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 of occurrences 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 the estimation of a linear integral functional of a distribution F, using i.i.d. observations which density is a mixture of a family of densities k(.,y) under F. We examine the asymptotic distribution of the estimator obtained by plugging the non parametric maximum likelihood estimator (NPMLE) of F in the functional. A problem here is that usually, the NPMLE does not dominate F.
Our main aim here is to show that this can be overcome by considering a convex combination of F and the NPMLE.

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 estimates of the Lévy distance between the distribution of the normalized sum and the standard normal.

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincaré and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced from this approach.

We present a very simple proof of the existence of the value for 'Big Match' first shown by Blackwell and Ferguson (1968).

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 the cycles 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 points gone 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 a systematic 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 estimates in the non homogeneous case, which are corollaries of already published works by Catoni (1992) and Trouvé (1992, 1996a).

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 number of 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 “unrestricted solid 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 the inverse 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 the original Kawasaki dynamics.

In this paper a multi-scaled diffusion-approximation theorem is proved so as to unify various applications in wave propagation in random media: transmission of optical modes through random planar waveguides; time delay in scattering for the linear wave equation; decay of the transmission coefficient for large lengths with fixed output and phase difference in weakly nonlinear random media.

We study the evolution of a multi-component system which is modeled by a semi-Markov process. We give formulas for the avaibility and the reliability of the system. In the r-positive case, we prove that the quasi-stationary probability on the working states is the normalised left eigenvector of some computable matrix and that the asymptotic failure rate is equal to the absolute value of the convergence parameter r.

Consider a one dimensional nonlinear reaction-diffusion equation (KPP equation) with non-homogeneous second order term, discontinuous initial condition and small parameter. For points ahead of the Freidlin-KPP front, the solution tends to 0 and we obtain sharp asymptotics (i.e. non logarithmic). Our study follows the work of Ben Arous and Rouault who solved this problem in the homogeneous case. Our proof is probabilistic, and is based on the Feynman-Kac formula and the large deviation principle satisfied by the related diffusions. We use the Laplace method on Wiener space. The main difficulties come from the nonlinearity and the possibility for the endpoint of the optimal path to lie on the boundary of the support of the initial condition.

We study in this paper the convergence rate of the Swendsen-Wang dynamics towards its equilibrium law, when the energy belongs to a large family of energies used in image segmentation problems. We compute the exponential equivalents of the transitions which control the process at low temperature, as well as the critical constant which gives its convergence rate. We give some theoretical tools to compare this dynamics with Metropolis, and develop an experimental study in order to calibrate both dynamics performances in image segmentation problems.

In this paper, we address the problem of testing hypotheses using maximum likelihood statistics in non identifiable models. We derive the asymptotic distribution under very general assumptions. The key idea is a local reparameterization, depending on the underlying distribution, which is called locally conic. This method enlights how the general model induces the structure of the limiting distribution in terms of dimensionality of some derivative space. We present various applications of the theory. The main application is to mixture models. Under very general assumptions, we solve completely the problem of testing the size of the mixture using maximum likelihood statistics. We derive the asymptotic distribution of the maximum likelihood statistic ratio which takes an unexpected form.

We generalize the results of Komlós, Major and Tusnády concerning the strong approximation of partial sums of independent and identically distributed random variables with a finite r-th moment to the case when the parameter set is two-dimensional. The most striking result is that the rates of convergence are exactly the same as in the one-dimensional case.

We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix, are extended to bounded, uniformly elliptic and Lipschitz continuous matrices. As an application, we show that within each class, a particular choice of the coefficients leads to a probabilistic interpretation of generalizations of Burgers' equation.

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 with an infinite number of jumps (asymptotically Δ-accessible points); and the set of points that the process cannot reach by jumping (Δ-inaccessible points).

We establish optimal uniform upper estimates on heat kernels whose generators satisfy a logarithmic Sobolev inequality (or entropy-energy inequality) with the optimal constant of the Euclidean space. Off-diagonals estimates may also be obtained with however a smaller d istance involving harmonic functions. In the last part, we apply these methods to study some heat kernel decays for diffusion operators of the type Laplacian minus the gradient of a smooth potential with a given growth at infinity.