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July 2003 The depth first processes of Galton--Watson trees converge to the same Brownian excursion
Jean-François Marckert, Abdelkader Mokkadem
Ann. Probab. 31(3): 1655-1678 (July 2003). DOI: 10.1214/aop/1055425793


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.


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Jean-François Marckert. Abdelkader Mokkadem. "The depth first processes of Galton--Watson trees converge to the same Brownian excursion." Ann. Probab. 31 (3) 1655 - 1678, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1049.05026
MathSciNet: MR1989446
Digital Object Identifier: 10.1214/aop/1055425793

Primary: 05C05 , 60F99 , 60G50 , 60J80

Keywords: Brownian excursion , depth , Galton--Watson trees , ladder variable , moderate deviations. , simple tree , subtree

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 3 • July 2003
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