SURFACE ORDER SCALING IN STOCHASTIC GEOMETRY

• 成果类型: Article
• 署名作者: Yukich, J. E.
• 署名单位: Lehigh University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP992
• 发表日期: 2015
• 页码: 177-210
• 关键词: gaussian limits; large numbers; voronoi; records; LAWS

摘要

Let P lambda := P lambda kappa denote a Poisson point process of intensity lambda kappa on [0,1](d), d >= 2, with K a bounded density on [0,1](d) and A e (0, co). Given a closed subset M c [0, lid of Hausdorff dimension (d 1), we consider general statistics E 7,2, (x, PA, M), where the score function 4 vanishes unless the input x is close to M and where satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics ExEPA. (X x ' lld Al/divt. as co. When.A4 is of class C2, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order Vol(Al/dM). We use the general results to deduce variance asymptotics and central limit theorems for statistics arising in stochastic geometry, including Poisson Voronoi volume and surface area estimators, answering questions in Heveling and Reitzner [Ann. App!. Probab. 19 (2009) 719-736] and Reitzner, Spodarev and Zaporozhets [Adv. in App!. Probab. 44 (2012) 938-953]. The general results also yield the limit theory for the number of maximal points in a sample.