zbMATH — the first resource for mathematics

Functionals of spatial point process having density with respect to the Poisson process. (English) Zbl 1316.60069
Summary: $$U$$-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-Itō 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.

MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F05 Central limit and other weak theorems
Full Text:
References:
 [1] Baddeley, A.: Spatial point processes and their applications. Stochastic geometry. Lecture Notes in Math. 1892 (2007), 1-75. · Zbl 1127.62086 · doi:10.1007/978-3-540-38175-4_1 [2] Decreusefond, L., Flint, I.: Moment formulae for general point processes. C. R. Acad. Sci. Paris, Ser. I (2014), 352, 357-361. · Zbl 1297.60031 · doi:10.1016/j.jfa.2014.04.014 [3] Kaucky, J.: Combinatorial Identities (in Czech). Veda, Bratislava 1975. [4] Last, G., Penrose, M. D.: Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150 (2011), 663-690. · Zbl 1233.60026 · doi:10.1007/s00440-010-0288-5 · arxiv:0909.3205 [5] Last, G., Penrose, M. D., Schulte, M., Thäle, Ch.: Moments and central limit theorems for some multivariate Poisson functionals. Adv. Appl. Probab. 46 (2014), 2, 348-364. · Zbl 1350.60020 · doi:10.1239/aap/1401369698 [6] Møller, J., Helisová, K.: Power diagrams and interaction processes for unions of disc. Adv. Appl. Probab. 40 (2008), 321-347. · Zbl 1146.60322 · doi:10.1239/aap/1214950206 [7] Møller, J., Waagepetersen, R.: Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton 2004. · Zbl 1044.62101 [8] Peccati, G., Taqqu, M. S.: Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi Univ. Press, Springer, Milan 2011. · Zbl 1231.60003 · doi:10.1007/978-88-470-1679-8 [9] Peccati, G., Zheng, C.: Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15 (2010), 48, 1487-1527. · Zbl 1228.60031 · emis:journals/EJP-ECP/_ejpecp/viewarticlec5ad.html [10] Reitzner, M., Schulte, M.: Central limit theorems for $$U$$-statistics of Poisson point processes. Ann. Probab. 41 (2013), 3879-3909. · Zbl 1293.60061 · doi:10.1214/12-AOP817 · arxiv:1104.1039 [11] Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin 2008. · Zbl 1175.60003 · doi:10.1007/978-3-540-78859-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.