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

Author

Listed:
  • Richard Nickl

    (University of Vienna)

  • Benedikt M. Pötscher

    (University of Vienna)

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.

Suggested Citation

  • Richard Nickl & Benedikt M. Pötscher, 2007. "Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type," Journal of Theoretical Probability, Springer, vol. 20(2), pages 177-199, June.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:2:d:10.1007_s10959-007-0058-1
    DOI: 10.1007/s10959-007-0058-1
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    References listed on IDEAS

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    1. Arcones, Miguel A., 1994. "The central limit theorem for U-processes indexed by Hölder's functions," Statistics & Probability Letters, Elsevier, vol. 20(1), pages 57-62, May.
    2. van der Vaart, Aad, 1994. "Bracketing smooth functions," Stochastic Processes and their Applications, Elsevier, vol. 52(1), pages 93-105, August.
    3. Yukich, J. E., 1986. "Rates of convergence for classes of functions: The non-i.i.d. case," Journal of Multivariate Analysis, Elsevier, vol. 20(2), pages 175-189, December.
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    Cited by:

    1. Söhl, Jakob & Trabs, Mathias, 2012. "A uniform central limit theorem and efficiency for deconvolution estimators," SFB 649 Discussion Papers 2012-046, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    2. Denis Belomestny & Tobias Hübner & Volker Krätschmer, 2022. "Solving optimal stopping problems under model uncertainty via empirical dual optimisation," Finance and Stochastics, Springer, vol. 26(3), pages 461-503, July.
    3. Nickl, Richard & Reiß, Markus, 2012. "A Donsker theorem for Lévy measures," SFB 649 Discussion Papers 2012-003, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    4. Fallahgoul, Hasan & Franstianto, Vincentius & Lin, Xin, 2024. "Asset pricing with neural networks: Significance tests," Journal of Econometrics, Elsevier, vol. 238(1).

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