Multivariate Normal Approximation for Functionals of Random Polytopes

Jens Grygierek ()
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Jens Grygierek: Osnabrück University

Journal of Theoretical Probability, 2021, vol. 34, issue 2, 897-922

Abstract: Abstract Consider the random polytope that is given by the convex hull of a Poisson point process on a smooth convex body in $$\mathbb {R}^d$$ R d . We prove central limit theorems for continuous motion invariant valuations including the Wills functional and the intrinsic volumes of this random polytope. Additionally we derive a central limit theorem for the oracle estimator that is an unbiased and minimal variance estimator for the volume of a convex set. Finally we obtain a multivariate limit theorem for the intrinsic volumes and the components of the $$\mathbf {f}$$ f -vector of the random polytope.

Keywords: Central limit theorem; Multivariate limit theorem; Intrinsic volumes; f-vector; Random polytope; Random convex hull; Stochastic geometry; Poisson point process; Oracle estimator; Volume estimation; 52A22; 60D05; 60F05 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-00985-3

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