  We consider a one dimensional random walk in a random environment (RWRE) with
a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_\alpha>0$. Gantert and
Zeitouni showed that if the environment has both positive and negative local
drifts then the quenched slowdown probabilities $P_\omega(X_n < xn)$ with $x
\in (0,v_\alpha)$ decay approximately like $\exp\{-n^{1-1/s}\}$ for a
deterministic $s > 1$. More precisely, they showed that $n^{-\gamma} \log
P_\omega( X_n < x n)$ converges to $0$ or $-\infty$ depending on whether
$\gamma > 1-1/s$ or $\gamma < 1-1/s$. In this paper, we improve on this by
showing that $n^{-1+1/s} \log P_\omega( X_n < x n)$ oscillates between $0$ and
$-\infty$, almost surely. This had previously been shown by Gantert only in a
very special case of random environments.
