Branching Random Walks, Stable Point Processes and Regular Variation

Extremes of multitype branching random walks: heaviest tail wins

“…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.…”

“…The HL convergence in M 0 = M ( R0 ) \ {∅} is discussed in and used by [26,21] in the context of large deviation for point processes and in [10] in the context of branching random walk.…”

“…A point process on R0 is an M ( R0 )-valued random variable defined on (Ω, F, P) that does not charge any mass to ±∞. The definition of strict stability of such point processes is introduced by [18] and has been shown to be connected to regular variation on M 0 in [10]. We quote the following result from [10], which gives a sufficient condition for a point process to be in the superposition domain of attraction of a StαS point process.…”

“…Then we follow the core of the idea used in Bhattacharya et al [10], but here we carry out their arguments with larger complexity. The proof of the main result is broken up into four basic steps.…”

A note on randomly scaled scale-decorated Poisson point processes

“…As a consequence, we obtain a characterization for ScDPPP based on scaled-Laplace functional. This characterization also has been used in Bhattacharya et al [2017b] and Bhattacharya et al [2016] to verify that the limiting extremal process is SScDPPP in the context of branching random walk with displacements having regularly varying tail. In the former article, this characterization has also been used to establish that the approximately scaled superposition of regularly varying point processes converges to StαS point process.…”