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2022 Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes
Carina Betken, Matthias Schulte, Christoph Thäle
Author Affiliations +
Electron. J. Probab. 27: 1-47 (2022). DOI: 10.1214/22-EJP805


This paper deals with the union set of a stationary Poisson process of cylinders in Rn having an (nm)-dimensional base and an m-dimensional direction space, where m{0,1,,n1} and n2. 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.

Funding Statement

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.


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Carina Betken. Matthias Schulte. Christoph Thäle. "Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes." Electron. J. Probab. 27 1 - 47, 2022.


Received: 8 November 2021; Accepted: 1 June 2022; Published: 2022
First available in Project Euclid: 17 June 2022

MathSciNet: MR4441146
zbMATH: 1492.60028
Digital Object Identifier: 10.1214/22-EJP805

Primary: 60D05
Secondary: 52A22 , 53C65 , 60F05

Keywords: Berry-Esseen bound , central limit theorem , geometric functional , intrinsic volume , multivariate central limit theorem , Poisson cylinder process , second-order Poincaré inequality , Stochastic geometry , variance asymptotics

Vol.27 • 2022
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