Contributions

Edouard Daviaud
Dynamical approximation by ergodic systems
The metric number theory is the field of mathematics which aims at describing finely the size (in terms of Hausdorff dimension or measure) of sets of points approximable at a "certain speed" by a sequence of points of particular interest. The original motivation for such study lies in understanding the approximation of real numbers by rationals. Although developed in the context of number theory, these approximation results play an important role in various area of mathematics. For instance, given a signal f, the regularity of f around a point t sometimes depends on how well the number t is approximable by certain points naturally connected with the definition of the signal f. As an illustration, if f is a random signal defined using uniform i.i.d sequences, then the multifractal analysis of f should rely on (almost sure) approximation results of numbers or vectors by uniform i.i.d. sequences of random variable. For such reasons (and also because the question is very natural), the approximation theory with respect to random sequences and dynamical sequences has known many developments in the past 20 years. Very recently, Järvenpää, Järvenpää, Myllyoja and Stenflo established a certain number of general results in the case of appropximation by i.i.d. sequences of random variables. A very natural extension, which should be useful in practice, is to consider the case where the random variables are no longer independent and an approach one can adopt is to consider the dynamical analog of these results. In this talk, we will extend these results to dynamical approcimations by ergodic systems satisfying the so-called sigma mixing assumption of the space L¹ against the sapce of bounded variation mappings. We will also provide exemple aiming at justifying that the result we present should be optimal in general in a sense we will precise. 

Thelma Lambert (Université de Liège)
Régularité des séries aléatoires d'ondelettes en termes de p-exposants
Nous étudions la régularité des séries aléatoires d'ondelettes et plus précisément leur p-spectre de singularités, c'est-à-dire la dimension de Hausdorff des ensembles de points partageant un p-exposant donné. Les séries aléatoires d'ondelettes ont été introduites par J.-M. Aubry et S. Jaffard en 2002 et sont obtenues en tirant aléatoirement et indépendamment des coefficients d'ondelettes à chaque échelle selon une suite de lois de probabilité fixée. L'objectif est d'étendre les résultats connus depuis lors sur leur spectre de Hölder (correspondant au cas p = +∞) en laissant p prendre n’importe quelle valeur dans (0,+∞). L’exposé est basé sur un travail commun avec C. Esser et B. Vedel. 

Giuseppe Lamberti (Université de Bordeaux)
Separated determinantal point processes and generalized Fock spaces
We consider determinantal point processes Λ_ϕ arising from generalized Fock spaces F_ϕ, defined by a doubling subharmonic weight ϕ. We provide a complete characterization of when these processes are almost surely separated with respect to the euclidean metric on the plane. As a consequence, we also obtain a characterization of when these processes are almost surely interpolating for the classical Fock spaces. Furthermore, we emphasize the role of intrinsic repulsion in determinantal processes by comparing Λ_ϕ with the Poisson process of the same first intensity. (This is a joint work with X. Massaneda.)

Barbara Pascal (CNRS - École Centrale Nantes)
Detection of change in cancer breast tissues from fractal indicators: A brief introduction
It has been known for decades that mammograms, consisting in X-ray images of breasts, exhibit a scale-invariance, characteristic of fractal textures akin to fractional Brownian fields. Recent works have thus leveraged fractal features to characterize the microscopic structure of breast tissues. An overarching goal of these studies is early detection of tissue disruption, which is an evidence of homeostasis loss impairing the ability of the tissue to suppress precancerous lesions. Local fractal features are estimated through sliding window analysis, performed on thousands patches extracted from the mammogram to categorized three types of tissues based on their Hölder exponent: fatty, dense or disrupted. Based on the amount of disrupted tissues in the breast and on the asymmetry between the two breasts, this procedure is capable to detect and quantify tumor-associated loss of homeostasis in mammograms. Intensive numerical experiments on large mammograms datasets have proven the statistical power of tests based on the proposed aggregated indicators for cancer risk assessment. This presentation is a survey of the research conducted at CompuMAINE on this topic during the past years and of the perspective it opens for the image processing and stochastic geometry communities. I will first recall the general principles of the multifractal formalism and of the Wavelet Transform Modulus Maxima method. Then, the datasets, test methodologies and conclusions drawn from the analysis performed at CompuMAINE Laboratory will be discussed. Finally, complementary experiments and perspectives will be presented.

Francesca Pistolato (Université du Luxembourg)
Limit theorems for p-domain functionals of stationary Gaussian random fields
Integral functionals of stationary Gaussian random fields provide a unified framework to describe, for instance, the qth variation of stochastic processes, geometric functionals such as the excursion volume, and even sums of random variables. The topic of the talk is the p-domain functionals of stationary Gaussian random fields, a particular class of integral functionals where the domain of integration is a product of lower-dimensional domains, and their asymptotic behavior as the size of the integration domain grows. A cornerstone in the understanding of such functionals is Breuer-Major theorem, which links the short-range dependence of the integrand field, that is, the integrability of its covariance function, together with the standardized functional satisfying a Central Limit Theorem (CLT). In this talk, under further assumptions on the covariance function, we will show how the asymptotic behavior of p-domain functionals can be simply obtained from that of 1-domain functionals, explaining in a new light and in a more systematic way some results from the recent literature. For instance, the result provides a new proof of the convergence of the qth variation of the fractional Wiener sheet to a Gaussian distribution, proved by A. Réveillac, M. Stauch and C. Tudor in 2012, and improves their estimate on the rate of convergence. Moreover, it allows us to produce new examples of long-range dependent fields satisfying CLTs. The talk is based on a joint work with N. Leonenko, L. Maini and I. Nourdin.

Gauthier Quilan (Université Bretagne Sud)
Convex hull peeling
Le convex hull peeling d’un nuage de points est obtenu en construisant l’enveloppe convexe de ces points, puis en retirant les points extrémaux du nuage et en construisant la nouvelle enveloppe convexe des points restants et ainsi de suite. On appelle couche d’ordre n la frontière de l’enveloppe convexe obtenue à l’étape n de la procédure. Dans cet exposé, on s’intéresse à l’étude de fonctions (nombre de points extrémaux, de faces k-dimensionnelles et volume défaut) des couches successives du convex hull peeling d’un ensemble de points indépendants et uniformes dans la boule unité et dans un polytope simple. On rappellera dans un premier temps des thérèmes limites classiques sur le nombre de k-faces de la première couche, c’est-à-dire de l’enveloppe convexe. Ensuite nous présenterons nos résultats qui étendent ces théorèmes limites à toutes les premières couches.

Stéphane Seuret (Université Paris-Est Créteil)
Analyse multifractale bivariée de cascades aléatoires
Dans de nombreux phénomènes, les données à étudier sont multivariées, constituées de plusieurs signaux obtenus en parallèle et à analyser conjointement. Faire l'analyse multifractale multivariée de signaux est un exercice très difficile, peu de fondements théoriques sont à disposition. Dans cet exposé, on présente un résultat général sur la possibilité (ou non, en l'occurence) d'estimer le spectre multifractal multivarié par un formalisme multifractal, et nous étudions un modèle bivarié particulier de deux mesures construites comme cascades aléatoires, pour lequel le spectre bivarié obtenu peut prendre plusieurs formes différentes suivant le choix des paramètres impliqués.