Branching random walks with random environments in time

We consider a branching random walk on R with a random environment in time (denoted by ξ). Let Z_n be the counting measure of particles of generation n, and let Z ?n(t) be its Laplace transform. We show the convergence of the free energy n^-1logZ ?n(t), large deviation principles, and central limit theorems for the sequence of measures {Z_n}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Z ?n(t)/EZ ?n(t)|ξ].