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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 \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) 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 \(Z_n\).

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