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

• 成果类型: Article
• 署名作者: Matzinger, H
• 署名单位: University of Bielefeld
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000972
• 发表日期: 2005
• 页码: 778-819
• 关键词: mixing properties; colors

摘要

Let {xi(n)}(nis an element ofZ) be a two-color random scenery, that is, a random coloring of 76 in two colors, such that the xi(i)'s are i.i.d. Bernoulli variables with parameter 1/2. Let {S(n)}(nEN) be a symmetric random walk starting at 0. Our main result shows that a.s., xi o S (the composition of xi and S) determines xi 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 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.