A random cell splitting scheme on the sphere

Deuß, Christian; Hörrmann, Julia; Thäle, Christoph

doi:10.1016/j.spa.2016.08.010

Cited by 6 publications

(11 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Altogether, this yields that M 2,2 ( f (0) ) ≤ c t 4 (e r + e r + e r + re r ) ≤ c t 4 re r , where the exponent of t follows from 4 (0) ) with the lower variance bound provided by Lemma 9 we deduce from (14)…”

Section: We Computementioning

confidence: 70%

“…we already see one reason for the dependence of our results on the dimension of the hyperbolic space. In order to establish the normal approximation bounds of Theorem 5, which are based on the general bound provided in (14), we have to deal with further integrals of Crofton-type, as described in Sect. 6.2.…”

Section: Remark 2 Since Theorem 2 Follows From the General Fock Spacementioning

confidence: 99%

“…The normal approximation bound (14) for general U-statistics of Poisson processes is given by a sum involving terms of the type M u,v , which are defined in (12) and (13) and which in turn are given as sums of integrals over partitions σ ∈ Π con ≥2 (u, u, v, v). In applying these normal approximation bounds to the Euclidean counterparts of the functionals F (i) r ,t it was possible to extract a common scaling factor from each of the integrals in M u,v and to treat the number of terms, that is, the number of elements of Π con ≥2 (u, u, v, v) as a constant, see [36,58].…”

Section: The Case Of Growing Windows: Proof Of Theoremmentioning

confidence: 99%

“…As indicated above, we will use the bound (14) in combination with (12) and (13). We distinguish the cases i = 0 and i = 1.…”

Section: The Planar Case D = 2: Proof Of Theorem 5 (A)mentioning

confidence: 99%

“…Again more specifically, the papers [9,17,47,50] study various aspects of random geometric graphs in hyperbolic spaces, including central limit theorems for a number of parameters. Random tessellations of the unit sphere by great hyperspheres are the content of [2,24,25,44], while so-called random splitting tessellations in spherical spaces were introduced and investigated in [14,26]. The paper [12] is concerned with properties of Poisson-Voronoi tessellations on general Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Herold

Hug

Thäle

2021

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

Poisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

show abstract

Section: We Computementioning

confidence: 70%

Section: Remark 2 Since Theorem 2 Follows From the General Fock Spacementioning

confidence: 99%

Section: The Case Of Growing Windows: Proof Of Theoremmentioning

confidence: 99%

“…As indicated above, we will use the bound (14) in combination with (12) and (13). We distinguish the cases i = 0 and i = 1.…”

Section: The Planar Case D = 2: Proof Of Theorem 5 (A)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Herold

Hug

Thäle

2021

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

show abstract

Random inscribed polytopes in projective geometries

2021

Self Cite

View full text Add to dashboard Cite

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and a Berry–Esseen bound for functionals of independent random variables.

show abstract

The Typical Cell of a Voronoi Tessellation on the Sphere

Kabluchko

Thäle

2021

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

The typical cell of a Voronoi tessellation generated by $$n+1$$ n + 1 uniformly distributed random points on the d-dimensional unit sphere $$\mathbb {S}^d$$ S d is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in $$\mathbb {R}^d$$ R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases $$d\in \{2,3,4\}$$ d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well.

show abstract

A random cell splitting scheme on the sphere

Cited by 6 publications

References 17 publications

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Random inscribed polytopes in projective geometries

The Typical Cell of a Voronoi Tessellation on the Sphere

Contact Info

Product

Resources

About