Probabilities and Statistic seminar

Wei Lu est/sera maître de conférences à Nancy dans notre équipe à partir/depuis le 1er septembre 2025.

Matteo D’achille est/sera maître de conférences à Metz dans notre équipe à partir/depuis le 1er septembre 2025.

Les systèmes de particules sont des outils souvent utilisés pour modéliser des populations interagissant entre elles et/ou avec un environnement extérieur. Je présenterai un modèle appelé N-BMP (N-processus de Markov branchant) dans lequel les particules ont des trajectoires indépendantes sur la droite réelle, et se séparent en deux (on parle de branchement) à des instants aléatoires successifs indépendants et distribués selon une loi exponentielle. Pour garder une population de taille constante au cours du temps, la particule la plus basse est tuée à chaque instant de branchement. Je poserai la question de la limite hydrodynamique de ce processus, c’est-à-dire le comportement du processus obtenu quand le nombre N de particules tend vers l’infini. Le cas où les particules ont des trajectoires browniennes a notamment été étudié depuis 2017, et mis en relation avec un problème à frontière libre associé à l’équation de la chaleur. Je présenterai les résultats que nous avons obtenus, qui donnent un cadre plus général dans lequel le N-BMP a une limite hydrodynamique, en faisant intervenir une frontière analogue à celle qui apparaît dans le cas brownien. Travail en collaboration avec Jean Bérard.

Statistical methodology for spatial point patterns has traditionally focused on Euclidean data and planar surfaces. However, with recent advances in 3D biological imaging technologies targeting protein molecules on a cell’s plasma membrane, spatial point patterns are now being observed on complex shapes and manifolds whose geometry must be respected for principled inference. Consequently, there is now a demand for tools that can analyse these data for important scientific studies in cellular and micro-biology. Motivated by studying the spatial distribution of LPS proteins on the surface of E-Coli, we develop the fundamental functional summary statistics for the analysis of point patterns to general convex bounded shapes and demonstrate how they can be used to test for complete spatial randomness. We then develop their multi-type extensions, together with a test for independence of the component marginal processes. To support these methods, we introduce a plug-in estimator for the intensity of a spatial point process on a manifold. We conclude with a discussion on how these methods can readily be extended to a class of non-convex shapes. This talk will aim to provide an accessible overview of the references below.

References:

Ward, E.A.K. Cohen, N. M. Adams. Testing for complete spatial randomness on 3-dimensional bounded convex shapes. Spatial Statistics, Vol. 41, 2021.

Ward, H. S. Battey and E. A. K. Cohen. Nonparametric estimation of the intensity function of a spatial point process on a Riemannian manifold. Biometrika, Vol. 110, 2023.

Kumar, P. Inns, S. Ward, V. Lagage, J. Wang, R. Kaminska, S. Uphoff, E. A. K. Cohen, G. Mamou and C. Kleanthous. Immobile lipopolysaccharides and outer membrane proteins differentially segregate in growing Escherichia coli. Proceedings of the National Academy of Sciences, 122 (10), 2025

Ward, E. A. K. Cohen and N. M. Adams. Functional summary statistics and testing for independence in marked point patterns on the surface of three-dimensional convex shapes. Spatial Statistics, Vol. 67, 2025

We study a system of partial differential equations (PDEs) with interconnected obstacles and Neumann-type boundary conditions on a smooth bounded domain D. This system is the Hamilton-Jacobi-Bellman system of equations associated with multidimensional switching problem in finite horizon when the state process is constrained to live in the domain D. We prove the existence of a unique continuous viscosity solution. The existence of a viscosity solution is obtained using a probabilistic approach which connects the system of PDEs to a system of backward stochastic differential equations, where randomness is constrained to stay in the domain D. The second main result consists in verifying the maximum principle, which ensures the comparison between any viscosity sub-solution and super-solution of the PDEs system. This guarantees the uniqueness and continuity of the solution.

In this talk, we consider the problem of testing for the sphericity of a collection of random vectors. It is well known that in the classical elliptical model, testing for rotational symmetry of the underlying distribution is equivalent to testing that a scatter parameter is a multiple of the identity matrix. We consider the more general model of random vectors with elliptical directions introduced by R.H. Randles and present a few scenarios where testing for sphericity is still equivalent to testing that the scatter parameter is a multiple of the identity. These new scenarios include, for instance, non-classical settings where some dependence of a rather general form studied here for the first time may be present between observations. We study, under these new assumptions, the behavior of the classical spatial sign test and show that under certain mild assumptions, the test is asymptotically valid and has the same local asymptotic power as in the classical elliptical scenario. We then show that, contrary to some commonly held belief, the spatial sign test enjoys some local asymptotic optimality properties when it comes to testing for sphericity when the underlying distribution is strongly heavy-tailed.

(L’exposé sera en français, avec des slides en anglais.)