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 $$ \alpha $$ ‐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 ≥ 1 $$ s\ge 1 $$ on the unit cube [ 0, 1 ] d $$ {\left[0, 1\right]}^d $$ for d ≥ 3 $$ d\ge 3 $$ . 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 $$ \alpha =1 $$, with a rate of convergence of the order ( log s ) − ( d − 2 ) / 4 ( log log s ) ( d + 1 ) / 2 $$ {\left(\log s\right)}^{-\left(d-2\right)/4}{\left(\mathrm{loglog}s\right)}^{\left(d+1\right)/2} $$ . In this article, we extend these results and prove a central limit theorem in any dimension d ≥ 3 $$ d\ge 3 $$ for any α > 0 $$ \alpha >0 $$ . 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 $$ {\left(\log s\right)}^{-\left(d-2\right)/2} $$ on the Wasserstein and the Kolmogorov distances between the distribution of the total α $$ \alpha $$ ‐powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.