Asymptotic normality for random simplices and convex bodies in high dimensions,Proceedings of the American Mathematical Society

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Asymptotic normality for random simplices and convex bodies in high dimensions
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-10-09 , DOI: 10.1090/proc/15232
D. Alonso-Gutiérrez , F. Besau , J. Grote , Z. Kabluchko , M. Reitzner , C. Thäle , B.-H. Vritsiou , E. Werner

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

-dimensional $\ell _p$ -ball. In particular, this includes the cone and the uniform probability measure.

中文翻译：

高维随机单形和凸形的渐近正态性

摘要：在高维状态下，获得了一类随机凸体的对数的中心极限定理。尤其是，研究了固定在原点处的随机单纯形和随机生成所有顶点的单纯形的情况。假设生成矢量的坐标是独立的，并且具有次指数尾巴，并且分布相同。另外，当跨度矢量根据三维球上的径向对称概率测度分布时，也为随机凸体（包括固定在原点的随机单纯形）建立渐近正态性。特别地，这包括圆锥和均匀概率测度。 $\ mathbb {R} ^ n$ $n \ to \ infty$

$\ ell _p$

更新日期：2020-11-12

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