Palm representation and approximation of the covariance of random closed sets
Stephan Böhm, Volker Schmidt · Advances in Applied Probability · 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 lim r →∞ 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).