Bracketing metric entropy rates and empirical central limit theorems for function classes of besov- and sobolev-type

Journal of Theoretical Probability

Volumn 20, Issue 2, 2007, Pages 177-199

  Nickl, Richard ^a   Pötscher, Benedikt M ^a  

^a UNIVERSITY OF VIENNA   (Austria)

Author keywords

Glivenko Cantelli; Metric entropy with bracketing; Sobolev, Besov, H lder, Triebel, and Bessel potential spaces; Uniform Donsker class; Uniform metric entropy

Indexed keywords

EID: 34249036798     PISSN: 08949840     EISSN: 15729230     Source Type: Journal    
DOI: 10.1007/s10959-007-0058-1     Document Type: Article

References (41)

1. Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic, New York (2003)
2. Alexander, K.S.: Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12, 1041-1067 (1984)
3. Andersen, N.T., Giné, E., Ossiander, M., Zinn, J.: The central limit theorem and the law of the iterated logarithm for empirical processes under local conditions. Probab. Theory Relat. Fields 77, 271-305 (1988)
4. Arcones, M.A.: The central limit theorem for U-processes indexed by Holder's functions. Stat. Probab. Lett. 20, 57-62 (1994)
5. Birgé, L., Massart, P.: Rates of convergence of minimum contrast estimators. Probab. Theory Relat. Fields 97, 113-150 (1993)
6. Birgé, L., Massart, P.: An adaptive compression algorithm in Besov spaces. Constr. Approx. 16, 1-36 (2000)
7. p. Mat. Sb. 73, 331-355 (1967)
8. Bourdaud, G., De Cristoforis, M.L., Sickel, W.: Superposition operators and functions of bounded p-variation. Rev. Mat. Iberoamer. 22, 455-487 (2006)
9. Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab. 6, 899-929 (1978);
10. Correction, ibid. 7, 909-911 (1979)
11. Dudley, R.M.: Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab. 20, 1968-1982 (1992)
12. Dudley, R.M.: Uniform Central Limit Theorems. Cambridge University Press, Cambridge (1999)
13. Dudley, R.M., Koltchinskii, V.I.: Envelope moment conditions and Donsker classes. Theor. Probab. Math. Stat. 51, 39-48 (1994)
14. Dudley, R.M., Giné, E., Zinn, J.: Uniform and universal Glivenko-Cantelli classes. J. Theor. Probab. 4, 485-510 (1991)
15. Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers and Differential Operators. Cambridge University Press, Cambridge (1996)
16. Giné, E.: Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev-norms. Ann. Stat. 3, 1243-1266 (1975)
17. Giné, E., Zinn, J.: Lectures on the central limit theorem for empirical processes. In: Lecture Notes in Math., vol. 1221, pp. 50-113. Springer, New York (1986)
18. Giné, E., Zinn, J.: Empirical processes indexed by Lipschitz functions. Ann. Probab. 14, 1329-1338 (1986)
19. Giné, E., Zinn, J.: Gaussian characterization of uniform Donsker classes. Ann. Probab. 19, 758-782 (1991)
20. Haroske, D., Triebel, H.: Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators I. Math. Nachr. 167, 131-156 (1994)
21. Haroske, D., Triebel, H.: Wavelet bases and entropy numbers in weighted function spaces. Math. Nachr. 278, 108-132 (2005)
22. Kolmogorov, A.N., Tihomirov, V.M.: ε-entropy and ε-capacity of sets in function spaces. In: AMS Translations, Series 2, vol. 17, pp. 277-364 (1961)
23. Lorentz, G.G., v.Golitschek, M., Makovoz, Y.: Constructive Approximation: Advanced Problems. Springer, Berlin (1996)
24. Marcus, D.J.: Relationships between Donsker classes and Sobolev spaces. Z. Wahrsch. Verw. Gebiete 69, 323-330 (1985)
25. Nickl, R.: Empirical and Gaussian processes on Besov classes. In: Giné, E., Koltchinskii, V., Li, W., Zinn, J. (eds.) High Dimensional Probability IV. IMS Lecture Notes, vol. 51, pp. 185-195 (2006)
26. 2 bracketing. Ann. Probab. 15, 897-919 (1987)
27. Sheehy, A., Wellner, J.A.: Uniform Donsker classes of functions. Ann. Probab. 20, 1983-2030 (1992)
28. Talagrand, M.: Vapnik-Chervonenkis type conditions and uniform Donsker classes of functions. Ann. Probab. 31, 1565-1582 (2003)
29. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)
30. Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)
31. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Barth, Heidelberg (1995)
32. van de Geer, S.: The entropy bound for monotone functions. Technical Report 91-10, University of Leiden (1991)
33. van de Geer, S.: Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Stat. 21, 14-44 (1993)
34. van de Geer, S.: Empirical Processes in M-Estimation. Cambridge University Press, Cambridge (2000)
35. van der Vaart, A.W.: Bracketing smooth functions. Stoch. Process. Appl. 52, 93-105 (1994)
36. van der Vaart, A.W.: New Donsker classes. Ann. Probab. 24, 2128-2140 (1996)
37. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)
38. Wong, W.H., Shen, X.: Probability inequalities for likelihood ratios and convergence rates of sieve MLE's. Ann. Stat. 23, 339-362 (1995)
39. Yukich, J.E.: Rates of convergence for classes of functions: the non-i.i.d. case. J. Multivar. Anal. 20, 175-189 (1986)
40. Yukich, J.E.: Some limit theorems for the empirical process indexed by functions. Probab. Theory Relat. Fields 74 71-90 (1987)
41. Ziemer, W.P.: Weakly Differentible Functions. Springer, New York (1989)