The Annals of Probability

The depth first processes of Galton--Watson trees converge to the same Brownian excursion

Jean-François Marckert and Abdelkader Mokkadem

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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.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1655-1678.

First available in Project Euclid: 12 June 2003

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Zentralblatt MATH identifier

Primary: 05C05: Trees 60F99: None of the above, but in this section 60G50: Sums of independent random variables; random walks 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

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


Marckert, Jean-François; Mokkadem, Abdelkader. The depth first processes of Galton--Watson trees converge to the same Brownian excursion. Ann. Probab. 31 (2003), no. 3, 1655--1678. doi:10.1214/aop/1055425793.

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