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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

Information

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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