2005/02/01 by Heinrich Matzinger · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Point processes and geometric inequalities #Stochastic processes and statistical mechanics #math.PR #msc:60G10. #msc:60L37
paper · pdf · doi:10.1214/105051604000000972
published as Annals of Applied Probability 2005, Vol. 15, No. 1B, 778-819 · Published at http://dx.doi.org/10.1214/105051604000000972 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2005/02/01 · arxiv created 2005/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.