## Bernoulli

• Bernoulli
• Volume 24, Number 4A (2018), 2811-2841.

### Concentration and moderate deviations for Poisson polytopes and polyhedra

#### Abstract

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to a class of zero cells associated with Poisson hyperplane mosaics. As a special case, this comprises the typical Poisson–Voronoi cell conditioned on having large inradius.

#### Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2811-2841.

Dates
Revised: January 2017
First available in Project Euclid: 26 March 2018

https://projecteuclid.org/euclid.bj/1522051226

Digital Object Identifier
doi:10.3150/17-BEJ946

Mathematical Reviews number (MathSciNet)
MR3779703

Zentralblatt MATH identifier
06853266

#### Citation

Grote, Julian; Thäle, Christoph. Concentration and moderate deviations for Poisson polytopes and polyhedra. Bernoulli 24 (2018), no. 4A, 2811--2841. doi:10.3150/17-BEJ946. https://projecteuclid.org/euclid.bj/1522051226

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