The Annals of Probability
- Ann. Probab.
- Volume 31, Number 3 (2003), 1655-1678.
The depth first processes of Galton--Watson trees converge to the same Brownian excursion
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.
Ann. Probab., Volume 31, Number 3 (2003), 1655-1678.
First available in Project Euclid: 12 June 2003
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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.)
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. https://projecteuclid.org/euclid.aop/1055425793