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Lévy-based Cox point processes

Part of: Geometric probability and stochastic geometry Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Gunnar Hellmund ,
Michaela Prokešová  and
Eva B. Vedel Jensen
Gunnar Hellmund*
Affiliation:
University of Aarhus
Michaela Prokešová*
Affiliation:
University of Aarhus
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
*
∗ Postal address: T. N. Thiele Centre for Applied Mathematics in Natural Science, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark.
∗∗ Current address: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 18675 Praha 8, Czech Republic.
∗ Postal address: T. N. Thiele Centre for Applied Mathematics in Natural Science, Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark.
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Abstract

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In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

Keywords

Cox processinfinitely divisible distributioninhomogeneitykernel smoothingLévy basislog Gaussian Cox processmixingproduct densityshot noise Cox process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 603 - 629
Copyright
Copyright © Applied Probability Trust 2008 

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