In “The Limiting Behavior of a One-dimensional Random Walk in a Random Medium” by Sinai, two theorems are proved in the last sections.
I fail to understand why the basic result follows from these theorems.
Basic result: It exists random variable $m(0)$ independent of the environment, such that $\forall\eta>0$: \begin{align} \mathbb{P}\left(\left|\frac{\mathsf{X}_{n}}{\log^{2}n}-m(0)\right|>\eta\right)\longrightarrow_{n\rightarrow\infty}0. \end{align}
Theorem 1: Let $M_{i1},M_{i2}\in\mathfrak{M}$ and $[M_{i1},M_{i2}]$ is a valley with the depth $d([M_{i1},M_{i2}])>1$ and $0\in[M_{i1},M_{i2}]$. Then: \begin{align*} \mathbb{P}\left(\mathsf{X}_{m}\in[M_{i1}\log^{2}n,M_{i2}\log^{2}n],\forall 0\leq m\leq n\right)\to_{n\to\infty}1. \end{align*}
Theorem 2: We consider an environment of $C_{n}$. Then for $\frac{1}{2}n\leq r\leq n$ \begin{align*} \mathbb{P}\left(\omega_{r}=m^{(0)}\right)\to_{n\to\infty}1 \end{align*} uniformly for the environment of $C_{n}$.
Does anyone know? An heuristic approach is absolutely fine!
For more details see the paper by Sinai: https://epubs.siam.org/doi/abs/10.1137/1127028