Edgeworth expansions for the distribution and density n1/2 (of \widehat{\theta}$ − θ0) to magnitude n−3/2 (or to n−2 for the symmetric case),where θ0 is the true parameter value and n is typically the sample size.Applications are given to least squares estimates for both real and complex models.An alternative approach is given when the linear parameters of the model are nuisance parameters.The methods are illustrated with the problem of estimating the frequencieswhen the signal consists of the sum of sinusoids of unknown amplitudes.

Let Tn be a Studentized U-statistic. It is proved that a Cramér typemoderate deviation P(Tn ≥ x)/(1 − Φ(x)) → 1 holds uniformly in x ∈ [0, o(n1/6))when the kernel satisfies some regular conditions.

Using integration by parts on Gaussian spacewe construct a Stein Unbiased Risk Estimator (SURE)for the drift of Gaussian processes, based on theirlocal and occupation times.By almost-sure minimization of the SURE risk ofshrinkage estimators we derive an estimation and de-noisingprocedure for an input signal perturbed by acontinuous-time Gaussian noise.

Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β-mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [E. Löcherbach, D. Loukianova and O. Loukianov, Ann. Inst. H. Poincaré Probab. Statist. 47 (2011) 425–449].

Following the recent investigations of Baik and Suidan in [Int. Math. Res. Not. (2005) 325–337] and Bodineau and Martin in [Electron. Commun. Probab. 10 (2005) 105–112 (electronic)], we prove large deviation properties for a last-passage percolation model in ℤ+2 whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in [J. Baik and T.M. Suidan, Int. Math. Res. Not. (2005) 325–337] and [T. Bodineau and J. Martin, Electron. Commun. Probab. 10 (2005) 105–112 (electronic)], on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.

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, we extend the results of Orey and Taylor [S. Orey and S.J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28 (1974) 174–192]relative to random fractals generated by oscillations of Wiener processes to a multivariate framework. We consider a setup where Gaussian processes are indexed by classes of functions.

The joint estimation of both drift and diffusion coefficient parameters is treatedunder the situation where the data are discretely observed from an ergodic diffusion processand where the statistical model may or may not include the true diffusion process.We consider the minimum contrast estimator,which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density.The asymptotic normality of the minimum contrast estimator is proved.In particular, the rate of convergence for the minimum contrast estimator of diffusion coefficient parameter in a misspecified modelis different from the one in the correctly specified parametric model.

Let, for each t ∈ T, ψ(t, ۔) be a random measure on theBorel σ-algebra in ℝd such that Eψ(t, ℝd)k < ∞ for all k and let $\widehat{\psi}$(t, ۔) beits characteristic function. We call the function$\widehat{\psi}$(t1,…, tl ; z1,…, zl) = ${\sf E}\prod^l_{j=1}\widehat{\psi}(t_j, z_j)$ of arguments l ∈ ℕ, t1, t2… ∈ T, z1, z2 ∈ ℝd the covaristic of the measure-valued random function (MVRF)ψ(۔, ۔). A general limit theorem for MVRF's interms of covaristics is proved and applied to functions of thekind ψn(t, B) = µ{x : ξn(t, x) ∈ B}, where μ is anonrandom finite measure and, for each n, ξn is atime-dependent random field.


In the companion paper [C. Maugis and B. Michel,A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM: P&S15 (2011) 41–68] , a penalized likelihoodcriterion is proposed to select a Gaussian mixture model among aspecific model collection. This criterion depends on unknownconstants which have to be calibrated in practical situations. A“slope heuristics” method is described and experimented to dealwith this practical problem. In a model-based clustering context,the specific form of the considered Gaussian mixtures allows us todetect the noisy variables in order to improve the data clusteringand its interpretation. The behavior of our data-driven criterionis highlighted on simulated datasets, a curve clustering exampleand a genomics application.

We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product.As an application we obtain expansions for P(a < Y < b) for Y ~ Np(0,V) andfor repeated integrals of the density of Y.When V−1y > 0 in R3 the expansion for P(Y < y) reduces toone given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of Np(0,V−1).This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials.These are given explicitly.

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can be extended to all dimensions: inspired by quantum gravity in dimension 2, we focus on the log normal case in dimension 2.

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].

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.