Small deviation for Random walk with random environment in time

Abstract

We give the random environment version of Mogul'ski\vı 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 ${\mu }_n.$ 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 $$\bfP_μ(\forall_{0\leq i\leq n} S_{f(n)+i}\in[g(i/n)n^α,h(i/n)n^α]|S_{f(n)}=x)$$ is $e^{-c{n}^{1-2\alpha }}$ almost surely as $n\to +\infty$, 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.

12 pages