The depth first processes of Galton--Watson trees converge to the same Brownian excursion
-
Jean-François Marckert [1] ; Abdelkader Mokkadem [1]
-
[1] Université de Versailles St-Quentin
-
- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 3, 2003, págs. 1655-1678
- Idioma: inglés
- DOI: 10.1214/aop/1055425793
- Texto completo no disponible (Saber más ...)
-
Resumen
In this paper, we show a strong relation between the depth first processes associated to Galton--Watson trees with finite variance, conditioned by the total progeny: the depth first walk, the depth first queue process, the height process; a consequence is that these processes (suitably normalized) converge to the same Brownian excursion. This provides an alternative proof of Aldous' one of the convergence of the depth first walk to the Brownian excursion which does not use the existence of a limit tree. The methods that we introduce allow one to compute some functionals of trees or discrete excursions; for example, we compute the limit law of the process of the height of nodes with a given out-degree, and the process of the height of nodes, root of a given subtree.
