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Almost sure behavior of functionals of stationary Gibbs point processes

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  • Coeurjolly, Jean-François

Abstract

This paper is concerned with the almost sure control of functionals of stationary Gibbs point processes. We apply Kahane–Khintchine’s inequality to derive an almost sure control of various functionals under very mild assumption on the spatial point process X. In particular, if X is a locally stable Gibbs point process with finite range observed in [−n,n]d, we obtain the bound N[−n,n]d(X)/(2n)d=ρ+Oa.s.(n−d/2logn3/2) as n→∞, where NW(X) is the number of points of X∩W for W⊂Rd and where ρ is the intensity parameter of X.

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  • Coeurjolly, Jean-François, 2015. "Almost sure behavior of functionals of stationary Gibbs point processes," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 241-246.
    • Handle: RePEc:eee:stapro:v:97:y:2015:i:c:p:241-246
    DOI: 10.1016/j.spl.2014.11.014
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    1. A. Baddeley & R. Turner & J. Møller & M. Hazelton, 2005. "Residual analysis for spatial point processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 617-666, November.
    2. El Machkouri, Mohamed, 2002. "Kahane-Khintchine inequalities and functional central limit theorem for stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 102(2), pages 285-299, December.
    3. A. Baddeley & J. Møller & A. Pakes, 2008. "Properties of residuals for spatial point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 627-649, September.
    4. Jean-François Coeurjolly & Frédéric Lavancier, 2013. "Residuals and goodness-of-fit tests for stationary marked Gibbs point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(2), pages 247-276, March.
    5. Jesper Møller & Rasmus P. Waagepetersen, 2007. "Modern Statistics for Spatial Point Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 643-684, December.
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    1. Coeurjolly, Jean-François & Reynaud-Bouret, Patricia, 2019. "A concentration inequality for inhomogeneous Neyman–Scott point processes," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 30-34.

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