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

Richard Nickl () and Benedikt Pötscher
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Richard Nickl: University of Vienna

Journal of Theoretical Probability, 2007, vol. 20, issue 2, 177-199

Abstract: 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, Hölder, 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.

Keywords: Metric entropy with bracketing; Uniform metric entropy; Sobolev; Besov; Hölder; Triebel; and Bessel potential spaces; Uniform Donsker class; Glivenko-Cantelli (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-007-0058-1

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