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Palm representation and approximation of the covariance of random closed sets

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Stephan Böhm  and
Volker Schmidt
Stephan Böhm*
Affiliation:
University of Ulm
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Department of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
∗ Postal address: Department of Stochastics, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
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Abstract

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).

Keywords

Stochastic geometryrandom closed setcovariancePalm calculus, exponential approximationBoolean model

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 295 - 302
Copyright
Copyright © Applied Probability Trust 2003 

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