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

We give an explicit description of the law terminal value W additive martingales in a remarkable branching stable process. show that right tail probability decays exponentially fast and left follows ?logP(W<x)?12(logx)2 as x?0+. These are sharp contrast with results literature such Liu (2000, 2001) Buraczewski (2009). further is self-decomposable, therefore, possesses unimodal density. specify asymptotic behavior at 0 +? latter.