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April 2003 A limit theorem for the contour process of condidtioned Galton--Watson trees
Thomas Duquesne
Ann. Probab. 31(2): 996-1027 (April 2003). DOI: 10.1214/aop/1048516543


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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Thomas Duquesne. "A limit theorem for the contour process of condidtioned Galton--Watson trees." Ann. Probab. 31 (2) 996 - 1027, April 2003.


Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1025.60017
MathSciNet: MR1964956
Digital Object Identifier: 10.1214/aop/1048516543

Primary: 05G05 , 60F17 , 60G17 , 60G52

Keywords: conditioned Galton--Watson tree , limit theorem , stable continuous random tree

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 2 • April 2003
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