Open Access

Lothar Heinrich, Malte Spiess

• A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed by a stationary Poisson process of k-flats (0 < k < d) which are dilated by independent 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 r W as the scaling factor r grows unboundedly. Due to the long-range dependences within the union set of cylinders, the variance of its d-volume in r W increases asymptotically proportional to the (d+k)th power of r. To obtain the exact asymptotic behaviour of this variance we need a distinction between discrete and continuous directional distributions of the typical k-flat.
• Es wird ein zentraler Grenzwertsatz für das Volumen einer stationären zufälligen Menge, die aus der Vereinigung von Poissonschen Zylindern mit unabhängigen identisch verteilten k-dimensionalen Richtungsräumen und zufälligen kompakten Zylinderbasen besteht, in einem Euklidischen Raum mit Dimension d > k bewiesen.