We give the random environment version of Mogul'skiǐ estimation in quenched sense.Assume that $\{μ\}_{n\in\bfN}$ (called environment) is a sequence of i.i.d. random probability measures on $\bfR.$~ Let $\{X_n\}_{n\in\bfN}$ be a sequence of independent random variables, where $X_n$ has law $μ_n.$ We set $S_n=\sum_{i=1}^{n}X_i.$ Under some integrability conditions, we show that on the log scale, for any power function $f$. the decay rate of $$\bfP_μ(\forall_{0\leq i\leq n} S_{f(n)+i}\in[g(i/n)n^α,h(i/n)n^α]|S_{f(n)}=x)$$ is $e^{-cn^{1-2α}}$ almost surely as $n\rightarrow+\infty$, where $c>0,α\in(0,\frac{1}{2}),$ $g,h\in\mathcal{C}[0,1]$ (the set of all continuous functions defined on $[0,1]),$ $g(s)<h(s), \forall s\in[0,1],$ and $x\in(g(0),h(0)).$ The main result of this paper is also a basic tool in the researching of Branching random walk in random environment with selection.