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 $n\in \mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$ and $\tilde Z_n(t)=\int e^{tx}Z_n(dx)$beitsLaplacetransform\mathrm{.}Weshowthe$L^p$ convergence rate and the uniform convergence of the martingale $\tilde Z_n(t)/\mathbb E[\tilde Z_n(t)|\xi]$,andestablishamoderatedeviationprincipleforthemeasures$Z_n$.