Long edges in Galton-Watson trees

TL;DR

This paper investigates the asymptotic behavior of the longest edges in genealogical trees generated by continuous-time Galton-Watson processes, extending previous results on pendant edges in birth-death models.

Contribution

It provides new results on the limiting behavior of various types of longest edges in Galton-Watson trees over large time scales.

Findings

Asymptotic distribution of longest pendant edges

Behavior of longest interior edges over time

Extension of previous birth-death process results

Abstract

In this article, we will establish a number of results concerning the limiting behaviour of the longest edges in the genealogical tree generated by a continuous-time Galton-Watson (GW) process. Separately, we consider the large time behaviour of the longest pendant edges, the longest (strictly) interior edges, and the longest of all the edges. These results extend the special case of long pendant edges of birth-death processes established in Bocharov, Harris, Kominek, Mooers, and Steel [1] .