The limit distribution of the maximum probability nearest-neighbour ball

Abstract

Let X 1 , . . . , X n be independent random points drawn from an absolutely continuous probability measure with density f in R d . Under mild conditions on f , we derive a Poisson limit theorem for the number of large probability nearest neighbor balls. Denoting by P n the maximum probability measure of nearest neighbor balls, this limit theorem implies a Gumbel extreme value distribution for nP n − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nP n − ln n, which does not depend on f .

Poisson process approximation under stabilization and Palm coupling

“…As mentioned before, we are not aware of any quantitative result for binomial point processes even with a weaker rate of convergence or under weaker approximation distance in the literature. For a recent Poisson approximation result of nearest neighbor balls (i.e., k = 1) of a binomial point process in R d see [19]. Theorem 6.5.…”

“…, E 6 from Theorem 5.1. Using the above expression for L, we obtain thatE 1 ≤ 2n B P β n−1 (B rn(x,b) (x)) ≤ k − 1 λ(x)dx ≤ C ′ 1 e −b 1 + (k − 1) log log n + b log n For n ∈ N and x ∈ X define U x = B rn(x,b 0 ) (x).It follows from (6.18) and (6 19…”

Poisson approximation of Poisson-driven point processes and extreme values in stochastic geometry

“…In the language of Section 2 the process ξ s describes the thinning that retains every point x ∈ η s ∩ [0, 1] d with distance larger than r s (x) to its (k + 1)-nearest neighbor. For k = 0 it was shown in [10] that for binomial input with increasing intensity the maximal volume content of a nearest neighbor ball is in the domain of attraction of a Gumbel distribution. Choosing t = 1, λ = λ s from (4.1) and…”

“…In Section 4 this result is applied to a thinning of points with large volume content of its k-nearest neighbor ball. In the case k = 1 a similar problem was studied for binomial input in [10]. Our proof uses geometric estimates that are also applied in Section 5, where we study cells in the Poisson-Voronoi mosaic that are large w.r.t.…”

Criteria for Poisson process convergence with applications to inhomogeneous Poisson-Voronoi tessellations

“…As a corollary of the previous theorems, we have the following generalization to the inhomogeneous case of the result obtained in [8, Theorem 1, Equation (2a)] for the stationary case; see also [11,Section 5] for the maximal inradius of a stationary Poisson-Voronoi tessellation and of a stationary Gauss-Poisson-Voronoi tessellation. For an underlying binomial point process, (3.4) was shown under similar assumptions in [15]. The related problem of maximal weighted r-th nearest neighbor distances for the points of a binomial point process was studied in [16]; see also [17].…”