Limit theorems for random simplices in high dimensions

2017/08/01 by Grote, Julian, Kabluchko, Zakhar, Thäle, Christoph
#52A22 #52A23 #60D05 #60F05 #60F10 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1708.00471

Abstract

Let r=r(n) be a sequence of integers such that r≤ n and let X₁,…,X_r+1 be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on ℝⁿ. Limit theorems for the log-volume and the volume of the random convex hull of X₁,…,X_r+1 are established in high dimensions, that is, as r and n tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-ϕ convergence are derived. The results heavily depend on the asymptotic growth of r relative to n. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if r=o(n) (respectively, r∼ αn for some 0 < α< 1).

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