Seneta-Heyde norming in the branching random walk

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 nth 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.