Generalization of the Nualart-Peccati criterion

The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable $N$ of a given sequence $\{X_n\}_{n\ge1}$ of multiple Wiener-It\^{o} integrals of fixed order, if $\mathbb {E}[X_n^2]\to1$ and $\mathbb {E}[X_n^4]\to \mathbb {E}[N^4]=3$. Since its appearance in 2005, the natural question of ascertaining which other moments can replace the fourth moment in the above criterion has remained entirely open. Based on the technique recently introduced in [J. Funct. Anal. 266 (2014) 2341-2359], we settle this problem and establish that the convergence of any even moment, greater than four, to the corresponding moment of the standard Gaussian distribution, guarantees the central convergence. As a by-product, we provide many new moment inequalities for multiple Wiener-It\^{o} integrals. For instance, if $X$ is a normalized multiple Wiener-It\^{o} integral of order greater than one, $$\forall k\ge2,\qquad \mathbb {E}\bigl[X^{2k}\bigr]>\mathbb {E} \bigl[N^{2k}\bigr]=(2k-1)!!.$$Comment: Published at http://dx.doi.org/10.1214/14-AOP992 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Groups of smooth diffeomorphisms of Cantor sets embedded in a line

Let K be a Cantor set embedded in the real line R. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of R. Higman-Thompson's groups Vn appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups Vn. We also prove that any finitely generated subgroup H without free subsemigroup on two generators is virtually abelian. The corresponding result for the groups Vn was unknown to our knowledge. As a consequence, those groups do not contain nilpotent groups which are not virtually abelian.Comment: The proof of the Burnside property has been changed in this versio

Groups with infinitely many ends acting analytically on the circle

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.Comment: We can now make a precise reference to Deroin's work arXiv:1811.10298. 54 pages, 2 figure

Superconvergence phenomenon in Wiener chaoses

We establish, in full generality, an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. Namely, for every sequence of chaotic random variables, convergence in law to the Gaussian distribution is automatically upgraded to superconvergence: the regularity of the densities increases along the convergence, and all the derivatives converges uniformly on the real line. Our findings strikingly strengthen known results regarding modes of convergence for normal approximation on Wiener chaoses. Our result is then extended to the multivariate setting, and for polynomial mappings of a Gaussian field provided the projection on the Wiener chaos of maximal degree admits a non-degenerate Gaussian limit. While our findings potentially apply to any context involving polynomial functionals of a Gaussian field, we emphasize, in this work, applications regarding: improved Carbery-Wright estimates near Gaussianity; normal convergence in entropy and in Fisher information; superconvergence for the spectral moments of Gaussian orthogonal ensembles; moments bounds for the inverse of strongly correlated Wishart-type matrices; superconvergence in the Breuer-Major Theorem. Our proofs leverage Malliavin's historical idea to establish smoothness of the density via the existence of negative moments of the Malliavin gradient, and we further develop a new paradigm to study this problem. Namely, we relate the existence of negative moments to some explicit spectral quantities associated with the Malliavin Hessian. This link relies on an adequate choice of the Malliavin gradient, which provides a novel decoupling procedure of independent interest. Previous attempts to establish convergence beyond entropy have imposed restrictive assumptions ensuring finiteness of negative moments for the Malliavin derivatives. Our analysis renders these assumptions superfluous.Comment: Revised version to cover the multivariate case, and added more application

Développement d’une vanne d’injection d’échantillons liquides pour la micro-chromatographie en phase gazeuse : applications à des problématiques industrielles

Improving process control is a continuous industrial need. Since the early twothousand years, a new concept online industrial analysis consisting in implementing standardized modular micro-systems implanted as close processes appeared. The gas microchromatography occupies a special place among all the qualitative and quantitative analytical systems used for process control. Until today, this technology can only be used for gaseous samples. The present work aimed to help overcome the technological barrier that constitutes the injection of liquids in µ-GC. To this end, we conducted a comprehensive study of the performance of liquid injection valves HPLIS, ROLSI and OLIS. Two of them (HPLIS and ROLSI) gave satisfaction. The OLIS valve also gave satisfactory results but is still developing. Among the surveyed industrial applications demonstrating the use of these valves, we can quote the qualitative analysis of light or heavy petroleum fractions, quantitative analysis of impurities in trace amounts in a heavy matrix, and more elaborate way, simultaneous quantification of CO/CO2 gas and liquid phase during an absorption reaction of a mixture of these compounds with an aqueous solution of MEAL'amélioration du contrôle des procédés est un besoin industriel permanent. Depuis les années deux-milles se développe un nouveau concept d'analyse industrielle en ligne consistant à mettre en oeuvre des micro-systèmes standardisés modulaires implantés au plus près des procédés. La micro-chromatographie en phase gazeuse occupe une place de choix parmi l'ensemble des systèmes analytiques qualitatifs et quantitatifs utilisés pour le contrôle des procédés. Jusqu'à ce jour, cette technologie n'est utilisable que pour des échantillons gazeux. Le présent travail, effectué dans le cadre du FUI INNOVAL, a pour objectif de lever le verrou technologique que constitue l'injection des liquides dans les μ-GC. Après une étude approfondie des performances des vannes commerciales d'injection de liquide (HPLIS, ROLSI) nous avons développé, avec la contribution de la Sté SRA, une nouvelle vanne nommée OLIS. Des applications industrielles illustrant l'utilisation de ces vannes ont été ensuite étudiées. Nous pouvons citer l'analyse qualitative de coupes pétrolières légères et lourdes, l'analyse quantitative d'impuretés à l'état de trace dans une matrice lourde d'un précurseur de synthèse du nylon et enfin la quantification simultanée du CO/CO2 en phase gazeuse et liquide (solution aqueuse de monoéthanolamine). Cette dernière application a été réalisée grâce à un banc d'essai simulant l'opération de captage du CO2 émis par les installations sidérurgique