Limit theory for the Gilbert graph
Autor(en): | Reitzner, Matthias Schulte, Matthias Thaele, Christoph |
Stichwörter: | Central limit theorem; Compound Poisson limit theorem; Concentration inequality; Covariogram; FUNCTIONALS; GAUSSIAN FLUCTUATIONS; Gilbert graph; Malliavin-Stein method; Mathematics; Mathematics, Applied; NORMAL APPROXIMATION; POISSON; Poisson point process; Random geometric graph; Stable limit theorem; Talagrand's convex distance; U-STATISTICS | Erscheinungsdatum: | 2017 | Herausgeber: | ACADEMIC PRESS INC ELSEVIER SCIENCE | Journal: | ADVANCES IN APPLIED MATHEMATICS | Volumen: | 88 | Startseite: | 26 | Seitenende: | 61 | Zusammenfassung: | For a given homogeneous Poisson point process in R-d two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behavior of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using the convex distance. (C) 2017 Elsevier Inc. All rights reserved. |
ISSN: | 01968858 | DOI: | 10.1016/j.aam.2016.12.006 |
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geprüft am 17.05.2024