  We consider the boundary case (in the sense of Biggins and Kyprianou
[Electron. J. Probab. 10 (2005) 609--631] in a one-dimensional super-critical
branching random walk, and study the additive martingale $(W_n)$. We prove
that, upon the system's survival, $n^{1/2}W_n$ converges in probability, but
not almost surely, to a positive limit. The limit is identified as a constant
multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in
Appl. Probab. 36 (2004) 544--581], of the derivative martingale.

Probability