Reconstructing a multicolor random scenery seen along a random walk path with bounded jumps

Kesten [12] noticed that the scenery reconstruction method proposed by Matzinger [18] relies heavily on the skip-free property of the random walk. He asked if one can still reconstruct an i.i.d. scenery seen along the path of a non-skip-free random walk. In this article, we positively answer this question. We prove that if there are enough colors and if the random walk is recurrent with at most bounded jumps, and if it can reach every integer, then one can almost surely reconstruct almost every scenery up to translations and reflections. Our reconstruction method works if there are more colors in the scenery than possible single steps for the random walk.