A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees

TL;DR

This paper provides a simple proof of Duquesne's theorem, showing that the rescaled contour process of certain conditioned Galton-Watson trees converges to a stable Lévy process excursion, using an innovative absolute continuity approach.

Contribution

It introduces a new, robust proof technique for convergence of contour processes of conditioned Galton-Watson trees, extending to trees with prescribed degree distributions.

Findings

Rescaled contour functions converge to stable Lévy process excursions.

The proof method is adaptable to trees with fixed degree constraints.

The approach simplifies and generalizes previous invariance theorems.

Abstract

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton-Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index $\theta \in (1,2]$, conditioned on having total progeny $n$, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive L\'evy process of index $\theta$. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly $n$ and the conditional probability of having total progeny at least $n$. This new method is robust and can be adapted to establish invariance theorems for Galton-Watson trees having $n$ vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.