Discrete-time Branching Random Walks and Biggins' Theorem

An initial particle at the origin of the real line gives birth to L particles,
each of which having random displacement from its parent. The process
reiterates: the particles in the new generation all give birth to
a random number of particles, like in the general Galton-Watson process,
with random displacements. This goes on forever or until the process dies.
For suitable values, we discuss a theorem on martingale convergence
on Branching Random Walks, under a "X log X" condition, and
provide a conceptual proof. A limit theorem on the Galton-Watson
process will be derived, with the use of size-biased trees, and possible
problem directions will be discussed.