Central limit theorems for the Wasserstein distance between the empirical and the true distributions

• 成果类型: Article
• 署名作者: Del Barrio, E; Giné, E; Matrán, C
• 署名单位: Universidad de Valladolid; University of Connecticut
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677394
• 发表日期: 1999
• 页码: 1009-1071
• 关键词: quantile processes; lp-norms; CONVERGENCE; SPACES

摘要

If X is integrable, F is its cdf and F-n is the empirical cdf based on an i.i.d. sample from F, then the Wasserstein distance between F-n and F, which coincides with the L-1 norm integral(-infinity)(infinity)\F-n(t) - F(t)\ dt of the centered empirical process, tends to zero a.s. The object of this article is to obtain rates of convergence and distributional limit theorems for this law of large numbers or, equivalently, stochastic boundedness and distributional limit theorems for the L-1 norm of the empirical process. Some limit theorems for the Ornstein-Uhlenbeck process are also derived as a by-product.