Skorokhod spaces and convergence of discontinuous processes.
Catégorie d'évènement : Groupe de travail Probabilités et Statistique Date/heure : 15 mai 2025 09:15-10:30 Lieu : Oratrice ou orateur : Virgile Brodu Résumé :What happens if we want to study the convergence of discontinuous real-valued stochastic processes, which is often the case for modelling purposes? For example, think of tracking the evolution of the population size of living species, where deaths are instantaneous negative jumps… In 1956, Skorokhod proposed a topology on the space of discontinuous functions, which is predominant today. The aim of this talk is to explain the simple and intuitive ideas underlying the construction of Skorokhod to facilitate its understanding, without going in the depth of technical proofs. If we have time, we will introduce measure-valued processes, with biological motivations, and explain how the Skorokhod construction can be generalized to more complex spaces such as these measure spaces.
Even if the present talk is self-contained, it can be seen as an introduction to the GdT of May, 22. I will also present my work about measure-valued processes during the GdT SIMBA of April, 24 (14h, Salle de Conférences). You are warmly welcome to attend one of these to discover some of my PhD research!
Modern Perspectives on Hyperspherical Uniformity Testing: Maximal Projections and Stein Characterizations
Catégorie d'évènement : Séminaire Probabilités et Statistique Date/heure : 15 mai 2025 10:45-11:45 Lieu : Salle de conférences Nancy Oratrice ou orateur : Bruno Ebner (Karlsruher Institut für Technologie) Résumé :We introduce a novel methodology for testing uniformity on the unit hypersphere $S^{d−1}$, based on maximal projections and recent developments in Stein’s method. The first framework provides a unified perspective that encompasses classical tests such as those of Rayleigh and Bingham, while also revealing connections to multivariate skewness and kurtosis. We derive the asymptotic null distribution using limit theorems for Banach space-valued stochastic processes and employ tools from spherical harmonics theory to simulate the corresponding limiting distributions. The test’s performance is analyzed under both contiguous and fixed alternatives, and consistency is established for a broad class of alternatives. Furthermore, we present Bahadur efficiency results for specific alternatives. Theoretical properties and empirical power are assessed through comprehensive Monte Carlo simulations. Complementing this, we explore a second, more recent ap- proach leveraging Stein characterizations to propose new testing procedures that extend the insights of the projection-based framework.
Keywords. uniformity tests, maximal projections, directional data, stochastic pro- cesses in Banach spaces, contiguous alternatives, Monte Carlo simulations, Stein’s method
Pseudogroups and geometric structures
Catégorie d'évènement : Séminaire Théorie de Lie, Géométrie et Analyse Date/heure : 15 mai 2025 14:15-15:15 Lieu : Salle de réunion Metz (ARC-027) Oratrice ou orateur : Francesco Cattafi (Würzburg) Résumé :This philosophy can be made precise at various levels of generality (depending on the definition of « geometric structure ») and using different tools/methods. In this talk I will present some aspects of a new framework, which includes previous formalisms (e.g. G-structures or Cartan geometries) and allows us to prove integrability theorems.
A main novelty of this point of view consists of the fact that it uncovers the (beautiful!) hidden structures behind Lie pseudogroups and geometric structures. Indeed, the relevant objects which make this approach work are Lie groupoids endowed with a multiplicative « PDE-structure », their principal actions, and the related Morita theory. Poisson geometry provides the guiding principle to understand those objects, which are directly inspired from, respectively, symplectic groupoids, principal Hamiltonian bundles, and symplectic Morita equivalence.
This is based on a forthcoming book written jointly with Luca Accornero, Marius Crainic and María Amelia Salazar.
An additive application of the resonance method
Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 15 mai 2025 14:30-15:30 Lieu : Salle Döblin Oratrice ou orateur : Athanasios Sourmelidis (CNRS, Lille) Résumé :In this talk I will describe a way to implement the resonance method in problems of analytic number theory which are not necessarily multiplicative in nature.
This extension of the method not only produces improved extreme results wherever Dirichelt’s approximation theorem has been usually employed but it also highlights its connection to Bohr’s and Jessen’s proof of Kronecker’s approximation theorem.