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

2022/09/20 by Arenas-Velilla, Santiago, Arizmendi, Octavio
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2209.09431

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 n²/6 and variance n³/45. Furthermore, we give an estimate for the Kolmogorov distance to a Gaussian distribution which implies a convergence rate of order n^-1/2.

Related

• On the number of crossings in a random labelled tree with vertices in convex position
• Crossings in Randomly Embedded Graphs
• Convergence rates of limit theorems in random chord diagrams
• Central limit theorem for crossings in randomly embedded graphs
• Weak convergence of the number of vertices at intermediate levels of\n random recursive trees