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

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dc.creatorMarckert, Jean-François
dc.date2008-01-22
dc.date.accessioned2026-07-07T08:56:25Z
dc.date.available2026-07-07T08:56:25Z
dc.descriptionWe 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 $+\infty$, ``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 $j$th one]. Some consequences concerning Galton--Watson trees conditioned by the size are also derived.
dc.descriptionPublished in at http://dx.doi.org/10.1214/07-AAP450 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/0801.3330
dc.identifierhttp://arxiv.org/abs/0801.3330
dc.identifierAnnals of Applied Probability 2008, Vol. 18, No. 1, 209-244
dc.identifierdoi:10.1214/07-AAP450
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/146602
dc.subjectProbability
dc.subject60J80, 60F17, 60J65 (Primary)
dc.titleThe lineage process in Galton--Watson trees and globally centered discrete snakes
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