A note on the critical barrier for the survival of α-stable branching random walk with absorption

We consider a branching random walk with an absorbing barrier, where the step of the associated one-dimensional random walk is in the domain of attraction of an $\alpha$-stable law with $1<\alpha<2$. We shall prove that there is a barrier $an^{\frac{1}{1+\alpha}}$ and a critical value $a_\alpha$ such that if $a<a_\alpha$, then the process dies; if $a>a_\alpha$, then the process survives. The results generalize previous results in literature for the case $\alpha=2$.