Large deviations and almost sure convergence for the extremes of branching Lévy processes with regularly varying tails

In this paper, we study the asymptotic behaviors of supercritical branching Markov processes {Xt,t≥0} with spatial motions being Lévy processes having regularly varying tails. Recently, Ren et al. [J. Appl. Probab. 61 (2024)] proved that, when renormalized by some appropriate function h(t), {Xt,t≥0} converges weakly to some random measure. In this paper, we first establish the large deviations of {Xt,t≥0}. More precisely, we prove that, (i) when renormalized by a function Λ(t) which grows faster than h(t), {Xt,t≥0}, conditioned on {Rt>Λ(t)}, converges weakly; (ii) when renormalized by a function Λ(t) which grows slower than h(t), {Xt,t≥0}, conditioned on {Rt<Λ(t)}, converges weakly, where Rt is the maximum of Xt. We also characterize the almost sure envelopes of Rt.