Gaussian approximation for rooted edges in a random minimal directed spanning tree

TL;DR

This paper proves a Gaussian limit theorem for the total alpha-powered length of rooted edges in a random minimal directed spanning tree on a Poisson process in high dimensions, extending previous results and providing explicit convergence bounds.

Contribution

It generalizes the central limit theorem for the total alpha-powered length to all dimensions d ≥ 3 and all alpha > 0, with explicit non-asymptotic bounds using Stein's method.

Findings

Proved a CLT for all dimensions d ≥ 3 and alpha > 0.

Derived explicit bounds on Wasserstein and Kolmogorov distances.

Extended previous results from specific cases to a general setting.

Abstract

We study the total $\alpha$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in Bhatt and Roy (2004) - on a Poisson process with intensity $s \ge 1$ on the unit cube $[0,1]^d$ for $d \ge 3$. While a Dickman limit was proved in Penrose and Wade (2004) in the case of $d=2$, in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when $\alpha=1$, with a rate of convergence of the order $(\log s)^{-(d-2)/4} (\log \log s)^{(d+1)/2}$. In this paper, we extend these results and prove a central limit theorem in any dimension $d \ge 3$ for any $\alpha>0$. Moreover, making use of recent results in Stein's method for region-stabilizing functionals, we provide presumably optimal non-asymptotic bounds of the order $(\log s)^{-(d-2)/2}$ on the Wasserstein and the Kolmogorov distances between the distribution of the total $\alpha$-powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.