Annals of Probability

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

Thomas Duquesne

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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. 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.

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Ann. Probab., Volume 31, Number 2 (2003), 996-1027.

First available in Project Euclid: 24 March 2003

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Primary: 60F17: Functional limit theorems; invariance principles 05G05 60G52: Stable processes 60G17: Sample path properties

Stable continuous random tree limit theorem conditioned Galton--Watson tree


Duquesne, Thomas. A limit theorem for the contour process of condidtioned Galton--Watson trees. Ann. Probab. 31 (2003), no. 2, 996--1027. doi:10.1214/aop/1048516543.

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