Reference : Gaussian approximation in random minimal directed spanning trees
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Gaussian approximation in random minimal directed spanning trees
Bhattacharjee, Chinmoy mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
Random Structures and Algorithms
John Wiley & Sons
Yes (verified by ORBilu)
[en] We study the total $\alpha$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in \cite{BR04} - 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 \cite{PW04} in the case of $d=2$, in dimensions three and higher, \cite{BLP06} 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

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