Donsker theorems for diffusions: Necessary and sufficient conditions

• 成果类型: Article
• 署名作者: van der Vaart, A; van Zanten, H
• 署名单位: Vrije Universiteit Amsterdam
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000152
• 发表日期: 2005
• 页码: 1422-1451
• 关键词: Efficiency

摘要

We consider the empirical process G(t) of a one-dimensional diffusion with finite speed measure, indexed by a collection of functions F. By the central limit theorem for diffusions, the finite-dimensional distributions of G(t) converge weakly to those of a zero-mean Gaussian random process G. We prove that the weak convergence G(t) double right arrow G takes place in l(infinity)(F) if and only if the limit G exists as a tight, Borel measurable map. The proof relies on majorizing measure techniques for continuous martingales. Applications include the weak convergence of the local time density estimator and the empirical distribution function on the full state space.