Convergence Rate for The Number of Crossing in a Random Labelled Tree

Santiago Arenas-Velilla; Octavio Arizmendi

Convergence Rate for The Number of Crossing in a Random Labelled Tree

Probability 2022-09-21 v1

Authors: Santiago Arenas-Velilla , Octavio Arizmendi

Abstract

We consider the number of crossings in a random labelled tree with vertices in convex position. We give a new proof of the fact that this quantity is asymptotically Gaussian with mean $n^2/6$ and variance $n^3/45$ . Furthermore, we give an estimate for the Kolmogorov distance to a Gaussian distribution which implies a convergence rate of order $n^{-1/2}$ .

Keywords

random labelled tree number of crossings asymptotic gaussian kolmogorov distance convergence rate convex position

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@article{arxiv.2209.09431,
  title  = {Convergence Rate for The Number of Crossing in a Random Labelled Tree},
  author = {Santiago Arenas-Velilla and Octavio Arizmendi},
  journal= {arXiv preprint arXiv:2209.09431},
  year   = {2022}
}

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9 pages. arXiv admin note: text overlap with arXiv:2205.03995

Convergence Rate for The Number of Crossing in a Random Labelled Tree

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