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  • Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

  • Part of
  • Lothar Heinrich, Malte Spiess
  • Journal:
    Advances in Applied Probability / Volume 45 / Issue 2 / June 2013
    Published online by Cambridge University Press:
    22 February 2016, pp. 312-331
    Print publication:
    June 2013
    • Article
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  • 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.