We consider a non-nestling random walk in a product random environment. We assume an exponential moment for the step of the walk, uniformly in the environment. We prove an invariance principle (functional central limit theorem) under almost every environment for the centered and diffusively scaled walk. The main point behind the invariance principle is that the quenched mean of the walk behaves...

Recently, in [11], the “coin-turning walk” was introduced on Z. It is a non-Markovian process where steps form (possibly) time-inhomogeneous Markov chain. In this article, we follow up investigation by introducing analogous processes Zd,d≥2: at time n direction of “updated” with probability pn; otherwise next step repeats previous one. We study some fundamental properties these walks, such as t...