The Annals of Applied Probability

The lineage process in Galton–Watson trees and globally centered discrete snakes

Jean-François Marckert

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Abstract

We consider branching random walks built on Galton–Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of “globally centered discrete snake” that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when n goes to +∞, “globally centered discrete snakes” converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton–Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node u is the vector indexed by (k, j) giving the number of ancestors of u having k children and for which u is a descendant of the jth one]. Some consequences concerning Galton–Watson trees conditioned by the size are also derived.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 1 (2008), 209-244.

Dates
First available in Project Euclid: 9 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1199890021

Digital Object Identifier
doi:10.1214/07-AAP450

Mathematical Reviews number (MathSciNet)
MR2380897

Zentralblatt MATH identifier
1140.60042

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles 60J65: Brownian motion [See also 58J65]

Keywords
Galton–Watson trees discrete snake Brownian snake limit theorem

Citation

Marckert, Jean-François. The lineage process in Galton–Watson trees and globally centered discrete snakes. Ann. Appl. Probab. 18 (2008), no. 1, 209--244. doi:10.1214/07-AAP450. https://projecteuclid.org/euclid.aoap/1199890021


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