Scaling limits of branching random walks and branching stable processes

TL;DR

This paper investigates the conditions under which branching random walks converge to branching-stable processes, expanding understanding of their scaling limits and contrasting with previous asymptotic results that involved shifting or demagnification.

Contribution

It provides explicit criteria for the convergence of branching random walks to branching-stable processes under proper rescaling, highlighting new scaling limits in the field.

Findings

Derived explicit conditions for convergence to branching-stable processes

Contrasted rescaling limits with previous shifting or demagnification results

Enhanced understanding of the asymptotic behavior of branching random walks

Abstract

Branching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanism for point processes. Here, we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results that have been obtained during the last decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.