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

TL;DR

This paper establishes a limit theorem for the contour process of conditioned Galton-Watson trees, showing convergence to a continuous process related to stable CSBPs, extending known results in the Brownian case.

Contribution

It introduces a new convergence result for the contour process of conditioned Galton-Watson trees to a stable CSBP-related process, generalizing Aldous's Brownian case.

Findings

Rescaled height process converges to a stable height process

Contour process converges to a limit related to the stable CSBP

Extends continuum random tree results to stable branching mechanisms

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.