2005/09/22 by Thomas Duquesne, Duquesne, Thomas · 2 citations
Mathematics · #FOS: Mathematics #G22 #G3 #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:G22 #msc:G3
paper · pdf · doi:10.48550/arxiv.math/0509522
30 pages; 2 figures; 2002
arxiv created 2005/09/22 · openalex publication_date 2005/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈ (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 \citeLGLJ1, 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.