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

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

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

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group $\mathsf{Diff}^\omega_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^\omega_+(\mathbb S^1)$ acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work.Comment: v3 36 pages, 5 figures; cosmetic change