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

We consider a branching Brownian motion on RR in which one particle splits into 1+X1+X children. There exists a critical value λ¯ in the sense that λ¯ 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 λ¯ is closely connected with the rescaled Laplace transform of the limit of the so-called derivative martingale ∂Wt(λ¯). Thus special interest is put on the property of its limit ∂W(λ¯). 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, ∂W(λ¯)>0 if EX(log+X)2+δ<+∞EX(log+X)2+δ<+∞ for some δ>0δ>0 while ∂W(λ¯)=0 if EX(log+X)2−δ=+∞EX(log+X)2−δ=+∞. It is conjectured that ∂W(λ¯) is non-degenerate if and only if EX(log+X)2<+∞EX(log+X)2<+∞. The purpose of this article is to prove this conjecture.