Central limit theorems for the radial spanning tree

TL;DR

This paper establishes central limit theorems with convergence rates for edge functionals, including total edge length, of the radial spanning tree constructed from a Poisson point process in a convex set.

Contribution

It provides the first asymptotic expectation, variance, and CLT results with convergence rates for radial spanning trees in Euclidean space.

Findings

Asymptotic expectation and variance formulas derived

Central limit theorems with explicit convergence rates proved

Results apply to total edge length and similar functionals

Abstract

Consider a homogeneous Poisson point process in a compact convex set in $d$-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.