Advertisement

Siberian Mathematical Journal

, Volume 32, Issue 4, pp 578–588 | Cite as

Accuracy of the approximation of an empirical process by a Brownian bridge

  • V. I. Kolchinskii
Article
  • 50 Downloads

Keywords

Empirical Process Brownian Bridge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    J. Komlos, P. Major, and G. Tusnady, “An approximation of partial sums of independent RV's and the sample DF,” Z. Wahrscheinlichkeitstheor. Verw. Geb.,32, No. 2, 111–131 (1975).Google Scholar
  2. 2.
    I. S. Borisov, “Rate of convergence in invariance principle in linear spaces. Application to empirical measures,” Lect. Notes Math.,1021, 45–58 (1983).Google Scholar
  3. 3.
    I. S. Borisov, “A new approach to the problem of approximating distributions of sums of independent random variables in linear spaces,” Trudy Mat. Inst., Sib. Sec., Academy of Sciences of the USSR,5, 3–27 (1985).Google Scholar
  4. 4.
    I. S. Borisov, “Rate of convergence in the invariance principle for empirical measures,” in: Proc. 1st World Congress Bernoulli Soc., VNU Sci. Press, Amsterdam (1987), pp. 833–836.Google Scholar
  5. 5.
    P. Massart, “Rates of convergence in the central limit theorem for empirical processes,” Ann. Inst. Henri Poincaré,22, 381–423 (1986).Google Scholar
  6. 6.
    P. Massart, “Strong approximation for multivariate empirical and related processes, via KMT constructions,” Ann. Probab.,17, 266–291 (1989).Google Scholar
  7. 7.
    V. I. Kolchinskii (Kolčinski), “Rates of convergence in the invariance principle for empirical processes,” Festschrift, Yu. Prokhorov et al. (eds.), VSP/Mokslas (1991).Google Scholar
  8. 8.
    E. Gine and J. Zinn, “Some limit theorems for empirical processes,” Ann. Probab.,12, 929–998 (1984).Google Scholar
  9. 9.
    V. N. Vapnik and A. Ya. Chervonenkis, “The uniform convergence of frequencies of the appearance of events to their probabilities,” Teor. Veroyatn. Primen.,16, 264–279 (1971).Google Scholar
  10. 10.
    V. N. Vapnik and A. Ya. Chervonenkis, “Necessary and sufficient condition of the uniform convergence of empirical means,” Teor. Veroyatn. Primen.,26, 543–563 (1981).Google Scholar
  11. 11.
    V. I. Kolchinskii, “The central limit theorem for empirical measures,” Teor. Veroyatn. Mat. Stat., Kiev,24, 63–75 (1981).Google Scholar
  12. 12.
    V. I. Kolchinskii, “Functional limit theorems and empirical entropy. I,” Teor. Veroyatn. Mat. Stat., Kiev,33, 35–45 (1985).Google Scholar
  13. 13.
    V. I. Kolchinskii, “Functional limit thoerems and empirical entropy. II,” Teor. Veroyatn. Mat. Stat., Kiev,34, 81–93 (1986).Google Scholar
  14. 14.
    L. Le Cam, “A remark on empirical processes,” in: Festschrift for E. L. Lehmann in Honor of His Sixty-Fifth Birthday, Belmont, California, Wadsworth (1983), pp. 305–327.Google Scholar
  15. 15.
    E. Gine and J. Zinn, “Lectures on the central limit theorem for empirical processes,” Lect. Notes Math.,1221, 50–113 (1986).Google Scholar
  16. 16.
    M. Talagrand, “Classes de Donsker et ensemble pulverisés,” C. R. Acad. Sci. Paris, Ser. I, 161–163 (1985).Google Scholar
  17. 17.
    R. M. Dudley, “Central limit theorems for empirical measures,” Ann. Probab.,6, 899–929 (1978).Google Scholar
  18. 18.
    A. N. Zhdanov and E. A. Sevast'yanov, “Approximative and differential properties of measurable sets,” Mat. Sb.,121, 403–422 (1983).Google Scholar
  19. 19.
    R. M. Dudley, “Metric entropy of some classes of sets with differentiable boundaries,” J. Approx. Theory,10, 227–236 (1974).Google Scholar
  20. 20.
    S. Csörgö, “Limit behaviour of the empirical characteristic function,” Ann. Probab.,9, 130–144 (1981).Google Scholar
  21. 21.
    M. B. Marcus and W. Philipp, “Almost sure invariance principles for sums of B-valued random variables with applications to random Fourier series and the empirical characteristic process,” Trans. Am. Math. Soc.,269, No. 1, 67–90 (1982).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • V. I. Kolchinskii

There are no affiliations available

Personalised recommendations