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Asymptotic normality for random simplices and convex bodies in high dimensions


Authors: D. Alonso-Gutiérrez, F. Besau, J. Grote, Z. Kabluchko, M. Reitzner, C. Thäle, B.-H. Vritsiou and E. Werner
Journal: Proc. Amer. Math. Soc. 149 (2021), 355-367
MSC (2010): Primary 52A22, 52A23, 60D05, 60F05
DOI: https://doi.org/10.1090/proc/15232
Published electronically: October 9, 2020
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Abstract: Central limit theorems for the log-volume of a class of random convex bodies in $ \mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $ n\to \infty $. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is also established for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $ n$-dimensional $ \ell _p$-ball. In particular, this includes the cone and the uniform probability measure.


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Additional Information

D. Alonso-Gutiérrez
Affiliation: Department of Mathematics, University of Zaragoza, Pedro Cerbuna no. 12, Zaragoza, Spain
Email: alonsod@unizar.es

F. Besau
Affiliation: Faculty of Mathematics, Vienna University of Technology, Oskar-Morgenstern-Platz1, 1090 Wien, Austria
Email: florian.besau@tuwien.ac.at

J. Grote
Affiliation: Faculty of Mathematics and Economics, University of Ulm, 89069 Ulm, Germany
Email: julian.grote@uni-ulm.de

Z. Kabluchko
Affiliation: Faculty of Mathematics and Computer Science, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: zakhar.kabluchko@wwu.de

M. Reitzner
Affiliation: School of Mathematics/Computer Science, University of Osnabrück, Albrechtstr. 28a, 49069 Osnabrück, Germany
Email: matthias.reitzner@uni-osnabrueck.de

C. Thäle
Affiliation: Department of Mathematics, Ruhr University Bochum, Universitätsstrasse 150, 44801 Bohum, Germany
Email: christoph.thaele@rub.de

B.-H. Vritsiou
Affiliation: Department of Mathematics, University of Alberta in Edmonton, Edmonton, Alberta, Canada T6G 2G1 Canada
Email: vritsiou@ualberta.ca

E. Werner
Affiliation: Department of Mathematics, Applied Mathematics and Statistics, Yost Hall, Case Western Reserve University, 2049 Martin Luther King Jr. Drive, Cleveland, Ohio 44106-7058
Email: elisabeth.werner@case.edu

DOI: https://doi.org/10.1090/proc/15232
Keywords: Central limit theorem, high dimensions, $\ell_p$-ball, random convex body, random determinant, random parallelotope, random polytope, random simplex, stochastic geometry.
Received by editor(s): July 15, 2019
Received by editor(s) in revised form: February 26, 2020
Published electronically: October 9, 2020
Additional Notes: The first author was partially supported by MICIN Project PID2019-105979GB-I00 and DGA Project E48_20R
The second author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) grant BE 2484/5-2.
The third author was supported by DFG via RTG 2131 “High-dimensional Phenomena in Probability – Fluctuations and Discontinuity”.
The fourth athor was supported by DFG under Germany’s Excellence Strategy EXC 2044 - 390685587
The eighth author was partially supported by NSF grant DMS-1811146.
Communicated by: Deane Yang
Article copyright: © Copyright 2020 American Mathematical Society