Some nonasymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests

We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities.Comment: Published in at http://dx.doi.org/10.1214/08-AOS667; http://dx.doi.org/10.1214/08-AOS668 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniform growth of groups acting on Cartan-Hadamard spaces

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $\Gamma$ acting on $X$, then either $\Gamma$ is virtually nilpotent or the algebraic entropy $Ent (\Gamma) \geq C(n,a)$

Observations sur le captage et la croissance de l'huître creuse ouest-africaine, Crassostrea gasar, en Casamance, Sénégal

Ces observations ont été faites dans le but de définir les modalités d'élevage de #Crassostrea gasar$ en Casamance, sur deux sites, retenus pour leurs caractéristiques environementales opposées à l'intérieur de la zone de distribution locale de cette huître. L'un est situé à proximité de l'Océan, sur l'île de Carabane, et l'autre, en amont à 60 km par voie d'eau de l'embouchure du fleuve Casamance, près du village de Djivent. Sur le premier site l'élévation de la température de l'eau commande le captage du naissain de mars à octobre avec un maximum en juillet; en raison d'une forte concurrence biologique, la survie des huîtres n'est possible que dans la zone interdidale. Sur le second site, c'est la chute de la salinité, d'août à novembre, soit à la fin de la saison des pluies, qui provoque la ponte et la fixation des jeunes huîtres; celles-ci peuvent survivre aussi bien en immersion permanente, grâce à une faible concurrence biologique dûe aux fortes variations de salinité, que dans la zone interdidale. Sur les deux sites, la croissance est linéaire sur plusieurs années mais elle s'infléchit durant la période de reproduction. La différence des résultats obtenus implique deux techniques d'élevage distinctes qui pourront servir de référence. (Résumé d'auteur

Differentiable Rigidity under Ricci curvature lower bound

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of $(X,g_0)$. We show that there exists a number $\varepsilon:=\varepsilon (n, d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by $-n(n-1)$ and its volume satisfies \vol_g (Y)\leqslant (1+\varepsilon) \vol_{g_0} (X) then the manifolds are diffeomorphic. The proof relies on Cheeger-Colding's theory of limits of Riemannian manifolds under lower Ricci curvature bound.Comment: 33 pages, 1 dessi

Functional quantization-based stratified sampling methods

In this article, we propose several quantization-based stratified sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of stratification lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with stratified sampling. We first put the emphasis on the consistency of quantization for partitioning the state space in stratified sampling methods in both finite and infinite dimensional cases. We show that the proposed quantization-based strata design has uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein-Uhlenbeck processes. We derive in detail the case of Ornstein-Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction facto