Extremes of multitype branching random walks: heaviest tail wins

Abstract

We consider a branching random walk on a multi(Q)-type, supercritical Galton-Watson tree which satisfies 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 that of particles of type Q. In this article, 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 (SScDPPP) using the tools developed in [10]. As a consequence, we shall obtain the asymptotic distribution of the position of the rightmost particle in the n th generation.

Weak convergence of the extremes of branching Lévy processes with regularly varying tails

“…Recently, the extremal processes of the branching random walks with regularly varying steps were studied in [8, 9], where it was proved that the point random measure , where is the position of v , converges weakly to a Cox cluster process, which is quite different from the case with exponential moments. See also [10, 25] for related works on branching random walks with heavy-tailed displacements.…”

Weak convergence of the extremes of branching Lévy processes with regularly varying tails

“…In [8,9], it was proved that the point random measures |v|=n δ a −1 n Sv , where S v is the position of v, converges weakly to a Cox cluster process, which are quite different from the case with exponential moments. See also [10,24] for related works on branching random walks with heavy-tailed displacements.…”

Extreme positions of regularly varying branching random walk in random environment

“…We can also consider a multi-type BRW with branching process in i.i.d. environment (see [14], [13]).…”