Approximation, metric entropy and small ball estimates for Gaussian measures

• 成果类型: Article
• 署名作者: Li, WV; Linde, W
• 署名单位: University of Delaware; Friedrich Schiller University of Jena
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1999
• 页码: 1556-1578
• 关键词: numbers; Operators; Duality; VALUES; SPACE

摘要

A precise link proved by Kuelbs and Li relates the small ball behavior of a Gaussian measure mu on a Banach space E with the metric entropy behavior of K mu, the unit ball of the reproducing kernel Hilbert space of mu in E. We remove the main regularity assumption imposed on the unknown function in the link. This enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete Gaussian processes. Moreover, we show that the small ball behavior of a Gaussian process is also tightly connected with the speed of approximation by finite rank processes.