Limit theorems for stabilising functionals
Description Phase 2
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. Stabilisation means that the score of a point only depends on the other points of the point process in a random neighbourhood. 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 stabilising functionals can have different orders, which is crucial for their analysis. Limit theorems for stabilising 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 stabilising functionals in several directions: (i) multivariate quantitative central limit theorems, (ii) discretised 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 stabilising 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.