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 $+\infty$, ``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.