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 nth 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 nth generation.
|Keywords||Cox process, Extreme value, Multitype branching random walk, Point process, Regular variation, Rightmost point|
|Journal||Advances in Applied Probability|
Bhattacharya, A, Maulik, K, Palmowski, Z, & Roy, P. (2019). Extremes of multitype branching random walks: Heaviest tail wins. Advances in Applied Probability, 51(2), 514–540. doi:10.1017/apr.2019.20