UNBIASED SHIFTS OF BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Last, Guenter; Moerters, Peter; Thorisson, Hermann
• 署名单位: Helmholtz Association; Karlsruhe Institute of Technology; University of Bath; University of Iceland
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP832
• 发表日期: 2014
• 页码: 431-463
• 关键词: stationary random measures; distributions; times

摘要

Let B = (B-t)(t epsilon R) be a two-sided standard Brownian motion. An unbiased shift of B is a random time T, which is a measurable function of B, such that (BT+t - B-T)(t epsilon R) is a Brownian motion independent of B-T. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of B. For any probability distribution nu on R we construct a stopping time T >= 0 with the above properties such that B-T has distribution nu. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on R. Another new result is an analogue for diffuse random measures on R of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.