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

✍ Scribed by Heinrich Matzinger

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 212 KB
• Volume: 15
• Category: Article
• ISSN: 1042-9832
• DOI: 10.1002/(sici)1098-2418(199909)15:2<196::aid-rsa5>3.0.co;2-#

✦ Synopsis

Let n be a three-color random scenery, i.e., a random coloration of ‫ޚ‬ ng ‫ޚ‬ Ä Ž .4 in three colors, such that the colors of the different points in ‫ޚ‬ are i.i.d. Let S n be a

ng ‫ގ‬ Ž symmetric random walk starting at 0. Our main result shows that a.s., ( S the composition . of and S determines up to translation and reflection. In other words, by observing the scenery along the random walk path S, we can a.s. reconstruct up to translation and reflection. This result allows us to give a positive answer to the question of H. Kesten of whether one can a.s. detect a single defect in a three-color random scenery by observing it only along a random walk path.