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Transactions of the American Mathematical Society

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Expected volumes of Gaussian polytopes, external angles, and multiple order statistics


Authors: Zakhar Kabluchko and Dmitry Zaporozhets
Journal: Trans. Amer. Math. Soc. 372 (2019), 1709-1733
MSC (2010): Primary 60D05; Secondary 52A22, 60G15, 52A23, 60G70, 51M20
DOI: https://doi.org/10.1090/tran/7708
Published electronically: November 27, 2018
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Abstract: Let $ X_1,\ldots ,X_n$ be a standard normal sample in $ \mathbb{R}^d$. We compute exactly the expected volume of the Gaussian polytope $ \mathrm {conv}\, [X_1,\ldots ,X_n]$, the symmetric Gaussian polytope $ \mathrm {conv}\, [\pm X_1,\ldots ,\pm X_n]$, and the Gaussian zonotope $ [0,X_1]+\cdots +[0,X_n]$ by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson's formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including $ \mathrm {conv}\,[l_1X_1,\ldots ,l_nX_n]$ and $ \mathrm {conv}\, [\pm l_1 X_1,\ldots , \pm l_n X_n]$, where $ l_1,\ldots ,l_n\geq 0$ are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the $ k$th intrinsic volume of the regular simplex $ \Delta ^{n-1}$ to the expected maximum of independent standard Gaussian random variables $ \xi _1,\ldots ,\xi _n$ given that the maximum has multiplicity $ k$. Namely, we show that

$\displaystyle V_k(\Delta ^{n-1}) = \frac {(2\pi )^{\frac k2}} {k!} \cdot \lim _... ...ots ,\xi _n\} \mathbbm {1}_{\{\xi _{(n)} - \xi _{(n-k+1)}\leq \varepsilon \}}],$

where $ \xi _{(1)} \leq \cdots \leq \xi _{(n)}$ denote the order statistics. A similar result holds for the cross-polytope if we replace $ \xi _1,\ldots ,\xi _n$ with their absolute values.

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Additional Information

Zakhar Kabluchko
Affiliation: Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans–Ring 10, 48149 Münster, Germany
Email: zakhar.kabluchko@uni-muenster.de

Dmitry Zaporozhets
Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191011 St. Petersburg, Russia
Email: zap1979@gmail.com

DOI: https://doi.org/10.1090/tran/7708
Keywords: Gaussian polytope, symmetric Gaussian polytope, expected volume, regular simplex, regular cross-polytope, intrinsic volumes, external angles, asymptotics, order statistics, extreme-value theory, Burgers festoon
Received by editor(s): August 14, 2017
Received by editor(s) in revised form: May 15, 2018
Published electronically: November 27, 2018
Article copyright: © Copyright 2018 American Mathematical Society