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July 1999 Approximation, Metric Entropy and Small Ball Estimates for Gaussian Measures
Wenbo V. Li, Werner Linde
Ann. Probab. 27(3): 1556-1578 (July 1999). DOI: 10.1214/aop/1022677459


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.


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Wenbo V. Li. Werner Linde. "Approximation, Metric Entropy and Small Ball Estimates for Gaussian Measures." Ann. Probab. 27 (3) 1556 - 1578, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0983.60026
MathSciNet: MR1733160
Digital Object Identifier: 10.1214/aop/1022677459

Primary: 60G15
Secondary: 47D50 , 47G10 , 60F99

Keywords: approximation number , Gaussian process , Metric entropy , small deviation

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
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