Small deviation for Random walk with random environment in time

Abstract: We give the random environment version of Mogul'ski\v{\i} estimation in quenched sense.Assume that ${\mu}{n\in\bfN}$ (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 $\mu_n.$ 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 $$\bfP_\mu(\forall_{0\leq i\leq n} S_{f(n)+i}\in[g(i/n)n^{\alpha},h(i/n)n^{\alpha}]|S_{f(n)}=x)$$ is $e^{-cn^{1-2\alpha}}$ almost surely as $n\rightarrow+\infty$, where $c>0,\alpha\in(0,\frac{1}{2}),$ $g,h\in\mathcal{C}[0,1]$ (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.