For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of iterated logarithm. As a consequence, conditioned on the random scenery, the one dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.
"The quenched limiting distributions of a one-dimensional random walk in random scenery." Electron. Commun. Probab. 18 1 - 7, 2013. https://doi.org/10.1214/ECP.v18-2918