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

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.