The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions

Gusakova, Anna; Kabluchko, Zakhar; Thäle, Christoph

Research article (journal) | Peer reviewed

Abstract

The β-Delaunay tessellation in Rd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a β-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in Rd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. 2010 Mathematics Subject Classification. 52A22, 52A40, 60D05, 60F05, 60F10.

Details about the publication

JournalLatin American Journal of Probability and Mathematical Statistics (ALEA)
Volume19
Page range23-50
StatusPublished
Release year2022
Language in which the publication is writtenEnglish
DOI10.30757/ALEA.v19-02
Link to the full texthttps://alea.impa.br/articles/v19/19-02.pdf
KeywordsBeta-Delaunay tessellation; central limit theorem; cumulant method; large deviations; moderate deviations; mod-phi convergence; Kendall’s problem; stochastic geometry; typical cell; weighted typical cell.

Authors from the University of Münster

Gusakova, Anna
Juniorprofessorship of applied mathematics (Prof. Gusakova)
Kabluchko, Zakhar
Professorship for probability theory (Prof. Kabluchko)