Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Kabluchko, Zakhar; Marynych, Alexander; Temesvari, Daniel; Thaele, Christoph

doi:10.48550/arxiv.1801.08008

Cited by 6 publications

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“…We emphasize that the result of Theorem 1.1 is in sharp contrast to the recent developments [7,34] around random spherical convex hulls on half-spheres. In fact, if in Theorem 1.1 the set K is a closed half-sphere, then the central limit theorem breaks down.…”

Section: Central Limit Theorems In Spherical Spacesmentioning

confidence: 66%

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Asymptotic normality for random polytopes in non-Euclidean geometries

Besau¹,

Thäle²

2019

Preprint

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Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak's dual Brunn-Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein's method for normal approximation and geometric properties of weighted floating bodies.

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Section: Central Limit Theorems In Spherical Spacesmentioning

confidence: 66%

“…. is a sequence of independent random points distributed according to the normalized spherical Lebesgue measure on the half-sphere S d x d+1 , x ∈ R d \ {o}, with respect to the Lebesgue measure, see [34,Theorem 2.6]. Clearly, the limiting random variable on the right hand side in (1.1) is non-Gaussian.…”

Section: Central Limit Theorems In Spherical Spacesmentioning

confidence: 99%

Asymptotic normality for random polytopes in non-Euclidean geometries

Besau¹,

Thäle²

2019

Preprint

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“…The statement is trivial if C is a linear subspace, see (15), hence we exclude this possibility in the following. By Proposition 5.5 and by (13),…”

Section: Properties Of Convex Conesmentioning

confidence: 76%

“…This formula was obtained by Glasauer [9]; see [24, p. 263] and also [13,Lemma 5.1] for the conic version stated here. In particular, we have…”

Section: 1mentioning

confidence: 80%

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Grassmann angles and absorption probabilities of Gaussian convex hulls

Götze¹,

Kabluchko²,

Zaporozhets³

2019

Preprint

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Let M be an arbitrary subset in R n with a conic (or positive) hull C. Consider its Gaussian image AM , where A is a k× n-matrix whose entries are independent standard Gaussian random variables. We show that the probability that the convex hull of AM contains the origin in its interior coincides with the k-th Grassmann angle of C. Also, we prove that the expected Grassmann angles of AC coincide with the corresponding Grassmann angles of C. Using the latter result, we show that the expected sum of j-th Grassmann angles at ℓ-dimensional faces of a Gaussian simplex equals the analogous angle-sum for the regular simplex of the same dimension.

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Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

Forrester

Zhang

2020

J Stat Phys

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A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices X, the full Lyapunov spectrum for products of random matrices I N + 1 c X is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent.

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Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Cited by 6 publications

References 0 publications

Asymptotic normality for random polytopes in non-Euclidean geometries

Asymptotic normality for random polytopes in non-Euclidean geometries

Grassmann angles and absorption probabilities of Gaussian convex hulls

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

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