Probability Theory and Related Fields

, Volume 101, Issue 2, pp 173–192

Small values of Gaussian processes and functional laws of the iterated logarithm

  • Ditlev Monrad
  • Holger Rootzén
Article

Summary

We estimate small ball probabilities for locally nondeterministic Gaussian processes with stationary increments, a class of processes that includes the fractional Brownian motions. These estimates are used to prove Chung type laws of the iterated logarithm.

Mathematics Subject Classification (1991)

60F15 60G15 60G17 60G18 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    deAcosta, A.: Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Probab.11, 78–101 (1983)Google Scholar
  2. 2.
    Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc.6, 170–176 (1955)Google Scholar
  3. 3.
    Berman S.M.: Gaussian sample functions: uniform dimension and Hölder conditions nowhere. Nagoya Math. J.46, 63–86 (1972)Google Scholar
  4. 4.
    Berman S.M.: Local nondeterminism and local times for Gaussian processes. Indiana Univ. J. Math.23, 69–94 (1973)Google Scholar
  5. 5.
    Berman S.M.: Gaussian processes with biconvex covariances. J. Multivar. Anal.8, 30–44 (1978)Google Scholar
  6. 6.
    Borell, C.: A note on Gauss measures which agree on small balls. Ann. Inst. Henri Poincaré, Sect. B, XIII-3, 231–238 (1977)Google Scholar
  7. 7.
    Chung, K.L.: On the maximum partial sums of sequences of independent random variables. Trans. Am. Math. Soc.64, 205–233 (1948)Google Scholar
  8. 8.
    Csáki, E.: A relation between Chung's and Strassen's laws of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.54, 287–301 (1980)Google Scholar
  9. 9.
    Cuzick, J., Du Preez, J.: Joint continuity of Gaussian local times. Ann. Probab.10, 810–817 (1982)Google Scholar
  10. 10.
    Goodman, V., Kuelbs, J.: Rates of clustering for some Gaussian self-similar processes. Probab. Theory Relat. Fields88, 47–75 (1991)Google Scholar
  11. 11.
    Grill, K.: A lim inf result in Strassen's law of the iterated logarithm. Probab. Theory Relat. Fields89, 149–157 (1991)Google Scholar
  12. 12.
    Jain, N.C., Marcus, M.B.: Continuity of subgaussian processes. Adv. Probab.4, 81–196 (1978)Google Scholar
  13. 13.
    Jain, N.C., Pruitt, W.E.: The other law of the iterated logarithm. Ann. Probab.3, 1046–1049 (1975)Google Scholar
  14. 14.
    Kuelbs, J.: The law of the iterated logarithm and related strong convergence theorems for Banach space valued random variables. (Lect. Notes Math., 539, pp. 224–314) Berlin Heidelberg New York: Springer (1976)Google Scholar
  15. 15.
    Kuelbs, J., Li, W.V., Talagrand, M.: Lim inf results for Gaussian samples and Chung's functional LIL. (to appear 1992)Google Scholar
  16. 16.
    Loeve, M.: Probability Theory I. Berlin Heidelberg New York: Springer 1977Google Scholar
  17. 17.
    Mandelbrot, B.B., van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Review10, 422–437 (1968)Google Scholar
  18. 18.
    Marcus, M.B.: Gaussian processes with stationary increments possessing discontinuous sample paths. Pac. J. Math.26, 149–157 (1968)Google Scholar
  19. 19.
    Oodaira, H.: On Strassen's version of the law of the iterated logarithm for Gaussian processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.21, 289–299 (1972)Google Scholar
  20. 20.
    Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. J. Math.27, 309–330 (1978)Google Scholar
  21. 21.
    Pitt, L.D., Tran, L.T.: Local sample path properties of Gaussian fields. Ann. Probab.7, 477–493 (1979)Google Scholar
  22. 22.
    Šidák, Z.: On multivariate normal probabilities of rectangles: their dependence on correlations. Ann. Math. Stat.39, 1425–1434 (1968)Google Scholar
  23. 23.
    Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 211–226 (1964)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Ditlev Monrad
    • 1
    • 3
  • Holger Rootzén
    • 2
    • 3
  1. 1.Department of StatisticsUniversity of IllinoisChampaignUSA
  2. 2.Department of MathematicsChalmers UniversityGoteborgSweden
  3. 3.Center for Stochastic ProcessesUniversity of North Carolina at Chapel HillChapel HillUSA

Personalised recommendations