In the discrete-time supercritical branching random walk, there is a
Kesten-Stigum type result for the martingales formed by the Laplace transform
of the $n$th generation positions. Roughly, this says that for suitable values
of the argument of the Laplace transform the martingales converge in mean
provided an "$X \log X$" condition holds. Here it is established that when this
moment condition fails, so that the martingale ..converges to zero, it is
possible to find a (Seneta-Heyde) renormalization of the martingale that
converges in probability to a finite nonzero limit when the process survives.
As part of the proof, a Seneta-Heyde renormalization of the general
(Crump-Mode-Jagers) branching process is obtained; in this case the convergence
holds almost surely. The results rely heavily on a detailed study of the
functional equation that the Laplace transform of the limit must satisfy.