In this paper, we give sufficient conditions to establish central limittheorems and moderate deviation principle for a class of support estimates ofempirical and Poisson point processes. The considered estimates are obtained bysmoothing some bias corrected extreme values of the point process. We show howthe smoothing permits to obtain Gaussian asymptotic limits and thereforepointwise confidence intervals. Some unidimensional and multidimensionalexamples are provided.

Sur un ensemble fini, on s'intéresse aux minorations linéaires du trou spectral d'un noyau markovien réversible, en terme de la constante isopérimétrique associée. On montre que la constanteoptimale est l'inverse du cardinal de l'ensemble moins un, mais on verra aussi comment il est possible de l'améliorer dans certaines situations particulières (arbres pointés radiaux à nombre fini de générations). Une application des inégalités précédentes est de retrouver immédiatement le comportement à basse température des trous spectraux associés aux algorithmes de recuit simulé généralisés finis. Une approche similaire permet également d'obtenir très simplement une caractérisation de l'énergie virtuelle due à Trouvé.

Soit U une fonction définie sur un ensemble fini E muni d'unnoyau markovien irréductible M. L'objectif du papier est de comparerthéoriquement deux procédures stochastiques de minimisation globale de U :le recuit simulé et un algorithme génétique.Pour ceci on se placera dans la situation idéalisée d'une infinité de particules disponibleset nous feronsune hypothèse commode d'existence de suffisamment de symétries du cadre (E,M,U).On verra notamment que contrairement au recuit simulé, toute évolution logarithmiquede l'inverse de la température conduit asymptotiquement à concentrer les dynamiques de Feynman-Kac recuitessurcertains minima globaux de U (du moins si M ne fait pas sortir de leur ensemble en un temps presque immédiat).Par contre, ces dernières ont tendance à oublier plus difficilement leur condition initiale et on quantifiera précisément cette propriété. Enfin on présentera les conjectures correspondantes dans une situation plus générale.

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].

We propose a feature selection method for density estimation withquadratic loss. This method relies on the study of unidimensionalapproximation models and on the definition of confidence regions forthe density thanks to these models. It is quite general and includescases of interest like detection of relevant wavelets coefficientsor selection of support vectors in SVM. In the general case, weprove that every selected feature actually improves the performanceof the estimator. In the case where features are defined bywavelets, we prove that this method is adaptative near minimax (upto a log term) in some Besov spaces. We end the paper bysimulations indicating that it must be possible to extend theadaptation result to other features.


Particle filter algorithms approximate a sequence of distributions by a sequence of empirical measures generated by a population of simulated particles. In the context of Hidden Markov Models (HMM), they provide approximations of the distribution of optimal filters associated to these models. For a given set of observations, the behaviour of particle filters, as the number of particles tends to infinity, is asymptotically Gaussian, and the asymptotic variance in the central limit theorem depends on the set of observations. In this paper we establish, under general assumptions on the hidden Markov model, the tightness of the sequence of asymptotic variances when considered as functions of the random observations as the number of observations tends to infinity. We discuss our assumptions on examples and provide numerical simulations.

Asymmetric or partial information in financial markets may berepresented by different filtrations. We consider the case of alarger filtration F – the natural filtration of the“model world” – and a subfiltration $\hat{\mathcal F}$ thatrepresents the information available to an agent in the “realworld”. Given a price system on the larger filtration that isrepresented by a martingale measure Q and an associated numeraire S, we show that there is a canonical and nontrivial numeraire Ŝ such that the price system generated by(Ŝ,Q, $\hat{\mathcal F}$ ) is consistent, in a sense to be madeprecise, with the price system generated by (S,Q,F).

For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb{R}^n$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If $\widehat b_{1}^{(n)},\ldots, \widehat b_p^{(n)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors $r_j^{(n)} = \Vert \widehat b^{(n)}_{n-j+1}\Vert^2 / \Vert \widehat b^{(n)}_{n-j} \Vert^2$ , j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^{(n)}-1,j\geq 1)$ tends in distribution in some sense to an explicit process $({\mathcal R}_j -1,j\geq 1)$ ; some properties of the latter are provided. The probability that a random random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.

Consider a strong Markov process in continuous time, taking values in some Polish statespace. Recently, Douc et al. [Stoc. Proc. Appl.119, (2009) 897–923] introduced verifiable conditions in terms ofa supermartingale property implying an explicit control of modulated moments of hittingtimes. We show how this control can be translated into a control of polynomial moments ofabstract regeneration times which are obtained by using the regeneration method ofNummelin, extended to the time-continuous context. As a consequence, if ap-th moment of the regeneration times exists, we obtain non asymptoticdeviation bounds of the form

\begin{equation*} P_{\nu} \left(\left|\frac1t\int_0^tf(X_s){\rm d}s-\mu(f)\right|\geq\ge\right)\leqK(p)\frac1{t^{p- 1}}\frac 1{\ge^{2(p-1)}}\|f\|_\infty^{2(p-1)} ,\quad p \geq 2.\end{equation*}${\mathit{P}}_{\mathit{\nu }}\left(\left|\frac{\mathrm{1}}{\mathit{t}}{\mathrm{\int }}_{\mathrm{0}}^{\mathit{t}}\mathit{f}\mathrm{\left(}{\mathit{X}}_{\mathit{s}}\mathrm{\right)}\mathrm{d}\mathit{s}\mathrm{-}\mathit{\mu }\mathrm{\left(}\mathit{f}\mathrm{\right)}\right|\mathrm{\ge }\mathit{\epsilon }\right)\mathrm{\le }\mathit{K}\mathrm{\left(}\mathit{p}\mathrm{\right)}\frac{\mathrm{1}}{{\mathit{t}}^{\mathit{p}\mathrm{-}\mathrm{1}}}\frac{\mathrm{1}}{{\mathit{\epsilon }}^{\mathrm{2}\mathrm{(}\mathit{p}\mathrm{-}\mathrm{1}\mathrm{)}}}\mathrm{\parallel }\mathit{f}{\mathrm{\parallel }}_{\mathrm{\infty }}^{\mathrm{2}\mathrm{(}\mathit{p}\mathrm{-}\mathrm{1}\mathrm{)}}\mathit{,}\mathit{p}\mathrm{\ge }\mathrm{2.}$

