2021/11/08 by Carina Betken, Betken, Carina, Matthias Schulte +3
Mathematics · #53C65 #60F05 #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities #Primary 60D05 #Probability (math.PR) #Secondary 52A22
paper · pdf · doi:10.48550/arxiv.2111.04608
openalex publication_date 2021/11/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This paper deals with the union set of a stationary Poisson process of cylinders in ℝn having an (n-m)-dimensional base and an m-dimensional direction space, where m∈\0,1,…,n-1\ and n≥ 2. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or m-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of geometric functionals, including the intrinsic volumes. In this context the asymptotic variance constant is analysed in detail, which in contrast to the Boolean model leads to a new degeneracy phenomenon. A quantitative central limit theory is developed in a multivariate set-up as well.