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1 results

  • Palm representation and approximation of the covariance of random closed sets

  • Part of
  • Stephan Böhm, Volker Schmidt
  • Journal:
    Advances in Applied Probability / Volume 35 / Issue 2 / June 2003
    Published online by Cambridge University Press:
    22 February 2016, pp. 295-302
    Print publication:
    June 2003

  • The covariance C(r), r ≥ 0, of a stationary isotropic random closed set Ξ is typically complicated to evaluate. This is the reason that an exponential approximation formula for C(r) has been widely used in the literature, which matches C(0) and C(1)(0), and in many cases also limr→∞C(r). However, for 0 < r < ∞, the accuracy of this approximation is not very high in general. In the present paper, we derive representation formulae for the covariance C(r) and its derivative C(1)(r) using Palm calculus, where r ≥ 0 is arbitrary. As a consequence, an explicit expression is obtained for the second derivative C(2)(0). These results are then used to get a refined exponential approximation for C(r), which additionally matches the second derivative C(2)(0).