Abstract
AbstractWe study the total ‐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 on the unit cube for . While a Dickman limit was proved in Penrose and Wade (2004) in the case of , in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when , with a rate of convergence of the order . In this article, we extend these results and prove a central limit theorem in any dimension for any . Moreover, making use of recent results in Stein's method for region‐stabilizing functionals, we provide presumably optimal non‐asymptotic bounds of the order on the Wasserstein and the Kolmogorov distances between the distribution of the total ‐powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.
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