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

TL;DR

This paper studies the convergence of globally centered discrete snakes derived from Galton--Watson trees to the Brownian snake, extending classical models by analyzing lineage structures and their probabilistic limits.

Contribution

It introduces the concept of globally centered discrete snakes and proves their convergence to the Brownian snake under certain conditions, extending existing models.

Findings

Globally centered discrete snakes converge to the Brownian snake.

Lineage analysis links Galton--Watson trees to multinomial processes.

Derived consequences for conditioned Galton--Watson trees.

Abstract

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.