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

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".