Invariance principles for pruning processes of Galton-Watson trees

TL;DR

This paper establishes a limit theory connecting pruning processes of Galton-Watson trees with Levy trees, solving an open problem and demonstrating convergence of scaled processes in the Skorohod topology.

Contribution

It proves the convergence of pruning processes of Galton-Watson forests to Levy forests, bridging two previously separate areas of study.

Findings

Convergence of Galton-Watson pruning processes to Levy forests.

Separate treatment of pruning at branch points and edges.

Application to ascension times and Kesten trees.

Abstract

Pruning processes $(\mathcal{F}(\theta),\theta\geq 0)$ have been studied separately for Galton-Watson trees and for L\'evy trees/forests. We establish here a limit theory that strongly connects the two studies. This solves an open problem by Abraham and Delmas, also formulated as a conjecture by L\"ohr, Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson forests $\mathcal{F}_n$, $n\geq 1$, in the domain of attraction of a L\'evy forest $\mathcal{F}$, suitably scaled pruning processes $(\mathcal{F}_n(\theta),\theta\geq 0)$ converge in the Skorohod topology on cadlag functions with values in the space of (isometry classes of) locally compact real trees to limiting pruning processes. We separately treat pruning at branch points and pruning at edges. We apply our results to study ascension times and Kesten trees and forests.