A limit theorem for the contour process of conditioned Galton-Watson trees

Abstract

In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index $\alpha \in (1,2]$. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced \cite{LGLJ1, LGLJ1}. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one ancestor and conditioned on the total progeny, converges in a functional sense, to a new process: the normalized excursion of the continuous height process associated with the $\alpha$-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case $\alpha =2$, the limiting process is the normalized Brownian excursion that codes the continuum random tree: the result is due to Aldous who used a different method.

30 pages; 2 figures; 2002

Cited by in corpus (11)

• Random trees and applications
• Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
• Small-time behavior of beta coalescents
• Random stable looptrees
• Local limits of conditioned Galton-Watson trees I: the infinite spine case
• The gap between Gromov-vague and Gromov-Hausdorff-vague topology
• A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees
• Triangulating stable laminations
• Convergence of bi-measure R-tree and the pruning process
• Tree limits and limits of random trees
• Critical trees are neither too short nor too fat