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Limit laws for large $k$th-nearest neighbor balls

Part of: Limit theorems Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  06 July 2022

Nicolas Chenavier ,
Norbert Henze  and
Moritz Otto
Nicolas Chenavier*
Affiliation:
Université du Littoral Côte d’Opale
Norbert Henze*
Affiliation:
Karlsruhe Institute of Technology (KIT)
Moritz Otto*
Affiliation:
Otto von Guericke University Magdeburg
*
*Postal address: 50 rue Ferdinand Buisson, 62228 Calais, France. Email address: nicolas.chenavier@univ-littoral.fr
**Postal address: Englerstr. 2, D-76133 Karlsruhe, Germany. Email address: Norbert.Henze@kit.edu
***Postal address: Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark. Email address: otto@math.au.dk
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Abstract

Let $X_1,X_2, \ldots, X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density f. Under some conditions on f, we obtain a Poisson limit theorem, as $n \to \infty$ , for the number of large probability kth-nearest neighbor balls of $X_1,\ldots, X_n$ . Our result generalizes Theorem 2.2 of [11], which refers to the special case $k=1$ . Our proof is completely different since it employs the Chen–Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.

Keywords

Binomial point processlarge kth nearest neighbor ballsChen–Stein methodPoisson convergenceGumbel distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 880 - 894
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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