Reconstruction of periodic sceneries seen along a random walk

This article was motivated by a question of Kesten. Kesten asked for which random walks periodic sceneries can be reconstructed. Among others, he asked the question for random walks which at each step can move by one or two units to the right. Previously, Howard [C.D. Howard, Distinguishing certain random sceneries on ZZ via random walks, Statist. Probab. Lett. 34 (2) (1997) 123–132] proved that all periodic sceneries can be reconstructed provided they are observed along the path of a simple random walk.We prove that for a large class of random walks it is possible to reconstruct periodic sceneries. Among others, we consider the random walk that can only move by one or two units to the right at each step. We show that in this case, the scenery can be reconstructed, provided that the two unit step is less likely than the one unit step.