Open Access
2023 Large deviations for the volume of k-nearest neighbor balls
Christian Hirsch, Taegyu Kang, Takashi Owada
Author Affiliations +
Electron. J. Probab. 28: 1-27 (2023). DOI: 10.1214/23-EJP965


This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of M0-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.

Funding Statement

CH would like to acknowledge the financial support of the CogniGron research center and the Ubbo Emmius Funds (Univ. of Groningen). TO’s research was partially supported by the AFOSR grant FA9550-22-0238 and the NSF grant DMS-1811428.


The authors are very grateful for valuable comments received from an anonymous referee and an anonymous Associate Editor. These comments helped the authors to improve the quality of the paper. The third author is thankful for fruitful discussions with Yogeshwaran D. and Z. Wei, which has helped him to deduce the desired M0-convergence in Theorem 3.1 by means of [1, Theorem 6.4].


Download Citation

Christian Hirsch. Taegyu Kang. Takashi Owada. "Large deviations for the volume of k-nearest neighbor balls." Electron. J. Probab. 28 1 - 27, 2023.


Received: 24 October 2022; Accepted: 24 May 2023; Published: 2023
First available in Project Euclid: 2 June 2023

MathSciNet: MR4596351
zbMATH: 1519.60040
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP965

Primary: 60F10
Secondary: 60D05 , 60G55

Keywords: k-nearest neighbor ball , large deviation principle , M0-convergence , point process , Stochastic geometry

Vol.28 • 2023
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