Limit theorems for stabilising functionals

• Applicant: Professor Dr. Matthias Schulte
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 531541167

Many random geometric systems are constructed from point processes. Their global quantities can frequently be expressed as sums of contributions of the underlying points, so-called scores. Stabilization means that the score of a point only depends on the other points of the point process in a random neighborhood. Such sums of scores are called stabilising functionals and play an important role in stochastic geometry. They arise, for example, in the context of spatial random graphs, random tessellations or random polytopes. This project concerns stabilising functionals of underlying Poisson or binomial point processes and studies their asymptotic behaviour as the number of points goes to infinity. Here the variances of stabilizing functionals can have different orders, which is crucial for their analysis. Limit theorems for stabilizing functionals are an active topic of research for more than twenty years and can be applied to many different problems. The goal of this project is to significantly extend and complement the limit theory for stabilizing functionals in several directions: (i) multivariate quantitative central limit theorems, (ii) discretized normal approximation, (iii) normal approximation in total variation distance and (iv) functional central limit theorems. It is planned to derive abstract results that are valid for large classes of stabilizing functionals and, in particular, for different variance orders. These findings will be applied to many prominent examples from stochastic geometry such as edge-length functionals and component counts of spatial random graphs, edge-length functionals of Voronoi tessellations, the volume of the Voronoi set approximation, numbers of maximal points as well as intrinsic volumes and numbers of k-faces of random polytopes. The proofs of some of the intended results will rely on Stein’s method or its combination with Malliavin calculus.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2265: Random geometric systems