The Annals of Applied Probability

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

Heinrich Matzinger

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Let {ξ(n)}n∈ℤ be a two-color random scenery, that is, a random coloring of ℤ in two colors, such that the ξ(i)’s are i.i.d. Bernoulli variables with parameter ½. Let {S(n)}n∈ℕ be a symmetric random walk starting at 0. Our main result shows that a.s., ξ○S (the composition of ξ and S) determines ξ up to translation and reflection. In other words, by observing the scenery ξ along the random walk path S, we can a.s. reconstruct ξ 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.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 778-819.

First available in Project Euclid: 1 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60L37
Secondary: 60G10: Stationary processes

Scenery reconstruction random walk observations made by random walk


Matzinger, Heinrich. Reconstructing a two-color scenery by observing it along a simple random walk path. Ann. Appl. Probab. 15 (2005), no. 1B, 778--819. doi:10.1214/105051604000000972.

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