Central limit theorems for volume and surface content of stationary Poisson cylinder processes in expanding domains (Q2837749)

Infinitely long cylinders can be introduced as follows. For a space \(L\in G(d, k)\) (the Grassmannian of \(k\)-dimensional subspaces of \({\mathbb R}^d\)), \(k=0,\dots, d-1\), and a set \(B\) in the orthogonal complement \(L^\bot\), we define a cylinder as the Minkowski sum \(L\oplus B\), where \(L\) is called the direction space and \(B\) the base of the cylinder. In the literature, convex bases \(B\) are mostly considered, but also polyconvex bases are taken into account. In this paper, the orientation of the direction space \(L\) is suppressed and the restriction on the polyconvexity of \(B\) is dropped, thus allowing the base to be compact. Since it is a natural choice for the modelling of material, the paper considers cylinder processes (CPs) which are driven by a Poisson process, so-called Poisson cylinder processes (PCPs). Under the condition that the exponential moment of the \((d-k)\)-volume of the typical cylinder base exists, \textit{L. Heinrich} and \textit{M. Spiess} [Lith. Math. J. 49, No. 4, 381--398 (2009; Zbl 1186.60017)] have derived a central limited theorem with Berry-Esseen bounds. If the second moment of the \((d-k)\)-volume of the typical cylinder base exists, the authors prove asymptotic normality of the \(d\)-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain \(\rho W\) as \(\rho\) grows unboundedly. As the second main result, the authors prove formulae for the asymptotic variance in the formula \({{|\Xi\cap \rho W|_d - E|\Xi\cap \rho W|_d}\over{\sqrt{\operatorname{var}|\Xi\cap \rho W|_d}}}\) (\(\Xi\) is the union set of the PCP and \(|\Xi\cap \rho W|_d\) denotes the volume of the intersection of \(\Xi\) and \(\rho W\)) by distinguishing between discrete and continuous directional distributions. A corresponding central limit theorem for the surface content is stated at the end.