Reconstructing a two-color scenery by observing it along a simple random walk path
Matzinger, Heinrich
Ann. Appl. Probab., Tome 15 (2005) no. 1A, p. 778-819 / Harvested from Project Euclid
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.
Publié le : 2005-02-14
Classification:  Scenery reconstruction,  random walk,  observations made by random walk,  60L37,  60G10
@article{1107271668,
     author = {Matzinger, Heinrich},
     title = {Reconstructing a two-color scenery by observing it along a simple random walk path},
     journal = {Ann. Appl. Probab.},
     volume = {15},
     number = {1A},
     year = {2005},
     pages = { 778-819},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1107271668}
}
Matzinger, Heinrich. Reconstructing a two-color scenery by observing it along a simple random walk path. Ann. Appl. Probab., Tome 15 (2005) no. 1A, pp.  778-819. http://gdmltest.u-ga.fr/item/1107271668/