Showing papers in "Esaim: Probability and Statistics in 2008"

Analysis of the Rosenblatt process

Abstract: We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Major (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Ito multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.

Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence

Abstract: We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x|^a, a in [1/2,1) In that case, we study the rate of convergence of a symmetrized version of the Euler scheme This symmetrized version is easy to simulate on a computer We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz

Large population limit and time behaviour of a stochastic particle model describing an age-structured population

Abstract: 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 Meleard, 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.

Deviation bounds for additive functionals of markov processes

Abstract: In this paper we derive non asymptotic deviation bounds for P� � � � 1 Z t 0 V (X s)ds Z V dµ � � � 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 Poincare etc...).

Parametric inference for mixed models defined by stochastic differential equations

Abstract: Non-linear mixed models defined by stochastic differential equations (SDEs) are consid- ered: 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 measure- ment 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.

Metastable behaviour of small noise lévy-driven diffusions

Abstract: We consider a dynamical system in R driven by a vector fieldU ,w hereU is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Levy noise of small intensity and such that the heaviest tail of its Levy 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-tail nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

Dependent Lindeberg central limit theorem and some applications

Abstract: In this paper, a very useful lemma (in two versions) is proved: it simplifies notably the essential step to establish a Lindeberg central limit theorem for dependent processes. Then, applying this lemma to weakly dependent processes introduced in Doukhan and Louhichi (1999), a new central limit theorem is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these central limit theorems are provided. All the usual causal or non causal time series: Gaussian, associated, linear, ARCH(1), bilinear, Volterra processes,. . ., enter this frame.

Distortion mismatch in the quantization of probability measures

Abstract: We elucidate the asymptotics of the L s -quantization error induced by a sequence of L r - optimal n-quantizers of a probability distribution P on R d when s>r . In particular we show that under natural assumptions, the optimal rate is preserved as long as s

On EM algorithms and their proximal generalizations

Abstract: In this paper, we analyze the celebrated EM algorithm from the point of view of proximal point algorithms. More precisely, we study a new type of generalization of the EM procedure introduced in [Chretien and Hero (1998)] and called Kullback-proximal algorithms. The proximal framework allows us to prove new results concerning the cluster points. An essential contribution is a detailed analysis of the case where some cluster points lie on the boundary of the parameter space.

Exponential inequalities for VLMC empirical trees.

Abstract: A seminal paper by Rissanen, published in 1983, introduced the class of Variable Length Markov Chains and the algorithm Context which estimates the probabilistic tree generating the chain. Even if the subject was recently considered in several papers, the central question of the rate of convergence of the algorithm remained open. This is the question we address here. We provide an exponential upper bound for the probability of incorrect estimation of the probabilistic tree, as a function of the size of the sample. As a consequence we prove the almost sure consistency of the algorithm Context. We also derive exponential upper bounds for type I errors and for the probability of underestimation of the context tree. The constants appearing in the bounds are all explicit and obtained in a constructive way.

Estimation of anisotropic Gaussian fields through Radon transform

Abstract: We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quite accurately.

Stochastic differential equations driven by processes generated by divergence form operators II: convergence results

Abstract: We have seen in a previous article how the theory of "rough paths" allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Ito integrals already constructed for stochastic processes generated by divergence form operators by using time-reversal techniques.

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Abstract: Let ( X t , t ≥ 0 ) be a Levy process started at 0 , with Levy measure ν . We consider the first passage time T x of ( X t , t ≥ 0 ) to level x > 0 , and K x := X Tx - x the overshoot and L x := x- X T x - the undershoot. We first prove that the Laplace transform of the random triple ( T x ,K x ,L x ) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that converges in distribution as x → ∞, where denotes a suitable renormalization of T x .

Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré

Abstract: In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow Royer's ap- proach (Royer, 1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box (�n,n) d (with free boundary condi- tions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincare. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

Multidimensional limit theorems for smoothed extreme value estimates of point processes boundaries

Abstract: In this paper, we give sufficient conditions to establish central limit theorems and moderate deviation principle for a class of support estimates of empirical and Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing permits to obtain Gaussian asymptotic limits and therefore pointwise confidence intervals. Some unidimensional and multidimensional examples are provided.

Logarithmic sobolev inequalities for inhomogeneous markov semigroups

Abstract: We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.

Density estimation with quadratic loss: a confidence intervals method

Abstract: We propose a feature selection method for density estimation with quadratic loss. This method relies on the study of unidimensional approximation models and on the definition of confidence regions for the density thanks to these models. It is quite general and includes cases of interest like detection of relevant wavelets coefficients or selection of support vectors in SVM. In the general case, we prove that every selected feature actually improves the performance of the estimator. In the case where features are defined by wavelets, we prove that this method is adaptative near minimax (up to a log term) in some Besov spaces. We end the paper by simulations indicating that it must be possible to extend the adaptation result to other features.

Random thresholds for linear model selection

Abstract: A method is introduced to select the significant or non null mean terms among a collection of independent random variables. As an application we consider the problem of recovering the signifi- cant coefficients in non ordered model selection. The method is based on a convenient random centering of the partial sums of the ordered observations. Based on L-statistics methods we show consistency of the proposed estimator. An extension to unknown parametric distributions is considered. Simulated examples are included to show the accuracy of the estimator. An example of signal denoising with wavelet thresholding is also discussed.

Théorèmes limites avec poids pour les martingales vectorielles à temps continu

Abstract: On developpe une approche generale du theoreme limite centrale presque-sure pour les martingales vectorielles quasi-continues a gauche convenablement normalisees dont on degage une extension quadratique et un nouveau theoreme de la limite centrale. L'application de ce resultat a l'estimation de la variance d'un processus a accroissements independants et stationnaires illustre l'usage qu'on peut en faire en statistique.

Concentration inequalities for semi-bounded martingales

Abstract: In this paper, we apply the technique of decoupling to obtain some exponential inequalities for semi-bounded martingale, which extend the results of de la Pena, Ann. probab. 27 (1999) 537–564.

Separation principle in the fractional Gaussian linear-quadratic regulator problem with partial observation

Abstract: In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time with partial observation. For a controlled linear system where both the state and observation processes are driven by fractional Brownian motions, we describe explicitly the optimal control policy which minimizes a quadratic performance criterion. Actually, we show that a separation principle holds, i.e., the optimal control separates into two stages based on optimal filtering of the unobservable state and optimal control of the filtered state. Both finite and infinite time horizon problems are investigated.

Minimax and bayes estimation in deconvolution problem

Abstract: We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function eh where h ∈ L2 (R1 ) and e is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii, IMS Lecture Notes Monograph Series 36 (2001) 419–433].