A Law of the Iterated Logarithm for Fractional Brownian Motions

  • Driss Baraka
  • Thomas Mountford
Part of the Lecture Notes in Mathematics book series (LNM, volume 1934)


We show that for a class of Gaussian processes indexed by one dimensional time, the local times obey the behavior conjectured by Xiao.

Key words

Local times Gaussian processes fractional Brownian motion Girsanov’s theorem 


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  1. 1.
    Ehm, W. (1981), Sample function properties of multiparameter stable processes. Zeit. Wahr. Theorie 56, 195–228.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Genian, D. and Horowitz, J. (1980), Occupation densities. Ann. Probab., 8, 1–67.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Genian, D., Horowitz, J. and Rosen, J. (1984), A local time analysis of intersections of Brownian paths in the plane. Ann. Probab., 12, 86–107.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Khoshenevisan, D., Xiao, Y. and Zhong, Y. (2003), Local times of additive Lévy processes. Stochastic Process. Appl., 104, no. 2, 193–216.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mountford, T and Nualart, E (2004), Level sets of multiparameter Brownian motions. Electron. J. Probab., 9, no. 20, 594–614.MathSciNetGoogle Scholar
  6. 6.
    Mueller, C. and Tribe, R. (2002), Hitting properties of a random string. Electron. J. Probab., 7, 1–29.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Perkins, E. (1981), The exact Hausdorff measure of the level sets of Brownian motion. Z. Wahrsch. verw. Gebiete, 58, 373–388.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Pitt, L.D. and Tran, L.T. (1979), Local Sample Path Properties of Gaussian Fields. Ann. Probab., 7, no. 3, 477–493.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Revuz, D. and Yor, M. (1999), Continuous martingales and Brownian motion. Third edition. Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  10. 10.
    Rogers, C.A. (1998), Housdorff measures. Cambridge University Press.Google Scholar
  11. 11.
    Samorodnitsky, G and Taqqu, M.S. (1994), Stable non-Gaussian random processes: Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York.MATHGoogle Scholar
  12. 12.
    Xiao, Y. (1997), Holder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Th. Rel. Fields, 109, 129–157.MATHCrossRefGoogle Scholar
  13. 13.
    Xiao, Y. (2003), The packing measure of the trajectories of multiparameter fractional Brownian motion. Math. Proc. Camb. Phil. Soc., 135, 349–375.MATHCrossRefGoogle Scholar
  14. 14.
    Xiao, Y. (2005), Strong Local Nondeterminism and Sample Path Properties of Gaussian Random Fields. Preprint.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Driss Baraka
    • 1
  • Thomas Mountford
    • 1
  1. 1.Département de MathématiquesÉcole Polytechnique FédéraleLausanneSwitzerland

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