A phase transition in random coin tossing

• 成果类型: Article
• 署名作者: Levin, DA; Pemantle, R; Peres, Y
• 署名单位: University of Connecticut; University of Wisconsin System; University of Wisconsin Madison; Hebrew University of Jerusalem; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2001
• 页码: 1637-1669
• 关键词: scenery

摘要

Suppose that a coin with bias 9 is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let we be the distribution of the observed sequence of coin tosses, and let u(n) denote the chance of a renewal at time n. Harris and Keane showed that if Sigma(n=1)(infinity) u(n)(2)=infinity, then u(theta) and u(0) are singular, while if Sigma(n=1)(infinity) u(n)(2) < infinity and theta is small enough, then u(theta) is absolutely continuous with respect to go. They conjectured that absolute continuity should not depend on B, but only on the square-summability of {u(n)}. We show that in fact the power law governing the decay of {u(n)} is crucial, and for some renewal sequences fun 1, there is a phase transition at a critical parameter theta(c) is an element of (0, 1): for \theta\ < theta(c) the measures u(theta) and u(0) are mutually absolutely continuous, but for \theta\ > theta(c) they are singular. We also prove that when u(n) = O(n(-1)), the measures u(theta) for theta is an element of [-1, 1] are all mutually absolutely continuous.