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

Thomas Duquesne

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

Thomas Duquesne

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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 $α\in (1, 2]$. 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 $α$-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case $α=2$, 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. 2005-09-22. A limit theorem for the contour process of conditioned Galton-Watson trees. https://arxiv.org/abs/math/0509522

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