On the maximal displacement of subcritical branching random walk in random environment

In this paper, we consider the subcritical branching random walk in a random environment. We assume the branching and the step jump are independent; and the branching is in random envirenment, i.e., the particles in generation $n$ produce children according the probability measure $F_n\in \mathcal{P}\left({\mathbb{N}}_0\right)$, and the $F_n$, $n=1,2,\cdots ,$ are i.i.d under the $P_E$. ``subcritical" means that $a:=\text{E}[X_1]\in (-\mathrm{\infty },0)$, where $X_1:=\log {\bar{F}}_1$ and ${\bar{F}}_1$ is the mean of $F_1$.