Central Limit Theorem for the Volume of stationary Poisson Cylinder Processes in Expanding Domains (original) (raw)

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed by a stationary Poisson process of k-flats (1 ≤ k ≤ d − 1) which are dilated by i.i.d. 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 ̺ W as ̺ grows unboundedly. Due to the long-range dependences within the union set of cylinders, the variance of its d-volume in ̺ W increases asymptotically proportional to the (d + k)th power of ̺. To obtain the exact asymptotic behaviour of this variance we need a distinction between discrete and continuous directional distributions of the typical k-flat.