On the law of terminal value of additive martingales in a remarkable branching stable process

We give an explicit description of the law of terminal value $W$ of additive martingales in a remarkable branching stable process. We show that the right tail probability of the terminal value decays exponentially fast and the left tail probability follows that $-\log \mathbb{P}(W<x)\sim \frac{1}{2}(\log x{)}^2$ as $x\to 0+$. These are in sharp contrast with results in the literature such as Liu (2000, 2001) and Buraczewski (2009). We further show that the law of $W$ is self-decomposable, and therefore, possesses a unimodal density. We specify the asymptotic behavior at $0$ and at $+\mathrm{\infty }$ of the latter.