Branching random walks, stable point processes and regular variation

Abstract

Abstract. Using the language 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 then 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 [18] and verify that two related predictions of Brunet & Derrida [14] remain valid for this model.

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

“…We temporarily return to the case of arbitrary N and prove the functional theory part of our main result Theorem 1. (1) and (2) are satisfied, then the equations system (5) has a unique solution in the function class C θ , for each θ = (θ 1 , . .…”

“…Recent study of maximum of particles positions in the standard space-homogeneous BRW on real line was carried out in [16], [18] and [20] under condition of "light" tails of the distribution of the random walk jump. The case of "heavy" tails required other handling provided, e.g., in [2], [13] and [17].…”

Extremes of multitype branching random walks: heaviest tail wins

“…The point processes associated with a branching random walk and branching brownian motion have been extensively studied in recent years; see, e.g. [3], [4], [5], [6], [9], [10], and [33]. This approach provides more insight allowing identification of all order statistics of the position of the particle.…”

“…With motivations from [12], [33], and the works mentioned above, we analyze the point process induced by a branching random walk with heavy-tailed displacements, continuing the research initiated by Bhattacharya et al [10], and generalize it to the multitype case, allowing dependence between weights. More formally, in this paper we consider a multitype branching random walk with heavy tailed increments.…”

“…The main step of the proof is based on showing that the 'heaviest' displacement dominates the whole behaviour of our branching random walk. Then we follow the core of the idea used in [10], but here we carry out their arguments with larger complexity. The proof of the main result is broken up into four basic steps.…”