Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type

Abstract

We derive ℒ r (μ)-bracketing metric and sup-norm metric entropy rates of bounded subsets of general function spaces defined over ℝ d or, more generally, over Borel subsets thereof, by adapting results of Haroske and Triebel (Math. Nachr. 167, 131–156, 1994; 278, 108–132, 2005). The function spaces covered are of (weighted) Besov, Sobolev, Holder, and Triebel type. Applications to the theory of empirical processes are discussed. In particular, we show that (norm-)bounded subsets of the above mentioned spaces are Donsker classes uniformly in various sets of probability measures.