Aarhus Universitets segl

Large deviations for the volume of $k$-nearest neighbor balls

Takashi Owada (Purdue University)
Tirsdag 9. juli 2024 13:15–14:00 Aud. D4 (1531-219)
Stochastics seminar

This work develops the large deviations theory for the point process associated with the Euclidean and hyperbolic volume of k-nearest neighbor ($k$-NN) balls. In the Euclidean setting, the $k$-NN balls are centered around the points of a homogeneous Poisson process in the torus. In the hyperbolic setting, we consider a stationary Poisson point process of unit intensity in a growing sampling window in the hyperbolic space. Two different types of large deviation behaviors are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-NN balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes. Joint work with Christian Hirsch, Taegyu Kang, Moritz Otto, and Christoph Thäle.

Organiseret af: Stochastics Group
Kontakt: Andreas Basse-O'Connor Revideret: 20.06.2024