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

We consider a branching Brownian motion on R in which one particle splits into 1 + X children. There exists a critical value (lambda) under bar in the sense that (lambda) under bar 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 (lambda) under bar is closely connected with the rescaled Laplace transform of the limit of the so-called derivative martingale partial derivative W-t((lambda) under bar). Thus special interest is put on the property of its limit partial derivative W((lambda) under bar. Kyprianou [Kyprianou, A.E., 2004. Traveling wave solutions to the K-P-P equation: alternatives to Simon Harris' probability analysis. Ann. Inst. H. Poincare 40,53-72.] proved that, partial derivative W((lambda) under bar) > 0 if EX(log(+) X)(2+delta) < +infinity for some delta > 0 while partial derivative W ((lambda) under bar) = 0 if EX (log(+) X)(2-delta) = +infinity. It is conjectured that partial derivative W((lambda) under bar is non-degenerate if and only if EX (log(+) X)(2) < +infinity. The purpose of this article is to prove this conjecture. (C) 2010 Elsevier B.V. All rights reserved.