Branching random walks, stable point processes and regular variation
Stochastic Processes and their Applications
Using the theory of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is used to obtain the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. Because of heavy tails of the step size distribution, we can invoke a one large jump principle at the level of point processes to give an explicit representation of the limiting point process. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.
Bhattacharya, Ayan; Hazra, Rajat Subhra; and Roy, Parthanil, "Branching random walks, stable point processes and regular variation" (2018). Journal Articles. 1627.