Extremes for the inradius in the Poisson line tessellation

A Poisson line tessellation is observed within a window. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in the window in the limit as the window is scaled to infinity. We additionally prove that the limit shape of the cells minimising the inradius is a triangle

Compound Poisson process approximation under β\beta-mixing and stabilization

We establish Poisson and compound Poisson approximations for stabilizing statistics of $\beta$-mixing point processes and give explicit rates of convergence. Our findings are based on a general estimate of the total variation distance of a stationary $\beta$-mixing process and its Palm version. As main contributions, this article (i) extends recent results on Poisson process approximation to non-Poisson/binomial input, (ii) gives concrete bounds for compound Poisson process approximation in a Wasserstein distance and (iii) illustrates the applicability of the general result in an example on minimal angles in the stationary Poisson-Delaunay tessellation. The latter is among the first (nontrivial) situations in Stochastic Geometry, where compound Poisson approximation can be established with explicit extremal index and cluster size distribution.Comment: 21 pages, 1 figur

Some properties on extremes for transient random walks in random sceneries

Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(\xi(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(\xi(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of $(\xi(S_n))_{n\geq 0}$.Comment: arXiv admin note: text overlap with arXiv:2201.0568