Large deviations and almost sure convergence for the extremes of branching Lévy processes

In this paper, we investigate the asymptotic behavior of supercritical branching Markov processes $\{{\mathbb{X}}_t,t\ge 0\}$ whose spatial motions are Lévy processes with regularly varying tails. Recently, Ren et al. [Appl. Probab. 61 (2024)] studied the weak convergence of the extremes of $\{{\mathbb{X}}_t,t\ge 0\}$. In this paper, we establish the large deviation of $\{{\mathbb{X}}_t,t\ge 0\}$ as well as some almost sure convergence results of the maximum of ${\mathbb{X}}_t$.