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This paper deals with the union set of a stationary Poisson process ofcylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an$m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$.The concept simultaneously generalises those of a Boolean model and a Poissonhyperplane or $m$-flat process. Under very general conditions on the typicalcylinder base a Berry-Esseen bound for the volume of the union set within asequence of growing test sets is derived. Assuming convexity of the cylinderbases and of the window a similar result is shown for a broad class ofgeometric functionals, including the intrinsic volumes. In this context theasymptotic variance constant is analysed in detail, which in contrast to theBoolean model leads to a new degeneracy phenomenon. A quantitative centrallimit theory is developed in a multivariate set-up as well.

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