A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d-1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d-k)-dimensional orthogonal complement. If the second moment of the (d-k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k)th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.
"Central limit theorems for volume and surface content of stationary Poisson cylinder processes in expanding domains." Adv. in Appl. Probab. 45 (2) 312 - 331, June 2013. https://doi.org/10.1239/aap/1370870120