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

• 成果类型: Article
• 署名作者: Duquesne, T
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 996-1027
• 关键词: continuum random tree; Levy processes; branching-processes; excursion

摘要

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 a e (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 [21, 22]. 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 a-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.