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

• 成果类型: Article
• 署名作者: Marckert, Jean-Francois
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Bordeaux
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP450
• 发表日期: 2008
• 页码: 209-244
• 关键词: integrated superbrownian excursion; limit-theorem; embedded trees; CONVERGENCE; SPACES; tour; LAWS

摘要

We consider branching random walks built on Galton-Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their resealed 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 +infinity, 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.