Asymptotic normality for random simplices and convex bodies in high\n dimensions

2019/06/06 by David Alonso–Gutiérrez, Florian Besau, Alonso-Gutiérrez, David +13
Mathematics · #52A22 #52A23 #60D05 #60F05 #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1906.02471

openalex publication_date 2019/06/06 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Central limit theorems for the log-volume of a class of random convex bodies\nin \ℝⁿ are obtained in the high-dimensional regime, that is, as\nn\→\∞. In particular, the case of random simplices pinned at the origin\nand simplices where all vertices are generated at random is investigated. The\ncoordinates of the generating vectors are assumed to be independent and\nidentically distributed with subexponential tails. In addition, asymptotic\nnormality is established also for random convex bodies (including random\nsimplices pinned at the origin) when the spanning vectors are distributed\naccording to a radially symmetric probability measure on the n-dimensional\n\ℓ_p-ball. In particular, this includes the cone and the uniform\nprobability measure.\n

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