Entropic projections and dominating points are solutions to convexminimization problems related to conditional laws of largenumbers. They appear in many areas of applied mathematics such asstatistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugateduality and functional analysis, criteria are derived for theexistence of entropic projections, generalized entropicprojections and dominating points. Representations of thegeneralized entropic projections are obtained. It is shown thatthey are the “measure component" of the solutions to someextended entropy minimization problem. This approach leads to newresults and offers a unifying point of view. It also permits toextend previous results on the subject by removing unnecessarytopological restrictions. As a by-product, new proofs of alreadyknown results are provided.

We deal with a subject in the interplay between nonparametric statistics and geometricmeasure theory. The measure L0(G) of theboundary of a set G ⊂ ℝd (withd ≥ 2) can be formally defined, via a simple limit, bythe so-called Minkowski content. We study the estimation ofL0(G) from a sample of random points insideand outside G. The sample design assumes that, for each sample point, weknow (without error) whether or not that point belongs to G. Under thisdesign we suggest a simple nonparametric estimator and investigate its consistencyproperties. The main emphasis in this paper is on generality. So we are especiallyconcerned with proving the consistency of our estimator under minimal assumptions on theset G. In particular, we establish a mild shape condition onG under which the proposed estimator is consistent inL2. Roughly speaking, such condition establishes that theset of “very spiky” points at the boundary of G must be “small”. This isformalized in terms of the Minkowski content of such set. Several examples arediscussed.

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.

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.

We provide an alternative algebraic and geometric approach to the results of [I. Bailleul, Probab. Theory Related Fields141 (2008) 283–329] describing the asymptotic behaviour of the relativistic diffusion.

We consider a differential equation with a random rapidly varying coefficient.The random coefficient is aGaussian process with slowly decaying correlations and compete with a periodic component. In theasymptotic framework corresponding to the separation of scales present in theproblem, we prove that the solution of the differential equationconverges in distribution to the solution of a stochastic differential equationdriven by a classical Brownian motion in some cases, by a fractional Brownianmotion in other cases. The proofs of these results are based on the Lyons theory ofrough paths. Finally we discuss applications in two physical situations.

Let (\hbox{$\mathbb{S}^1, d_{\mathbb{S}^1}$}$S^{\mathrm{1}}\mathit{,}{\mathit{d}}_{S^{\mathrm{1}}}\frac{}{}$) be the unit circle inℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μ to admit a well defined Fréchet mean on (\hbox{$\mathbb{S}^1,d_{\mathbb{S}^1}$}$S^{\mathrm{1}}\mathit{,}{\mathit{d}}_{S^{\mathrm{1}}}\frac{}{}$). We derive a new sufficient condition of existence P(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

Multistable processes, that is, processes which are, at each “time”, tangent to a stableprocess, but where the index of stability varies along the path, have been recentlyintroduced as models for phenomena where the intensity of jumps is non constant. In thiswork, we give further results on (multifractional) multistable processes related to theirlocal structure. We show that, under certain conditions, the incremental moments display ascaling behaviour, and that the pointwise Hölder exponent is, as expected, related to thelocal stability index. We compute the precise value of the almost sure Hölder exponent inthe case of the multistable Lévy motion, which turns out to reveal an interestingphenomenon.

Consider a mean-reverting equation, generalized in the sense it is driven by a1-dimensional centeredGaussian process with Hölder continuous paths on [0,T] (T> 0). Taking thatequation in rough paths sense only gives local existence of the solution because thenon-explosion condition is not satisfied in general. Under natural assumptions, by usingspecific methods, we show the global existence and uniqueness of the solution, itsintegrability, the continuity and differentiability of the associated Itô map, and weprovide an Lp-convergingapproximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a largedeviation principle, and the existence of a density with respect to Lebesgue’s measure,for the solution of that generalized mean-reverting equation. Finally, we study ageneralized mean-reverting pharmacokinetic model.

