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

Abstract

Let {\xi (n)}_{n\in Z} be a two-color random scenery, that is, a random coloring of Z in two colors, such that the \xi (i)'s are i.i.d. Bernoulli variables with parameter \tfrac12. Let {S(n)}_{n\in N} be a symmetric random walk starting at 0. Our main result shows that a.s., \xi \circ S (the composition of \xi and S) determines \xi up to translation and reflection. In other words, by observing the scenery \xi along the random walk path S, we can a.s. reconstruct \xi up to translation and reflection. This result gives a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in almost every two-color random scenery by observing it only along a random walk path.