Skip to main content
SpringerLink
Log in

Fractional Lévy Fields

  • Chapter
  • First Online:
Lévy Matters II

Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2061))

Abstract

In this survey, we would like to summarize most of the results concerning the so-called fractional Lévy fields in a way as self-contained as possible. Beside the construction of these fields, we are interested in the regularity of their sample paths, and self-similarity properties of their distributions. It turns out that for applications, we often need non-homogeneous fields that are only locally self-similar. Then we explain how to identify those models from a discrete sample of one realization of the field. At last some simulation techniques are discussed.

Mathematics Subject Classification 2000: Primary: 60G10, 60G70, 60J10 Secondary: 91B28, 91B84

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Abry, P. Gonçalves, J. Lévy Véhel (eds.), in Scaling, Fractals and Wavelets. Digital Signal and Image Processing Series (ISTE, London, 2009)

    Google Scholar 

  2. S. Asmussen, J. Rosiński, Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38(2), 482–493 (2001)

    Google Scholar 

  3. A. Ayache, A. Benassi, S. Cohen, J.L. Véhel, in Regularity and Identification of Generalized Multifractional Gaussian Processes. Séminaire de Probabilité XXXVIII. Lecture Notes in Mathematics, vol. 1804 (Springer Verlag, 2004)

    Google Scholar 

  4. J.M. Bardet, Testing for the presence of self-similarity of gaussian time series having stationary increments. J. Time Ser. Anal. 21(5), 497–515 (2000)

    Google Scholar 

  5. J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe, M.S. Taqqu, in Generators of Long-Range Dependent Processes: A Survey. Theory and Applications of Long-Range Dependence (Birkhäuser, Boston, 2003), pp. 579–623

    Google Scholar 

  6. A. Benassi, S. Jaffard, D. Roux, Gaussian processes and pseudodifferential elliptic operators. Revista Mathematica Iberoamericana 13(1), 19–90 (1996)

    Google Scholar 

  7. A. Benassi, S. Cohen, J. Istas, Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8(1), 97–115 (2002)

    Google Scholar 

  8. J.-F. Coeurjolly, Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stochast. Process. 4(2), 199–227 (2001)

    Google Scholar 

  9. S. Cohen, J. Istas, Fractional fields and applications (2011), http://www.math.univ-toulouse.fr/~cohen

  10. S. Cohen, M.S. Taqqu, Small and large scale behavior of the Poissonized telecom process. Meth. Comput. Appl. Probab. 6(4), 363–379 (2004)

    Google Scholar 

  11. L. Debnath (ed.), in Wavelets and Signal Processing. Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, 2003)

    Google Scholar 

  12. M. Dekking, J.L. Véhel, E. Lutton, C. Tricot (eds.), Fractals: Theory and Applications in Engineering (Springer, Berlin, 1999)

    Google Scholar 

  13. M.E. Dury, Identification et simulation d’une classe de processus stables autosimilaires à accroissements stationnaires. Ph.D. thesis, Université Blaise Pascal, Clermont-Ferrand (2001)

    Google Scholar 

  14. P. Flandrin, P. Abry, in Wavelets for Scaling Processes. Fractals: Theory and Applications in Engineering (Springer, London, 1999), pp. 47–64

    Google Scholar 

  15. J. Istas, G. Lang, Quadratic variations and estimation of the Holder index of a gaussian process. Ann. Inst. Poincaré 33(4), 407–436 (1997)

    Google Scholar 

  16. A. Janssen, Zero-one laws for infinitely divisible probability measures on groups. Zeitschrift für Wahrscheinlichkeitstheorie 60, 119–138 (1982)

    Google Scholar 

  17. I. Kaj, M. Taqqu, in Convergence to Fractional Brownian Motion and to the Telecom Process: The Integral Representation Approach, ed. by M.E. Vares, V. Sidoravicius. Brazilian Probability School, 10th anniversary volume (Birkhauser, Boston, 2007)

    Google Scholar 

  18. O. Kallenberg, in Foundations of Modern Probability, 2nd edn. Probability and Its Applications (New York) (Springer, New York, 2002)

