Probability Theory and Related Fields

, Volume 80, Issue 3, pp 365–379 | Cite as

Upper classes for the increments of fractional Wiener processes

  • J. Ortega


Let (X(t), t≧0) be a centred Gaussian process with stationary increments andEX2(t)=C0t for someC0>0, 0<α<1, and let 0<a t t be a nondecreasing function oft witha t /t nonincreasing. The asymptotic behaviour of several increment processes constructed fromX anda t is studied in terms of their upper classes.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berman, S.M.: Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist.35, 502–516 (1964)Google Scholar
  2. 2.
    Berman, S.M.: An asymptotic bound for the distribution of the maximum of a Gaussian process. Ann. Inst. Henri Poincaré21, 47–57 (1985)Google Scholar
  3. 3.
    Berman, S.M.: The maximum of a Gaussian process with nonconstant variance. Ann. Inst. Henri Poincaré21, 383–391 (1985)Google Scholar
  4. 4.
    Csörgö, M., Révész, P.: How big are the increments of the Wiener process?. Ann. Probab.7, 731–743 (1979)Google Scholar
  5. 5.
    Orey, S.: Growth rate of certain Gaussian processes. In: Proc. Sixth Berkeley, Symposium Math. Stat. Prob., vol. 2, pp. 443–451. Berkeley: University of California Press 1971Google Scholar
  6. 6.
    Ortega, J., Wschebor, M.: On the increments of the Wiener process. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 329–339 (1984)Google Scholar
  7. 7.
    Ortega, J.: On the size of the increments of non-stationary Gaussian processes. Stochastic Proc. Appl.18, 47–56 (1984)Google Scholar
  8. 8.
    Ortega, J.: Comportamiento asintótico de los incrementos de los procesos de Wiener fraccionarios. Actas II Congreso Latinoamericano Prob. Est. Mat., pp. 195–215, Caracas: Equinoccio 1986Google Scholar
  9. 9.
    Plackett, R.L.: A reduction formula for normal multivariate integrals. Biometrika41, 351–360 (1954)Google Scholar
  10. 10.
    Qualls, C., Watanabe, H.: Asymptotic properties of Gaussian processes. Ann. Math. Statist.43, 580–596 (1972)Google Scholar
  11. 11.
    Qualls, C., Watanabe, H.: Asymptotic properties of Gaussian random fields. Trans. Am. Math. Soc.177, 155–171 (1973)Google Scholar
  12. 12.
    Rényi, A.: Probability theory. Amsterdam: North-Holland 1970Google Scholar
  13. 13.
    Révész, P.: On the increments of Wiener and related processes. Ann. Probab.10, 613–627 (1982)Google Scholar
  14. 14.
    Slepian, D.: The one-sided barrier problem for Gaussian noise. Bell System Tech. J.41, 463–501 (1962)Google Scholar
  15. 15.
    Watanabe, H.: An asymptotic property of Gaussian processes. Trans. Am. Math. Soc.148, 233–248 (1970)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • J. Ortega
    • 1
  1. 1.Depto. de MatemáticasInstituto Venezolano de Investigaciones CientificasCaracasVenezuela

Personalised recommendations