Orthogonality of Measures Induced by Random Walks with Scenery

Abstract

Thinking of a deterministic function s: ℤ → ℕ as ‘scenery’ on the integers, a simple random walk on ℤ generates a random record of scenery ‘observed’ along the walk. We address this question: If t:ℤ → ℕ is another scenery on the integers and we are handed a random scenery record obtained from either s or t, under what circumstances can the source be distinguished? We allow ourselves to use information about s and t together with information contained in the scenery record. It has been conjectured that it is sufficient for t to be neither a translate of s nor a translate of the reflection of s. We show that this condition is sufficient to ensure distinguishability if s−1(δ) is finite and non-empty for some δ ∈ℕ.