Approximation, Metric Entropy and Small Ball Estimates for Gaussian Measures

Abstract

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.