Limit theorem for derivative martingale at criticality w.r.t branching Brownian motion

We consider a branching Brownian motion on in which one particle splits into 1+X children. There exists a critical value in the sense that is the lowest velocity such that a traveling wave solution to the corresponding Kolmogorov-Petrovskii-Piskunov equation exists. It is also known that the traveling wave solution with velocity is closely connected with the rescaled Laplace transform of the limit of the so-called derivative martingale . Thus special interest is put on the property of its limit . Kyprianou [Kyprianou, A.E., 2004. Traveling wave solutions to the K-P-P equation: alternatives to Simon Harris' probability analysis. Ann. Inst. H. Poincaré 40, 53-72.] proved that, if EX(log+X)2+[delta]<+[infinity] for some [delta]>0 while if EX(log+X)2-[delta]=+[infinity]. It is conjectured that is non-degenerate if and only if EX(log+X)2<+[infinity]. The purpose of this article is to prove this conjecture.