Reconstructing a three-color scenery by observing it along a simple random walk path

Let {¿(n)}n¿Z be a three-color random scenery, i.e., a random coloration of Z in three colors, such that the colors of the different points in N are i.i.d. Let {S(n)}n¿N be a symmetric random walk starting at 0. Our main result shows that a.s., ¿°S (the composition of ¿ and S) determines ¿ up to translation and reflection. In other words, by observing the scenery along the random walk path S, we can a.s. reconstruct ¿ up to translation and reflection. This result allows us to give a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in a three-color random scenery by observing it only along a random walk path.