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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 765–780 | Cite as

The Hausdorff dimension of the range of the Lévy multistable processes

  • R. Le GuévelEmail author
Article
  • 29 Downloads

Abstract

We compute the Hausdorff dimension of the image X(E) of a non-random Borel set \(E \subset [0,1]\), where X is a Lévy multistable process in \(\mathbf{R}.\) This extends the case where X is a classical stable Lévy process by letting the stability exponent \(\alpha \) be a smooth function. Hence, we are considering here non-homogeneous processes with increments which are not stationary and not necessarily independent. Contrary to the situation where the stability parameter is a constant, the dimension depends on the version of the multistable Lévy motion when the process has an infinite first moment.

Keywords

Lévy processes Hausdorff dimension Multistable processes 

Mathematics Subject Classification

60K17 60K51 60K52 

References

  1. 1.
    Ayache, A.: Sharp estimates on the tail behavior of a multistable distribution. Stat. Probab. Lett. 83(3), 680–688 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Becker-Kern, P., Meerschaert, M., Scheffler, H.-P.: Hausdorff dimension of operator stable sample paths. Monatsh. Math. 14, 91–101 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blumenthal, R.M., Getoor, R.K.: A dimension theorem for sample functions of stable processes. Ill. J. Math. 4, 370–375 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blumenthal, R.M., Getoor, R.K.: Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10, 493–516 (1961)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Davies, R.O.: Subsets of finite measure in analytic sets. Indag. Math. 14(1952), 488–489 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (1990)zbMATHGoogle Scholar
  7. 7.
    Falconer, K., Lévy Véhel, J.: Multifractional, multistable, and other processes with prescribed local form. J. Theoret. Probab. 22, 375–401 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Falconer, K.J., Liu, L.: Multistable processes and localisability. Stoch. Models 28(2012), 503–526 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ferguson, T.S., Klass, M.J.: A representation of independent increment processes without Gaussian components. Ann. Math. Stat. 43, 1634–1643 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hawkes, J.: On the Hausdorff dimension of the intersection of the range of a stable process with a Borel set. Z. Wahrsch. Verw. Gebiete 19, 90–102 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hendricks, W.J.: Hausdorff dimension theorem in a processes with stable components—an interesting counterexample. Ann. Math. Stat. 43, 690–694 (1972)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kahane, J.-P.: (1985) Ensembles aléatoires et dimensions. In: Recent Progress in Fourier Analysis, El Escorial, pp 65–121. North-Holland, Amsterdam (1983)Google Scholar
  13. 13.
    Khoshnevisan, D., Xiao, Y.: Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33, 841–878 (2005).  https://doi.org/10.1214/009117904000001026 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khoshnevisan, D., Xiao, Y., Zhong, Y.: Measuring the range of an additive Lévy process. Ann. Probab. 31, 1097–1141 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Le Guével, R., Lévy Véhel, J.: A Ferguson–Klass–LePage series representation of multistable multifractional processes and related processes. Bernoulli 18(4), 1099–1127 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Le Guével, R., Lévy, Véhel J.: Incremental moments and Hölder exponents of multifractional multistable processes. ESAIM PS (2013).  https://doi.org/10.1051/ps/2011151 zbMATHGoogle Scholar
  17. 17.
    Le Guével R., Lévy Véhel. J.: Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes. Submitted. Arxiv arXiv:1412.0599 (2014)
  18. 18.
    Le Guével, R., Lévy-Véhel, J., Lining, L.: On two multistable extensions of stable Lévy motion and their semimartingale representation. J. Theoret. Probab. (2012).  https://doi.org/10.1007/s10959-013-0528-6 zbMATHGoogle Scholar
  19. 19.
    McKean, H.P. Jr.: Sample functions of stable processes. Ann. Math. Second Ser 61(3), 564–579 (1955)Google Scholar
  20. 20.
    Meerschaert, M., Xiao, Y.: Dimension results for sample paths of operator stable Lévy processes. Stoch. Process. App. 115, 55–75 (2005)CrossRefzbMATHGoogle Scholar
  21. 21.
    Millar, P.W.: Path behavior of processes with stationary independent increments. Z. Wahrsch. Verw. Gebiete 17, 53–73 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pruitt, W.E.: The Hausdorff dimension of the range of a process with stationary independent increments. J. Math. Mech. 19, 371–378 (1969)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Pruitt, W.E., Taylor, S.J.: Sample path properties of processes with stable components. Z. Wahrsch. Verw. Gebiete 12, 267–289 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rosinski, J.: On series representations of infinitely divisible random vectors. Ann. Probab. 18(1), 405–430 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994)zbMATHGoogle Scholar
  26. 26.
    Von Bahr, B., Essen, C.G.: Inequalities for the rth absolute moment of a sum of random variables, 1<=r<= 2. Ann. Math. Stat. 36(1), 299–303 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Univ Rennes, CNRS, IRMAR - UMR 6625RennesFrance

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