Večeřa, Jakub Central limit theorem for Gibbsian \(U\)-statistics of facet processes. (English) Zbl 1413.60014 Appl. Math., Praha 61, No. 4, 423-441 (2016). Summary: A special case of a Gibbsian facet process on a fixed window with a discrete orientation distribution and with increasing intensity of the underlying Poisson process is studied. All asymptotic joint moments for interaction \(U\)-statistics are calculated and the central limit theorem is derived using the method of moments. Cited in 2 Documents MSC: 60F05 Central limit and other weak theorems 60D05 Geometric probability and stochastic geometry 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) Keywords:central limit theorem; facet process; \(U\)-statistics PDFBibTeX XMLCite \textit{J. Večeřa}, Appl. Math., Praha 61, No. 4, 423--441 (2016; Zbl 1413.60014) Full Text: DOI arXiv Link References: [1] V. Beneš, M. Zikmundová: Functionals of spatial point processes having a density with respect to the Poisson process. Kybernetika 50 (2014), 896–913. · Zbl 1316.60069 [2] P. Billingsley: Probability and Measure. John Wiley & Sons, New York, 1995. · Zbl 0822.60002 [3] H.-O. Georgii, H. J. Yoo: Conditional intensity and Gibbsianness of determinantal point processes. J. Stat. Phys. 118 (2005), 55–84. · Zbl 1130.82016 · doi:10.1007/s10955-004-8777-5 [4] G. Last, M. D. Penrose: 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 [5] G. Last, M. D. Penrose, M. Schulte, C. Thäle: Moments and central limit theorems for some multivariate Poisson functionals. Adv. Appl. Probab. 46 (2014), 348–364. · Zbl 1350.60020 · doi:10.1017/S0001867800007126 [6] G. Peccati, M. S. Taqqu: Wiener chaos: Moments, Cumulants and Diagrams. A survey with computer implementation. Bocconi University Press, Milano; Springer, Milan, 2011. · Zbl 1231.60003 [7] M. Reitzner, M. Schulte: Central limit theorems for U-statistics of Poisson point processes. Ann. Probab. 41 (2013), 3879–3909. · Zbl 1293.60061 · doi:10.1214/12-AOP817 [8] T. Schreiber, J. E. Yukich: Limit theorems for geometric functionals of Gibbs point processes. Ann. Inst. Henri Poincaré, Probab. Stat. 49 (2013), 1158–1182. · Zbl 1308.60064 · doi:10.1214/12-AIHP500 [9] J. Večeřa, V. Beneš: Interaction processes for unions of facets, the asymptotic behaviour with increasing intensity. Methodol. Comput. Appl. Probab. (2016), DOI-10.1007/s11009-016-9485-8. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.