2019
DOI: 10.30757/alea.v16-06
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Limit theorems for random simplices in high dimensions

Abstract: Let r = r(n) be a sequence of integers such that r ≤ n and let X 1 , . . . , X r+1 be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on R n . Limit theorems for the log-volume and the volume of the random convex hull of X 1 , . . . , 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… Show more

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Cited by 58 publications
(78 citation statements)
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“…We need to recall the results on the moments of the -volume of a -simplex and the -volume of a -parallelotope spanned by random points distributed according to either the beta-or beta ′ -distribution. The following two results are due to Miles [26,Equations (72) [26] only considers integer moments, we present an extension to moments of arbitrary (non-negative) real order, which is also needed elsewhere, for example in [19].…”
Section: Random Simplices and Parallelotopesmentioning
confidence: 99%
“…We need to recall the results on the moments of the -volume of a -simplex and the -volume of a -parallelotope spanned by random points distributed according to either the beta-or beta ′ -distribution. The following two results are due to Miles [26,Equations (72) [26] only considers integer moments, we present an extension to moments of arbitrary (non-negative) real order, which is also needed elsewhere, for example in [19].…”
Section: Random Simplices and Parallelotopesmentioning
confidence: 99%
“…Moreover, since the uniform distribution on the unit sphere S n−1 arises as the weak limit of the beta distribution, as β → −1 (see for example the proof of Theorem 2.7 in [10]), the result of Theorem 2.4 in [14] can be recovered by Theorem 1.1. Corollary 1.3.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…Lately, the high-dimensional geometry of sets arising from these models of randomness have been studied extensively; for instance, in terms of properties of their volume [10], facet numbers [5] or intrinsic volumes [11]. Asymptotic estimates on the expected volume of the polytope P β N,n , as N → ∞, were derived by Affentranger [1] for any fixed dimension n and parameter β.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
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“…Let us stress that here, the origin is added to the convex hull. Applying (7) to standard Gaussian vectors (details are in Subsection 2.3) leads to the following representation.…”
Section: 2mentioning
confidence: 99%