Invariance principles for pruning processes of Galton-Watson trees

Pruning processes (F(θ), θ ≥ 0) have been studied separately for Galton-Watson trees and for Lévy 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 Löhr, Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson forests Fn, n ≥ 1, in the domain of attraction of a Lévy forest F , suitably scaled pruning processes (Fn(θ), θ ≥ 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. AMS 2010 subject classifications: Primary 60J80; Secondary 60J25, 60F17.