Detecting a single defect in a scenery by observing the scenery along a random walk path

Harry Kesten
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Harry Kesten: Cornell University, Department of Mathematics

A chapter in Itô’s Stochastic Calculus and Probability Theory, 1996, pp 171-183 from Springer

Abstract: Summary A scenery on ℤ is a map ξ: ℤ → {0,..., k − 1}; we think of ξ as a coloring of ℤ, which assigns to each point of ℤ one of k colors. For a given scenery ξ, denote by $$\hat{\xi }$$ a scenery obtained from ξ by changing ξ(0) only. Let {S n}n≥0 be a simple symmetric random walk on ℤ, starting at the origin. Assume that we observe one of the two sequences {ξ(S n)}n≥0 or $$\hat{\xi }$$ (S n)}n≥0, without being told which of the two sequences is observed. If ξ is known, can we decide (with zero probability of error) on the basis of these observations which of the two sequences was observed ? We prove that this can be done for ‘almost all’ ξ, when k≥5.

Date: 1996
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DOI: 10.1007/978-4-431-68532-6_11

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