For a general one-dimensional random walk with state-dependent transition probabilities, we present weak limits of the empirical moments of conductance along the path of the random walk. In particular we obtain remarkably simple quenched convergences under random conductance model.
"Limit theorems for random walks under irregular conductance." Proc. Japan Acad. Ser. A Math. Sci. 89 (8) 87 - 91, October 2013. https://doi.org/10.3792/pjaa.89.87