TRACKING A RANDOM WALK FIRST-PASSAGE TIME THROUGH NOISY OBSERVATIONS

• 成果类型: Article
• 署名作者: Burnashev, Marat V.; Tchamkerten, Aslan
• 署名单位: Russian Academy of Sciences; IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom Paris
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP815
• 发表日期: 2012
• 页码: 1860-1879
• 关键词: boundary; moments

摘要

Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time tau(l) of a given level l with a stopping time eta defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation in f(eta) E vertical bar eta - tau(l)vertical bar which become tight as l -> infinity. Interestingly, in this regime the estimation error does not get smaller if we allow eta to be an arbitrary function of the entire observation process, not necessarily a stopping time. In the particular case where there is no drift, we show that it is impossible to track tau(l) : inf(eta) E vertical bar eta - tau(l)/(P) = infinity for any l > 0 and p >= 1/2.