Random Geometric Structures

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Title: Random Geometric Structures
Authors: Grygierek, Jens Jan
ORCID of the author: https://orcid.org/0000-0002-2709-2115
Thesis advisor: Prof. Dr. Matthias Reitzner
Thesis referee: PD Dr. Matthias Schulte
Abstract: We construct and investigate random geometric structures that are based on a homogeneous Poisson point process. We investigate the random Vietoris-Rips complex constructed as the clique complex of the well known gilbert graph as an infinite random simplicial complex and prove that every realizable finite sub-complex will occur infinitely many times almost sure as isolated complex and also in the case of percolations connected to the unique giant component. Similar results are derived for the Cech complex. We derive limit theorems for the f-vector of the Vietoris-Rips complex on the unit cube centered at the origin and provide a central limit theorem and a Poisson limit theorem based on the model parameters. Finally we investigate random polytopes that are given as convex hulls of a Poisson point process in a smooth convex body. We establish a central limit theorem for certain linear combinations of intrinsic volumes. A multivariate limit theorem involving the sequence of intrinsic volumes and the number of i-dimensional faces is derived. We derive the asymptotic normality of the oracle estimator of minimal variance for estimation of the volume of a convex body.
URL: https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-202001302552
Subject Keywords: Malliavin-Stein; random simplicial complex; random polytope; limit theorems; Poisson point process
Issue Date: 30-Jan-2020
License name: Attribution 3.0 Germany
License url: http://creativecommons.org/licenses/by/3.0/de/
Type of publication: Dissertation oder Habilitation [doctoralThesis]
Appears in Collections:FB06 - E-Dissertationen

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