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.
Author: | Karl-Heinz Küfer |
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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) |