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

TL;DR

This paper proves that the number of crossings in a random labelled tree with vertices in convex position follows an asymptotically Gaussian distribution, providing a new proof and convergence rate estimate.

Contribution

It offers a new proof of the Gaussian limit law for crossings and quantifies the convergence rate to the Gaussian distribution.

Findings

Number of crossings is asymptotically Gaussian with mean n^2/6 and variance n^3/45

Convergence rate to Gaussian distribution is of order n^{-1/2}

Provides an estimate for the Kolmogorov distance to the Gaussian distribution

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 $n^{-1/2}$.