Martingale methods have played an important role in the theory of Galton-Watson processes and branching random walks. The (random) Fourier transform of the position of the particles in the $n$th generation, normalized by its mean, is a martingale. Under second moments assumptions on the branching this has been very useful to study the asymptotics of the branching random walk. Using a different normalization, we obtain a new martingale which is in $L^2$ under weak assumptions on the displacement of the particles and strong assumptions on the branching.
"A New Martingale in Branching Random Walk." Ann. Appl. Probab. 3 (4) 1145 - 1150, November, 1993. https://doi.org/10.1214/aoap/1177005276