Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space

Holger Sambale, Christoph Thäle and Tara Trauthwein

Stochastic Processes and their Applications, 2025, vol. 188, issue C

Abstract: Consider a stationary Poisson process η in the d-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set η as follows. First, each point x∈η is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until x is contained in the convex hull of the points already connected to x. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.

Keywords: Central limit theorem; Hyperbolic stochastic geometry; Nearest neighbour embracing graph; Poisson process; Random geometric graph; Stabilizing functional; Stochastic geometry (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104671

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