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 $d\ge 3$ 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.

Published in at http://dx.doi.org/10.1214/10-AAP699 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)