We consider the branching random walks in d-dimensional integer lattice with time–space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable. When d≥3 and the fluctuation of environment satisfies a certain uniform square integrability then it is nondegenerate and we prove a central limit theorem for the density of the population in terms of almost sure convergence.
"Almost sure central limit theorem for branching random walks in random environment." Ann. Appl. Probab. 21 (1) 351 - 373, February 2011. https://doi.org/10.1214/10-AAP699