文章基本信息

标题：Limit theorems for random simplices in high dimensions
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作者：Julian Grote ; Zakhar Kabluchko ; Christoph Thäle 等
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2019
卷号：XVI
期号：1
页码：141-177
DOI：10.30757/ALEA.v16-06
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：Let r = r(n) be a sequence of integers such that r n and letX1; : : : ;Xr+1 be independent random points distributed according to the Gaussian,the Beta or the spherical distribution on Rn. Limit theorems for the log-volumeand the volume of the random convex hull of X1; : : : ;Xr+1 are established in highdimensions, that is, as r and n tend to innity simultaneously. This includes Berry-Esseen-type central limit theorems, log-normal limit theorems, and moderate andlarge deviations. Also dierent types of mod- convergence are derived. The resultsheavily depend on the asymptotic growth of r relative to n. For example, weprove that the uctuations of the volume of the simplex are normal (respectively,log-normal) if r = o(n) (respectively, r n for some 0 < < 1).
关键词：Central limit theorem; large deviations; moderate deviations; mod-;convergence; random simplex; volume.