Globally centered discrete snakes

Details Globally centered discrete snakes Globally centered discrete snakes

2006-06-14

arxiv.org/abs/math/0606338v1

Mathematics

Probability

Scientific paper

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 $+\infty$, "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. Some consequences concerning Galton-Watson trees conditioned by the size are also derived.

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