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On the Approximation of a Ball by Random Polytopes

  • Let (\(a_i)_{i\in \bf{N}}\) be a sequence of identically and independently distributed random vectors drawn from the \(d\)-dimensional unit ball \(B^d\)and let \(X_n\):= convhull \((a_1,\dots,a_n\)) be the random polytope generated by \((a_1,\dots\,a_n)\). Furthermore, let \(\Delta (X_n)\) : = (Vol \(B^d\) \ \(X_n\)) be the deviation of the polytope's volume from the volume of the ball. For uniformly distributed \(a_i\) and \(d\ge2\), we prove that tbe limiting distribution of \(\frac{\Delta (X_n)} {E(\Delta (X_n))}\) for \(n\to\infty\) satisfies a 0-1-law. Especially, we provide precise information about the asymptotic behaviour of the variance of \(\Delta (X_n\)). We deliver analogous results for spherically symmetric distributions in \(B^d\) with regularly varying tail.

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Metadaten
Author:Karl-Heinz Küfer
URN:urn:nbn:de:hbz:386-kluedo-50509
Series (Serial Number):Preprints (rote Reihe) des Fachbereich Mathematik (250)
Document Type:Report
Language of publication:English
Date of Publication (online):2017/11/09
Year of first Publication:1994
Publishing Institution:Technische Universität Kaiserslautern
Date of the Publication (Server):2017/11/09
Page Number:17
Faculties / Organisational entities:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)