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Limit theorems for random points in a simplex

Part of: Limit theorems Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  21 June 2022

Anastas Baci ,
Zakhar Kabluchko ,
Joscha Prochno ,
Mathias Sonnleitner  and
Christoph Thäle
Anastas Baci*
Affiliation:
Ruhr University Bochum
Zakhar Kabluchko*
Affiliation:
University of Münster
Joscha Prochno*
Affiliation:
University of Graz and University of Passau
Mathias Sonnleitner*
Affiliation:
University of Graz and University of Passau
Christoph Thäle*
Affiliation:
Ruhr University Bochum
*
*Postal address: Faculty of Mathematics, Ruhr University Bochum, 44780 Bochum, Germany.
***Postal address: Faculty of Mathematics, University of Münster, 48149 Münster, Germany. Email: zakhar.kabluchko@uni-muenster.de
****Postal address: Institute of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria.
****Postal address: Institute of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria.
*Postal address: Faculty of Mathematics, Ruhr University Bochum, 44780 Bochum, Germany.
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Abstract

In this work the $\ell_q$ -norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry–Esseen bounds in the regime $1\leq q < \infty$ are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where $q=\infty$ . An application to the intersection volume of a regular simplex with an $\ell_p^n$ -ball is also carried out.

Keywords

Berry–Esseen boundcentral limit theoremhigh dimensionsmoderate deviationsℓq-normlarge deviationsregular simplex

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F10: Large deviations
Secondary: 52A23: Asymptotic theory of convex bodies 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 685 - 701
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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