On the slowdown of random walk in random environment with bounded jumps

In this paper we prove that under certain assumptions the transient random walk in random environment with bounded jumps (in $\mathbb{Z}$) grows much slower than the speed $n$. Precisely, there is $0<s<1$, such that although $X_n\text{rto}$ we have $\frac{X_n}{n^{s^{\mathrm{\prime }}}}\to 0$ for $0<s<s^{\mathrm{\prime }}$ almost surely.