Extremes of multitype branching random walks: heaviest tail wins

We consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index α , while the other types of particles have lighter tails than the particles of type Q . In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the n th generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the n th generation.