We investigate the optimal alignment of two independent random sequences of length n. We providea polynomial lower bound for the probability of the optimal alignmentto be macroscopically non-unique. We furthermore establisha connection between the transversal fluctuation andmacroscopic non-uniqueness.

In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (K. Dziedziul, M. Kucharska, B. Wolnik, Estimation of the smoothness parameter). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes possible for some classes of the density functions.

Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.

In a multiple testing context, we consider a semiparametric mixture model with two components where one component is known and corresponds to the distribution of p-values under the null hypothesis and the other component f is nonparametric and stands for the distribution under the alternative hypothesis. Motivated by the issue of local false discovery rate estimation, we focus here on the estimation of the nonparametric unknown component f in the mixture, relying on a preliminary estimator of the unknown proportion θ of true null hypotheses. We propose and study the asymptotic properties of two different estimators for this unknown component. The first estimator is a randomly weighted kernel estimator. We establish an upper bound for its pointwise quadratic risk, exhibiting the classical nonparametric rate of convergence over a class of Hölder densities. To our knowledge, this is the first result establishing convergence as well as corresponding rate for the estimation of the unknown component in this nonparametric mixture. The second estimator is a maximum smoothed likelihood estimator. It is computed through an iterative algorithm, for which we establish a descent property. In addition, these estimators are used in a multiple testing procedure in order to estimate the local false discovery rate. Their respective performances are then compared on synthetic data.

In this paper, we make use of the information measure introducedby Mokkadem (1997) for building a goodness-of-fit test forlong-range dependent processes.Our test statistic is performed in the frequency domain and writes asa non linear functional of the normalized periodogram. We establishthe asymptotic distribution of our statistic under the nullhypothesis. Under specific alternative hypotheses, we prove that the powerconverges to one. The performance of our test procedure isillustrated from different simulated series. In particular,we compare its size and its power with test of Chenand Deo.

Let a finite alphabet Ω. We consider a sequence of letters from Ωgenerated by a discrete time semi-Markov process $\{Z_{\gamma};\ \gamma\in \xN \}.$ Wederive the probability of a word occurrence in the sequence. We also obtain resultsfor the mean and variance of the number of overlapping occurrences of a word in afinite discrete time semi-Markov sequence of letters under certain conditions.

For 0 < α ≤ 2 and 0 < H < 1, anα-time fractional Brownian motion is an iterated processZ =  {Z(t) = W(Y(t)), t ≥ 0} obtained by taking a fractional Brownian motion {W(t), t ∈ ℝ} with Hurst index0 < H < 1 and replacing the time parameter with astrictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that suchprocesses have natural connections to partial differential equations and, whenY is a stable subordinator, can arise as scaling limit of randomlyindexed random walks. The existence, joint continuity and sharp Hölder conditions in theset variable of the local times of a d-dimensionalα-time fractional Brownian motionX = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ...,Xd(t)), where t ≥ 0 andX1, ..., Xdare independent copies of Z, are investigated. Our methods rely on thestrong local nondeterminism of fractional Brownian motion.

In this note we prove that the local martingale part of a convex functionf of a d-dimensional semimartingaleX = M + A can be written in terms ofan Itô stochastic integral∫H(X)dM, whereH(x) is some particular measurable choice ofsubgradient \hbox{$\sub$}$\overline{\mathrm{\nabla }}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ off at x, and M is the martingale partof X. This result was first proved by Bouleau in [N. Bouleau, C.R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here wepresent a new treatment of the problem. We first prove the result for\hbox{$\widetilde{X}=X+\epsilon B$}$\begin{array}{}{}_{\mathrm{􏽥}}\\ \mathit{X}\end{array}\mathrm{=}\mathit{X}\mathrm{+}\mathit{ϵB}$,ϵ > 0, where B is a standardBrownian motion, and then pass to the limit as ϵ → 0, using results in[M.T. Barlow and P. Protter, On convergence of semimartingales. In Séminaire deProbabilités, XXIV, 1988/89, Lect. Notes Math., vol. 1426.Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math.36 (1992) 420–427]. The former paper concerns convergence ofsemimartingale decompositions of semimartingales, while the latter studies a special caseof converging convex functions of semimartingales.

We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation of quantiles and recursive estimation of the mean.

In this paper, we prove that the laws of interacting Brownian particlesare characterized as Gibbs fields on pathspace associated to an explicit class ofHamiltonian functionals. More generally, we show that a large class of Gibbsfields on pathspace corresponds to Brownian diffusions. Some applications totime reversal in the stationary and non stationary case are presented.

We pursue the study of a random coloring first passage percolation model introduced byFontes and Newman. We prove that the asymptotic shape of this first passage percolationmodel continuously depends on the law of the coloring. The proof uses several couplings,particularly with greedy lattice animals.