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Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics

Part of: Stochastic processes Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 September 2020

Viktor Beneš ,
Christoph Hofer-Temmel ,
Günter Last  and
Jakub Večeřa
Viktor Beneš*
Affiliation:
Charles University in Prague
Christoph Hofer-Temmel*
Affiliation:
CWI & Dutch Defense Academy
Günter Last*
Affiliation:
Karlsruhe Institute of Technology
Jakub Večeřa*
Affiliation:
Charles University in Prague
*
*Postal address: Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 18675 Praha 8, Czech Republic.
***Postal address: CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands. Email address: math@temmel.me
****Postal address: Department of Mathematics, Karlsruhe Institute of Technology, Postfach 6980, D-76049 Karlsruhe, Germany. Email address: guenter.last@kit.edu
*Postal address: Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 18675 Praha 8, Czech Republic.
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Abstract

We study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation, we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A by-product of our approach is a new uniqueness result for Gibbs particle processes.

Keywords

Gibbs processparticle processpair potentialdisagreement percolationcorrelation functionscentral limit theoremU-statistics

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 928 - 955
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Copyright
© Applied Probability Trust 2020

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