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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Abstract

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

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

Dates
First available in Project Euclid: 12 June 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1055425793

Digital Object Identifier
doi:10.1214/aop/1055425793

Mathematical Reviews number (MathSciNet)
MR1989446

Zentralblatt MATH identifier
1049.05026

Subjects
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.)

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

Citation

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


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