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

Abstract

In this work, we study asymptotics of the genealogy of Gallon-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 [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 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.