Extremes of multitype branching random walks: heaviest tail wins

Abstract

AbstractWe consider a branching random walk on a multitype (withQtypes of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of typeQhave regularly varying tails of index$\alpha$, while the other types of particles have lighter tails than the particles of typeQ. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in thenth 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 thenth generation.