Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

We consider a branching random walk on ({mathbb {R}}) with a stationary and ergodic environment (xi =(xi _n)) indexed by time (nin {mathbb {N}}). Let (Z_n) be the counting measure of particles of generation n and (tilde{Z}_n(t)=int mathrm{e}^{tx}Z_n(mathrm{d}x)) be its Laplace transform. We show the (L^p) convergence rate and the uniform convergence of the martingale (tilde{Z}_n(t)/{mathbb {E}}[tilde{Z}_n(t)|xi ]), and establish a moderate deviation principle for the measures (Z_n).