Application of moderate deviation techniques to prove Sinai's Theorem on RWRE

We apply the techniques developed in Comets and Popov (2003) to present a new proof to Sinai's theorem (Sinai, 1982) on one-dimensional random walk in random environment (RWRE), working in a scale-free way to avoid rescaling arguments and splitting the proof in two independent parts: a quenched one, related to the measure $P_\omega$ conditioned on a fixed, typical realization $\omega$ of the environment, and an annealed one, related to the product measure $\mathbb{P}$ of the environment $\omega$. The quenched part still holds even if we use another measure (possibly dependent) for the environment.