Article (Scientific journals)
Gaussian approximation in random minimal directed spanning trees
Bhattacharjee, Chinmoy
2021In Random Structures and Algorithms, p. 1-31
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Abstract :
[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 variable.
Disciplines :
Mathematics
Author, co-author :
Bhattacharjee, Chinmoy ;  University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
External co-authors :
no
Language :
English
Title :
Gaussian approximation in random minimal directed spanning trees
Publication date :
2021
Journal title :
Random Structures and Algorithms
ISSN :
1042-9832
Publisher :
John Wiley & Sons, Hoboken, United States - New Jersey
Pages :
1-31
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBilu :
since 11 May 2021

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