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Bounds for the Probability Generating Functional of a Gibbs Point Process

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  22 February 2016

Kaspar Stucki  and
Dominic Schuhmacher
Kaspar Stucki*
Affiliation:
University of Bern
Dominic Schuhmacher*
Affiliation:
University of Bern
*
∗ Current address: Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstrasse 7, 37077 Göttingen, Germany
∗ Current address: Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstrasse 7, 37077 Göttingen, Germany
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Abstract

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We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics such as the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity, and higher-order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

Keywords

Gibbs processprobability generating functionalintensitycorrelation functionF functionG functionK functionPoisson saddlepoint approximationStein's method

MSC classification

Primary: 60G55: Point processes
Secondary: 62M30: Spatial processes
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 21 - 34
Copyright
© Applied Probability Trust 

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