Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Autor(en): Bachmann, Sascha
Reitzner, Matthias 
Stichwörter: COMPONENT; Concentration inequalities; Convex distance; FINE GAUSSIAN FLUCTUATIONS; FUNCTIONALS; Mathematics; Poisson point process; Random graphs; SPACE; Statistics & Probability; Stochastic geometry; Subgraph counts
Erscheinungsdatum: 2018
Herausgeber: ELSEVIER SCIENCE BV
Journal: STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volumen: 128
Ausgabe: 10
Startseite: 3327
Seitenende: 3352
Zusammenfassung: 
Concentration bounds for the probabilities P(N >= M r) and P(N <= M - r) are proved, where M is a median or the expectation of a subgraph count N associated with a random geometric graph built over a Poisson process. The lower tail bounds have a Gaussian decay and the upper tail inequalities satisfy an optimality condition. A remarkable feature is that the underlying Poisson process can have a.s. infinitely many points. The estimates for subgraph counts follow from tail inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques involving the convex distance for Poisson processes. (C) 2017 Elsevier B.V. All rights reserved.
ISSN: 03044149
DOI: 10.1016/j.spa.2017.11.001

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