Functionals of spatial point processes having a density with respect to the Poisson process

Viktor Beneš; Markéta Zikmundová

Kybernetika (2014)

  • Volume: 50, Issue: 6, page 896-913
  • ISSN: 0023-5954

Abstract

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U -statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener-Itô chaos expansion. In the second half we obtain more explicit results for a system of U -statistics of some parametric models in stochastic geometry. In the logarithmic form functionals are connected to Gibbs models. There is an inequality between moments of Poisson and non-Poisson functionals in this case, and we have a version of the central limit theorem in the Poisson case.

How to cite

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Beneš, Viktor, and Zikmundová, Markéta. "Functionals of spatial point processes having a density with respect to the Poisson process." Kybernetika 50.6 (2014): 896-913. <http://eudml.org/doc/262184>.

@article{Beneš2014,
abstract = {$U$-statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener-Itô chaos expansion. In the second half we obtain more explicit results for a system of $U$-statistics of some parametric models in stochastic geometry. In the logarithmic form functionals are connected to Gibbs models. There is an inequality between moments of Poisson and non-Poisson functionals in this case, and we have a version of the central limit theorem in the Poisson case.},
author = {Beneš, Viktor, Zikmundová, Markéta},
journal = {Kybernetika},
keywords = {difference of a functional; limit theorem; moments; U-statistics; spatial point processes; Poisson process; -statistics; central limit theorem; moments},
language = {eng},
number = {6},
pages = {896-913},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Functionals of spatial point processes having a density with respect to the Poisson process},
url = {http://eudml.org/doc/262184},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Beneš, Viktor
AU - Zikmundová, Markéta
TI - Functionals of spatial point processes having a density with respect to the Poisson process
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 6
SP - 896
EP - 913
AB - $U$-statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener-Itô chaos expansion. In the second half we obtain more explicit results for a system of $U$-statistics of some parametric models in stochastic geometry. In the logarithmic form functionals are connected to Gibbs models. There is an inequality between moments of Poisson and non-Poisson functionals in this case, and we have a version of the central limit theorem in the Poisson case.
LA - eng
KW - difference of a functional; limit theorem; moments; U-statistics; spatial point processes; Poisson process; -statistics; central limit theorem; moments
UR - http://eudml.org/doc/262184
ER -

References

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  8. Peccati, G., Taqqu, M. S., Wiener Chaos: Moments, Cumulants and Diagrams., Bocconi Univ. Press, Springer, Milan 2011. Zbl1231.60003MR2791919
  9. Peccati, G., Zheng, C., Multi-dimensional Gaussian fluctuations on the Poisson space., Electron. J. Probab. 15 (2010), 48, 1487-1527. Zbl1228.60031MR2727319
  10. Reitzner, M., Schulte, M., 10.1214/12-AOP817, Ann. Probab. 41 (2013), 3879-3909. Zbl1293.60061MR3161465DOI10.1214/12-AOP817
  11. Schneider, R., Weil, W., Stochastic and Integral Geometry., Springer, Berlin 2008. Zbl1175.60003MR2455326

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