  We consider the general branching random walk under minimal assumptions,
which in particular guarantee that the empirical particle distribution admits
an almost sure central limit theorem. For such a process, we study the large
time decay of the probability that the fraction of particles in a typical set
deviates from its typical value. We show that such probabilities decay doubly
exponentially with speed which is either linear in or the square root of the
number of generations, depending on the set and the magnitude of the deviation.
We also find the rate of decay in each case.
