# Convergence to a Continuous State Branching Process with jumps and Height Process

## TL;DR

This paper investigates the asymptotic behavior of Galton-Watson genealogies, demonstrating convergence of rescaled processes to a continuous state branching process with jumps and the associated height process.

## Contribution

It establishes the convergence of the rescaled height process of Galton-Watson trees to the continuous height process introduced by Le Gall and Le Jan.

## Key findings

- Rescaled Galton-Watson processes converge to CSBP with jumps.
- Rescaled height processes converge to the continuous height process.
- Provides a functional convergence result for genealogical structures.

## Abstract

In this work, we study asymptotics of the genealogy of Galton-Watson processes. Thus we consider a offspring distribution such that the rescaled Galton-Watson processes converges to a continuous state branching process (CSBP) with jumps. After we show that the rescaled height (or exploration) process of the corresponding Galton-Watson family tree, converges in a functional sense, to the continuous height process that Le Gall and Le Jan introduced in 1998 on their paper "branching processes in L\'evy processes : The exploration process".