A note on conditioned Galton-Watson trees I: the infinite spine case.

Romain Abraham (1) , Jean-François Delmas (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.

Mots clés: Conditioned Galton-Watson tree; Kesten's tree

Domaines: Probabilités [math.PR]

Romain Abraham, Jean-François Delmas. A note on conditioned Galton-Watson trees I: the infinite spine case.. 2013. ⟨hal-00813145v1⟩