Weak convergence of the extremes of branching Lévy processes with regularly varying tails

In this paper, we study the weak convergence of the extremes of supercritical branching Lévy processes $\{{\mathbb{X}}_t,t\ge 0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, ${\mathbb{X}}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of ${\mathbb{X}}_t$.