The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto distribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ, which will be compared to other recent estimators through some simulations. Asymptotic normality results are also proved. Our new estimator of ρ looks especially competitive when  |ρ|  is small.

Assume that (Xt )t∈Z is a real valued time seriesadmitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat.24 (1996) 508–539] propose near-minimax estimators $\widehat f_n$ based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators $\widehat f_n$ depend on weak dependence properties of the sequence (Xt )t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévymeasure ν. We consider the first passage time T x of(Xt, t ≥ 0) to level x > 0, and Kx := XTx - x theovershoot and Lx := x- XTx- the undershoot. We first provethat the Laplace transform of the random triple (Tx,Kx,Lx )satisfies some kind of integral equation. Second, assuming thatν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution asx → ∞, where $\widetilde{T_x}$ denotes a suitablerenormalization of T x .


In this paper, we consider ℝd-valued integrable processes which are increasing in the convex order, i.e. ℝd-valued peacocks in our terminology. After the presentation of some examples, we show that an ℝd-valued process is a peacock if and only if it has the same one-dimensional marginals as an ℝd-valued martingale. This extends former results, obtained notably by Strassen [Ann. Math. Stat. 36 (1965) 423–439], Doob [J. Funct. Anal. 2 (1968) 207–225] and Kellerer [Math. Ann. 198 (1972) 99–122].

We present a spectral theory for a class of operators satisfying a weak“Doeblin–Fortet" condition and apply it to a class of transition operators.This gives the convergence of the series ∑k≥0krPkƒ, $r \in \mathbb{N}$ ,under some regularity assumptions and implies the central limit theoremwith a rate in $n^{- \frac{1}{2} }$ for the corresponding Markov chain.An application to a non uniformly hyperbolic transformation on the interval is also given.

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

With direct and simple proofs, we establish Poincaré type inequalities (includingPoincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities),entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. Theproofs are efficient for non-local Dirichlet forms with general jump kernel, and also workfor Lp(p>1) settings. Our results yield a new sufficient condition forfractional Poincaré inequalities, which were recently studied in [P.T. Gressman,J. Funct. Anal. 265 (2013) 867–889. C. Mouhot, E. Russ andY. Sire, J. Math. Pures Appl. 95 (2011) 72–84.] To ourknowledge this is the first result providing entropy inequalities and Beckner-typeinequalities for measures more general than Lévy measures.

In this paper, a singular semi-linear parabolic PDE with locally periodiccoefficients is homogenized. We substantially weaken previous assumptions onthe coefficients. In particular, we prove new ergodic theorems. We show thatin such a weak setting on the coefficients, the proper statement of thehomogenization property concerns viscosity solutions, though we need abounded Lipschitz terminal condition.

Consider an autoregressive model with measurement error: we observe Zi =Xi +εi, where theunobserved Xi is a stationarysolution of the autoregressive equation Xi =gθ0(Xi− 1) + ξi. Theregression function gθ0 isknown up to a finite dimensional parameter θ0 to be estimated. The distributions ofξ1 and X0 are unknownand gθ belongs to a largeclass of parametric regression functions. The distribution of ε0 is completelyknown. We propose an estimation procedure with a new criterion computed as the Fouriertransform of a weighted least square contrast. This procedure provides an asymptoticallynormal estimator \hbox{$\hat \theta$}θ̂ of θ0, for a large class of regressionfunctions and various noise distributions.

We consider the segmentation problem of Poisson and negative binomial (i.e.overdispersed Poisson) rate distributions. In segmentation, an important issueremains the choice of the number of segments. To this end, we propose a penalized-likelihood estimator where the penaltyfunction is constructed in a non-asymptotic context following the works of L. Birgé and P.Massart. The resulting estimator is proved to satisfy an oracle inequality. Theperformances of our criterion is assessed using simulated and real datasets in the RNA-seqdata analysis context.

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.