    Google Scholar 

  19. A. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum. C. R. (Dokl.) Acad. Sci. URSS 26, 115–118 (1940)

    Google Scholar 

  20. H. Kunita, in Stochastic Flows and Stochastic Differential Equations, vol. 24 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1997). Reprint of the 1990 original

    Google Scholar 

  21. C. Lacaux, Real harmonizable multifractional Lévy motions. Ann. Inst. H. Poincaré Probab. Statist. 40(3), 259–277 (2004)

    Google Scholar 

  22. C. Lacaux, Series representation and simulation of multifractional Lévy motions. Adv. Appl. Probab. 36(1), 171–197 (2004)

    Google Scholar 

  23. M. Ledoux, M. Talagrand, in Probability in Banach Spaces, vol. 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Isoperimetry and Processes (Springer, Berlin, 1991)

    Google Scholar 

  24. S. Mallat, A theory of multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989)

    Google Scholar 

  25. B. Mandelbrot, J. Van Ness, Fractional Brownian motion, fractional noises and applications. Siam Rev. 10, 422–437 (1968)

    Google Scholar 

  26. T. Marquardt, Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12(6), 1099–1126 (2006)

    Google Scholar 

  27. R. Peltier, J. Lévy-Vehel, Multifractional Brownian motion: Definition and preliminary results (1996). Prepublication, http://www-syntim.inria.fr/fractales/

  28. V.V. Petrov, An estimate of the deviation of the distribution of independent random variables from the normal law. Soviet Math. Doklady 6, 242–244 (1982)

    Google Scholar 

  29. P. Protter, Stochastic Integration and Differential Equations (Springer, Berlin, 1990)

    Google Scholar 

  30. B.S. Rajput, J. Rosiński, Spectral representations of infinitely divisible processes. Probab. Theor. Relat. Fields 82(3), 451–487 (1989)

    Google Scholar 

  31. J. Rosinski, On path properties of certain infinitely divisible process. Stochast. Process. Appl. 33, 73–87 (1989)

    Google Scholar 

  32. J. Rosiński, On series representations of infinitely divisible random vectors. Ann. Probab. 18(1), 405–430 (1990)

    Google Scholar 

  33. J. Rosiński, in Series Representations of Lévy Processes from the Perspective of Point Processes. Lévy Processes (Birkhäuser, Boston, 2001), pp. 401–415

    Google Scholar 

  34. G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes (Chapman and Hall, London, 1994)

    Google Scholar 

  35. K.-I. Sato, in Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1999). Translated from the 1990 Japanese original, Revised by the author

    Google Scholar 

  36. E. Simoncelli, in Bayesian Denoising of Visual Images in the Wavelet Domain. Lect. Notes Stat., vol. 141 (Springer, Berlin, 1999), pp. 291–308

    Google Scholar 

  37. S. Stoev, M.S. Taqqu, Simulation methods for linear fractional stable motion and FARIMA using the fast Fourier transform. Fractals 12(1), 95–121 (2004)

    Google Scholar 

  38. S. Stoev, M.S. Taqqu, Stochastic properties of the linear multifractional stable motion. Adv. Appl. Probab. 36(4), 1085–1115 (2004)

    Google Scholar 

  39. S. Stoev, M.S. Taqqu, Path properties of the linear multifractional stable motion. Fractals 13(2), 157–178 (2005)

    Google Scholar 

  40. S. Stoev, M.S. Taqqu, How rich is the class of multifractional Brownian motion. Stochast. Process. Appl. 116(1), 200–221 (2006)

    Google Scholar 

  41. B. Vidakovic, Statistical Modeling by Wavelets (Wiley, New York, 1999)

    Google Scholar 

  42. W.B. Wu, G. Michailidis, D. Zhang, Simulating sample paths of linear fractional stable motion. IEEE Trans. Inform. Theor. 50(6), 1086–1096 (2004)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank Claudia Küppelberg for the friendly pressure she put on him. Without this help and her patience he is doubtful that this survey would have been ever written. I would also like to thank both referees for their careful reading that improved my first version.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, Université de Toulouse, 118 Route de Narbonne, 31062, Toulouse Cedex 9, France

    Serge Cohen

Authors
  1. Serge Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Cohen, S. (2012). Fractional Lévy Fields. In: Lévy Matters II. Lecture Notes in Mathematics(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31407-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation