Journal of Theoretical Probability

, Volume 30, Issue 3, pp 729–770 | Cite as

Couplings and Strong Approximations to Time-Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

  • Péter KeveiEmail author
  • David M. Mason


We define a time-dependent empirical process based on n i.i.d. fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.


Coupling inequality Fractional Brownian motion Strong approximation Time-dependent empirical process 

Mathematics Subject Classification (2010)

62E17 60G22 60F15 



The authors thank the Associate Editor for a comment that led to Remark 1. PK was partially supported by the Hungarian Scientific Research Fund OTKA PD106181, by the European Union and co-funded by the European Social Fund under the project ‘Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences’ of project number TÁMOP-4.2.2.A-11/1/KONV-2012-0073, and by a postdoctoral fellowship of the Alexander von Humboldt Foundation.


  1. 1.
    Arcones, M.A.: On the law of the iterated logarithm for Gaussian processes. J. Theor. Probab. 8, 877–903 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7, 29–54 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berthet, P., Mason, D. M.: Revisiting two strong approximation results of Dudley and Philipp. High dimensional probability, IMS Lecture Notes Monogr. Ser., 51, Inst. Math. Statist., Beachwood, OH, 155–172 (2006)Google Scholar
  4. 4.
    Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dudley, R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dudley, R.M.: Real Analysis and Probability. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1989)zbMATHGoogle Scholar
  7. 7.
    Einmahl, U., Mason, D.M.: Gaussian approximation of local empirical processes indexed by functions. Probab. Theory Relat. Fields 107, 283–311 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Einmahl, U., Mason, D.M.: An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab. 13, 1–37 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kevei, P., Mason, D. M.: Strong approximations to time dependent empirical and quantile processes based on independent fractional Brownian motions, arXiv:1308.4939
  10. 10.
    Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent rv’s and the sample df. I. Z. Wahrsch verw Gebiete 32, 111–131 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kosorok, M.R.: Introduction to empirical processes and semiparametric inference. Springer Series in Statistics. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kuelbs, J., Kurtz, T., Zinn, J.: A CLT for empirical processes involving time-dependent data. Ann. Probab. 41, 785–816 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kuelbs, J., Zinn, J.: Empirical quantile-clts for time-dependent data. High Dimensional Probability VI, Banff, AB, 2011, Progr. in Probab., vol. 66, pp. 167–194 (2013)Google Scholar
  14. 14.
    Kuelbs, J., Zinn, J.: Empirical quantile central limit theorems for some self-similar processes. J. Theoret. Probab. 28, 313–336 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Landau, H.J., Shepp, L.A.: On the supremum of a Gaussian process. Sankhyā Ser. A 32, 369–378 (1970)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ledoux, M., Talagrand, M.: Comparison theorems, random geometry and some limit theorems for empirical processes. Ann. Probab. 17, 596–631 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ledoux, M. and Talagrand, M.: Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23, Springer-Verlag, Berlin (1991)Google Scholar
  18. 18.
    LePage, R. D.: Log log law for Gaussian processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25, 103–108 (1972/73)Google Scholar
  19. 19.
    Lifshits, M.A.: Gaussian random functions. Mathematics and its Applications, vol. 322. Kluwer Academic Publishers, Dordrecht (1995)CrossRefzbMATHGoogle Scholar
  20. 20.
    Marcus, M. B., Shepp, L. A.: Sample behavior of Gaussian processes. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University California, Berkeley, CA, 1970/1971), Vol. II: Probability theory, pp. 423–441. University California Press, Berkeley, CA, (1972)Google Scholar
  21. 21.
    Philipp, W.: Invariance principles for independent and weakly dependent random variables. Dependence in probability and statistics (Oberwolfach, 1985), 225–268, Progr. Probab. Statist., 11, Birkhäuser Boston, Boston (1986)Google Scholar
  22. 22.
    Satô, H.: A remark on Landau-Shepp’s theorem. Sankhyā Ser. A 33, 227–228 (1971)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Swanson, J.: Weak convergence of the scaled median of independent Brownian motions. Probab. Theory Relat. Fields 138, 269–304 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, 28–76 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    van der Vaart, A.W.: Asymptotic statistics. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3. Cambridge University Press, Cambridge (1998)Google Scholar
  26. 26.
    van der Vaart, A.W., Wellner, J.A.: Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer, New York (1996)zbMATHGoogle Scholar
  27. 27.
    Wang, W.: On a functional limit result for increments of a fractional Brownian motion. Acta Math. Hungar. 93, 153–170 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zaitsev, AYu.: Estimates of the Lévy-Prokhorov distance in the multivariate central limit theorem for random variables with finite exponential moments. Theory Probab. Appl. 31, 203–220 (1987a)CrossRefGoogle Scholar
  29. 29.
    Zaitsev, AYu.: On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein’s inequality conditions. Probab. Theory Relat. Fields 74, 534–566 (1987b)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zaitsev, AYu.: The accuracy of strong Gaussian approximation for sums of independent random vectors Russian Math. Surveys 68, 721–761 (2013)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.MTA-SZTE Analysis and Stochastics Research Group, Bolyai InstituteSzegedHungary
  2. 2.Center for Mathematical Sciences, Technische Universität MünchenGarchingGermany
  3. 3.Department of Applied Economics and StatisticsUniversity of DelawareNewarkUSA

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