Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

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.

Subjects

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

DDC Class

600: Technik

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publishedVersion

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https://creativecommons.org/licenses/by/4.0/

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22-EJP805.pdf

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1.45 MB

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Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes