Tail probability of maximal displacement in critical branching Lévy process with stable branching

Consider a critical branching Lévy process $\{X_t,t\ge 0\}$ with branching rate $\beta >0,$ offspring distribution $\{p_k:k\ge 0\}$ and spatial motion $\{{\xi }_t,{\mathrm{\Pi }}_x\}$. For any $t\ge 0$, let $N_t$ be the collection of particles alive at time $t$, and, for any $u\in N_t$, let $X_u(t)$ be the position of $u$ at time $t$. We study the tail probability of the maximal displacement $M:={\sup }_{t>0}{\sup }_{u\in N_t}{X}_u(t)$ under the assumption ${\lim }_{n\to \mathrm{\infty }}{n}^{\alpha }{\sum }_{k=n}^{\mathrm{\infty }}{p}_k=\kappa \in (0,\mathrm{\infty })$ for some $\alpha \in (1,2)$, ${\mathrm{\Pi }}_0({\xi }_1)=0$ and ${\mathrm{\Pi }}_0(\mathrm{\mid }{\xi }_1{\mathrm{\mid }}^r)\in (0,\mathrm{\infty })$ for some $r>2\alpha \mathrm{/}(\alpha -1)$. Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of which are proved under the assumption ${\sum }_{k=0}^{\mathrm{\infty }}{k}^3{p}_k<\mathrm{\infty }$.