Almost sure central limit theorem for branching random walks in random environment

Abstract

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.