Identifying words with unexpected frequencies is an important problem in the analysis of long DNA sequences. To solve it, we need an approximation of the distribution of the number ofoccurrences N(W) of a word W. Modeling DNA sequences with m-order Markov chains, we use the Chen-Stein method to obtain Poisson approximations for two different counts. We approximate the “declumped” count of W by a Poisson variable and the number of occurrences N(W) by a compound Poisson variable. Combinatorial results are used to solve the general case of overlapping words and to calculate the parameters of these distributions.


We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound is established via exponential tightness, the difficulty being that the marginals of our measure-valued processes are not of bounded masses. The local minoration is proved by linking the trajectories of the action functional's domain to the solutions of perturbations of the PDE obtained in the large population limit. The use of Girsanov theorem then leads us to regularize these perturbations. As an application, we study the logistic age-structured population. In the super-critical case, the deterministic approximation admits a non trivial stationary stable solution, whereas the stochastic microscopic process gets extinct almost surely. We establish estimates of the time during which the microscopic process stays in the neighborhood of the large population equilibrium by generalizing the works of Freidlin and Ventzell to our measure-valued setting.


We establish new exponential inequalities for partial sums of random fields. Next, using classicalchaining arguments, we give sufficient conditions for partial sum processes indexed by large classes ofsets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, thecondition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixingrandom fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.

We extend the Lindeberg method for the central limit theorem to strongly mixing sequences. Here we obtain a generalization of the central limit theorem of Doukhan, Massart and Rio to nonstationary strongly mixing triangular arrays. The method also provides estimatesof the Lévy distance between the distribution of the normalized sum and the standard normal.

In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.

We study the exit path from a general domain after the last visit to a set of a Markov chain with rare transitions. We prove several large deviation principles for the law of the succession of thecycles visited by the process (the cycle path), the succession of the saddle points gone through to jump from cycle to cycle on the cycle path (the saddle path) and the succession of all the pointsgone through (the exit path). We estimate the time the process spends in each cycle of the cycle path and how it decomposes into the time spent in each point of the exit path. We describe asystematic method to find the most likely saddle paths. We apply these results to the reversible case of the Metropolis dynamics. We give in appendix the corresponding large deviation estimatesin the non homogeneous case, which are corollaries of already published works by Catoni (1992) and Trouvé (1992, 1996a).

In this paper we focus on the problem of estimating a boundeddensity using a finite combination of densities from a givenclass. We consider the Maximum Likelihood Estimator (MLE) and thegreedy procedure described by Li and Barron (1999)under the additional assumption of boundedness of densities. Weprove an $O(\frac{1}{\sqrt{n}})$ bound on the estimation errorwhich does not depend on the number of densities in the estimatedcombination. Under the boundedness assumption,this improves the bound of Li and Barron by removing the $\log n$ factor and also generalizes it to the base classes with convergingDudley integral.

In this paper, we develop bounds on the distribution function of the empiricalmean for general ergodic Markov processes having a spectral gap. Our approach isbased on the perturbation theory for linear operators, following the techniqueintroduced by Gillman.

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue ofSchrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine a class of modelsextending the hard obstacle modelof K. Burdzy, R. Holyst and P. March and including the Moran type scheme presented by the authors in a previous work. We provide precise uniform estimates with respect to the time parameter and we analyze the fluctuations of continuous time particle models.

A general theory is developed for the projection of martingale related processes onto smaller filtrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain their nature, but the case of local martingales is more delicate, as illustrated by an explicit case study for the inverse Bessel process. This has implications for the concept of No Free Lunch with Vanishing Risk, in Finance.

In this paper we derive non asymptotic deviation bounds for $${\mathbb P}_\nu (|\frac 1t\int_0^t V(X_s) {\rm d}s - \int V {\rm d} \mu | \geq R)$$ where X is a μ stationary and ergodic Markov process and V is some μ integrable function. These bounds are obtained under various moments assumptions for V, and various regularity assumptions for μ. Regularity means here that μ may satisfy various functional inequalities (F-Sobolev,generalized Poincaré etc.).

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.

Let (Xt ) be a diffusion on the interval (l,r) and Δn a sequence of positive numbers tending to zero. We define J i as the integral between iΔn and (i + 1)Δn of X s .We give an approximation of the law of (J0,...,Jn-1)by means of a Euler scheme expansion for the process (Ji ). In some special cases, an approximation by anexplicit Gaussian ARMA(1,1) process is obtained.When Δn = n-1 we deduce from this expansion estimatorsof the diffusion coefficient of X based on (Ji ). These estimatorsare shown to be asymptotically mixed normal as n tends to infinity.

In this paper we define and study self-similar rankedfragmentations. We first show that any ranked fragmentation is theimage of some partition-valued fragmentation, and that there is infact a one-to-one correspondence between the laws of these twotypes of fragmentations. We then give an explicit construction ofhomogeneous ranked fragmentations in terms of Poisson pointprocesses. Finally we use this construction and classical resultson records of Poisson point processes to study the small-timebehavior of a ranked fragmentation.

Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants.A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.

This work is concerned with the theory of initial and progressive enlargements of areference filtration \hbox{$\mathbb{F}$}$F$ with a random time τ. We provide, under anequivalence assumption, slightly stronger than the absolute continuity assumption ofJacod, alternative proofs to results concerning canonical decomposition of an \hbox{$\mathbb{F}$}$F$-martingalein the enlarged filtrations. Also, we address martingales’ characterization in theenlarged filtrations in terms of martingales in the reference filtration, as well aspredictable representation theorems in the enlarged filtrations.

Under regularity assumptions, we establish a sharp largedeviation principle for Hermitian quadratic forms ofstationary Gaussian processes. Our result is similar tothe well-known Bahadur-Rao theorem [2] on the samplemean. We also provide several examples of applicationsuch as the sharp large deviation properties ofthe Neyman-Pearson likelihood ratio test, of the sum of squares,of the Yule-Walkerestimator of the parameter of a stable autoregressive Gaussian process,and finally of the empirical spectral repartition function.

The EGARCH model of Nelson [29] is one of the most successfulARCH models which may exhibit characteristic asymmetries offinancial time series, as well as long memory. The paper studiesthe covariance structure and dependence properties of the EGARCHand some related stochastic volatility models. We show that thelarge time behavior of the covariance of powers of the (observed)ARCH process is determined by the behavior of the covariance ofthe (linear) log-volatility process; in particular, a hyperbolicdecay of the later covariance implies a similar hyperbolic decayof the former covariances. We show, in this case, that normalizedpartial sums of powers of the observed process tend to fractionalBrownian motion. The paper also obtains a (functional) CLT for thecorresponding partial sums' processes of the EGARCH model withshort and moderate memory. These results are applied to studyasymptotic behavior of tests for long memory using the R/Sstatistic.

The density of real-valued Lévy processes is studied in small time under the assumption that the process has many small jumps. We prove that the real line can be divided into three subsets on which the density is smaller and smaller: the set of points that the process can reach with a finite number of jumps (Δ-accessible points); the set of points that the process can reach withan infinite number of jumps (asymptotically Δ-accessible points); and the set of points that the process cannot reach by jumping (Δ-inaccessible points).

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.