In this paper, a moderate deviation theorem for one-dimensional stable random walks in random scenery is proved. The proof relies on the analysis of maximum local times of stable random walks, and the comparison of moments between random walks in random scenery and self-intersection local times of the underlying random walks.
"Moderate deviations for stable random walks in random scenery." J. Appl. Probab. 49 (1) 280 - 294, March 2012. https://doi.org/10.1239/jap/1331216847