Weakly supercritical branching process in unfavourable environment

Let $\{Z_{n}\}$ be a weakly supercritical branching process in a random environment, and $\{S_{n}\}$ be its associated random walk. We consider a natural martingale $W_{n}=Z_{n}\exp(-S_{n})$, where $n\geq 0$. We prove two limit theorems for the random process $W_{\lfloor nt\rfloor}$, where $t\in [0,1]$, which is considered either under the condition on the unfavourable environment $\{\max_{1\leq i\leq n}S_{i}\}$ or under the condition on the unfavourable environment $\{S_{n}\leq u\}$, where $u$ is some positive constant.