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

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  • Bachmann, Sascha
  • Reitzner, Matthias

Abstract

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.

Suggested Citation

  • Bachmann, Sascha & Reitzner, Matthias, 2018. "Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3327-3352.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:10:p:3327-3352
    DOI: 10.1016/j.spa.2017.11.001
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    References listed on IDEAS

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    1. Lachièze-Rey, Raphaël & Peccati, Giovanni, 2013. "Fine Gaussian fluctuations on the Poisson space II: Rescaled kernels, marked processes and geometric U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4186-4218.
    2. Bachmann, Sascha, 2016. "Concentration for Poisson functionals: Component counts in random geometric graphs," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1306-1330.
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    Cited by:

    1. Gusakova, Anna & Sambale, Holger & Thäle, Christoph, 2021. "Concentration on Poisson spaces via modified Φ-Sobolev inequalities," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 216-235.

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    2. Bachmann, Sascha, 2016. "Concentration for Poisson functionals: Component counts in random geometric graphs," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1306-1330.

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