This paper deals with the union set of a stationary Poisson process of cylinders in having an -dimensional base and an m-dimensional direction space, where and . 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.
CB was supported by the German Academic Exchange Service (DAAD) via grant 57468851, and CB and CT have been supported by the DFG priority program SPP 2265 Random Geometric Systems.
We are grateful to Claudia Redenbach (Kaiserslautern) for simulating Poisson cylinder processes for us that led to the pictures shown in Figure 1.
"Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes." Electron. J. Probab. 27 1 - 47, 2022. https://doi.org/10.1214/22-EJP805