Abstract
Kesten noticed that the scenery reconstruction method proposed by Matzinger in his PhD thesis 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.
Highlights
Introduction and ResultA scenery is a coloring ξ of the integers Z with C0 colors {1, . . . , C0}
Work on the scenery reconstruction problem started by Kesten’s question, whether one can recognize a single defect in a random scenery
He asked whether the result might still hold in the case of a random walk with jumps
Summary
A (one dimensional) scenery is a coloring ξ of the integers Z with C0 colors {1, . . . , C0}. As was shown in [19] the two color scenery reconstruction problem for a scenery which is i.i.d. is equivalent to the following problem: let (R(k))k∈Z and (S(k))k≥0 be two independent simple random walks on Z both starting at the origin and living on the same probability space. For sceneries that can be reconstructed Benjamini asked whether the reconstruction works in polynomial time This question was positively answered by Matzinger and Rolles [21] and [23] (see [20]) in the case of a two color scenery and a simple random walk with holding. The correct arrangement of modulo classes is a technically intricate step in the reconstruction procedure
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