Gaussian approximation for rooted edges in a random minimal directed spanning tree. Issue 3 (15th December 2021) - Bhattacharjee, Chinmoy
Abstract: We study the total α α ‐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 ≥ 1 s1 on the unit cube [ 0, 1 ] d [0, 1]^d for d ≥ 3 d3 . While a Dickman limit was proved in Penrose and Wade (2004) in the case of d = 2 d=2 , in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when α = 1 α =1 , with a rate of convergence of the order ( log s ) − ( d − 2 ) / 4 ( log log s ) ( d + 1 ) / 2 (s)^-(d-2)/4(loglogs)^(d+1)/2 . In this article, we extend these results and prove a central limit theorem in any dimension d ≥ 3 d3 for any α > 0 α >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 (s)^-(d-2)/2 on the Wasserstein and the Kolmogorov distances between the distribution of the total α α ‐powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.