  We consider a branching random walk on $\mathbb{R}$ with a stationary and
ergodic environment $\xi=(\xi_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$
be the counting measure of particles of generation $n$ and $\tilde Z_n(t)=\int
e^{tx}Z_n(dx)$ 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
$Z_n$.
