Journal of Theoretical Probability

, Volume 21, Issue 3, pp 672–686 | Cite as

Weak Convergence of Vervaat and Vervaat Error Processes of Long-Range Dependent Sequences



Following Csörgő, Szyszkowicz and Wang (Ann. Stat. 34, 1013–1044, 2006) we consider a long range dependent linear sequence. We prove weak convergence of the uniform Vervaat and the uniform Vervaat error processes, extending their results to distributions with unbounded support and removing normality assumption.


Vervaat process Long range dependence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bahadur, R.R.: A note on quantiles in large samples. Ann. Math. Stat. 37, 577–580 (1966) CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1987) Google Scholar
  3. 3.
    Csáki, E., Csörgő, M., Földes, A., Shi, Z., Zitikis, R.: Pointwise and uniform asymptotics of the Vervaat error process. J. Theor. Probab. 15, 845–875 (2002) CrossRefMATHGoogle Scholar
  4. 4.
    Csörgő, M., Kulik, R.: Reduction principles for quantile and Bahadur-Kiefer processes of long-range dependent linear sequences. Probab. Theory Relat. Fields (2007, accepted) Google Scholar
  5. 5.
    Csörgő, M., Révész, P.: Strong approximation of the quantile process. Ann. Stat. 6, 882–894 (1978) CrossRefMATHGoogle Scholar
  6. 6.
    Csörgő, M., Szyszkowicz, B.: Sequential quantile and Bahadur-Kiefer processes. In: Order Statistics: Theory and Methods, pp. 631–688. North-Holland, Amsterdam (1998) Google Scholar
  7. 7.
    Csörgő, M., Szyszkowicz, B., Wang, L.: Strong invariance principles for sequential Bahadur-Kiefer and Vervaat error processes of long-range dependence sequences. Ann. Stat. 34, 1013–1044 (2006) CrossRefGoogle Scholar
  8. 8.
    Csörgő, M., Szyszkowicz, B., Wang, L.: Correction note: strong invariance principles for sequential Bahadur-Kiefer and Vervaat error processes of long-range dependent processes. Ann. Stat. (2007, accepted) Google Scholar
  9. 9.
    Csörgő, M., Zitikis, R.: On the general Bahadur-Kiefer, quantile, and Vervaat processes: old and new. In: Limit Theorems in Probability and Statistics, Balatonlelle, 1999, vol. I, pp. 389–426. János Bolyai Math. Soc., Budapest (2002) Google Scholar
  10. 10.
    Dehling, H., Taqqu, M.: The empirical process of some long-range dependent sequences with an applications to U-statistics. Ann. Stat. 17, 1767–1783 (1989) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Ho, H.-C., Hsing, T.: On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Stat. 24, 992–1024 (1996) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Hsing, T.: Linear processes, long-range dependence and asymptotic expansions. Stat. Inference Stoch. Process. 3, 19–29 (2000) CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Kiefer, J.: Deviations between the sample quantile process and the sample df. In: Puri, M.L. (ed.) Nonparametric Techniques in Statistical Inference, pp. 349–357. Cambridge University Press, Cambridge (1970) Google Scholar
  14. 14.
    Mikosch, T., Samorodnitsky, G.: The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10, 1025–1064 (2000) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Parzen, E.: Nonparametric statistical data modeling. With comments by John W. Tukey, Roy E. Welsch, William F. Eddy, D.V. Lindley, Michael E. Tarter and Edwin L. Crow, and a rejoinder by the author. J. Am. Stat. Assoc. 74, 105–131 (1979) CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Vervaat, W.: Functional central limit theorem for processes with positive drift and their inverses. Z. Wahrsch. Verw. Geb. 23, 245–253 (1972) CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Wang, Q., Lin, Y.-X., Gulati, C.M.: Strong approximation for long memory processes with applications. J. Theor. Probab. 16, 377–389 (2003) CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Wu, W.B.: Empirical processes of long-memory sequences. Bernoulli 9, 809–831 (2003) CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Wu, W.B.: On the Bahadur representation of sample quantiles for dependent sequences. Ann. Stat. 4, 1934–1963 (2005) CrossRefGoogle Scholar
  20. 20.
    Zitikis, R.: The Vervaat process. In: Asymptotic Methods in Probability and Statistics, Ottawa, ON, 1997, pp. 667–694. North-Holland, Amsterdam (1998) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  3. 3.Mathematical InstituteWrocław UniversityWrocławPoland

Personalised recommendations