Conference Agenda

We study empirical variants of the halfspace (Tukey) depth of a probability measure $\mu$, which are obtained by replacing $\mu$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions, which are random convex bodies. In the special case of $\mu$ being the uniform distribution on a convex body $K$, the depth trimmed regions are convex floating bodies of $K$, and we obtain strong limit theorems for their empirical estimators.

In this talk we present a family of random Laguerre tessellations $\mathcal{L}_d(f)$ in $\mathbb{R}^d$ as well as their duals, generated by inhomogeneous Poisson point processes in $\mathbb{R}^d\times\mathbb{R}$ whose intensity measures have density of the form $(v,h)\mapsto \gamma f(h)$ under some natural restrictions on the functions $f$. We show that the construction provides a random tessellation and establish a connection to fractional calculus. This family includes the models introduced in [1] and [2], namley the $\beta$-Voronoi, $\beta'$-Voronoi and the Gaussain-Voronoi tessellations. We give the distribution of the typical cell of the corresponding dual tessellation explicitly in terms of $f$. Further we include the classical Poisson-Voronoi tessellation in this family as a limit case of the $\beta$-Voronoi tessellation.

[1] Gusakova, A., Kabluchko, Z., and Thäle, C. The $\beta$-Delaunay tessellation: Description of the model and geometry of typical cells. Adv. in Appl. Probab. 54, 4 (2022), 1252-1290.

[2] Gusakova, A., Kabluchko, Z., and Thäle, C. The $\beta$-Delaunay tessellation II: The Gaussian limit tessellation. Electron. J. Probab., 27 (2022), 1-33.

Spatial random graphs provide an important framework for the analysis of relations and interactions in networks. In particular, the random geometric graph has been intensively studied and applied in various frameworks like network modeling or percolation theory.

In this talk we focus on approximation results for a generalization of the random geometric graph that consists of vertices given by a Gibbs process and (conditionally) independent edges generated from a connection probability function. We introduce a new graph metric between finite spatial graphs of possibly different sizes that is built on the OSPA metric for point patterns, but penalizes both vertex and edge structures. We develop Stein's method for general integral probability metrics that compare the distributions of spatial random graphs. We then focus on the Wasserstein distance with respect to the new graph metric to obtain improved rates of convergence for a suitable type of convergence in distribution of spatial random graphs. Finally, we present an application of our approximation results to the percolation graph of large balls in a Boolean model.

Many functionals of point processes arising in stochastic geometry can be written as sums of scores, where each score represents the contribution of a point. If the score of a point depends only on the local point configuration in a random neighbourhood, the functional is called stabilising. One is often interested in the asymptotic behaviour of stabilising functionals as an underlying observation window is increased. For this situation several central limit theorems were shown in recent years. In this talk these results are complemented by functional central limit theorems.