  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 do- mains 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.
