Point process convergence for branching random walks with regularly varying steps

Abstract

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of Durrett (1983) in our framework.

22 pages, 2 figures, To appear in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Proof of Lemma 3.4 differs from previous version