Martingale ratio convergence in the branching random walk

We consider the boundary case in a one-dimensional supercritical branching random walk, and study two of the most important martingales: the additive martingale (Wn) and the derivative martingale (Dn). It is known that upon the system's survival, Dn has a positive almost sure limit (Biggins and Kyprianou [9]), whereas Wn converges almost surely to 0 (Lyons [22]). Our main result says that after a suitable normalization, the ratio Wn/Dn converges in probability, upon the system's survival, to a positive constant