Skip to main content
Log in

Fourier Series Approximation of Linear Fractional Stable Motion

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

An approximation of the linear fractional stable motion by a Fourier sum is presented. In the continuous sample path case precise error bounds are derived. This approximation method is used to develop a simulation method of the sample path of linear fractional stable motions.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Chambers, J.M., Mallows, C., Stuckcw, B.W.: A method for simulating stable random variables. J. Amer. Statist. Assoc. 71(354), 340–344 (1976). With a correction in J. Amer. Statist. Assoc. 82(398), 704

    Article  MATH  MathSciNet  Google Scholar 

  2. Cohen, S., Lacaux, C., Ledoux, M.: A general framework for simulation of fractional fields. Stoch. Process. Appl. (2008, in press)

  3. Dietrich, C.R., Newsam, G.N.: Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18(4), 1088–1107 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Doukhan, P., Oppenheim, G., Taqqu, M.S.: Theory and Applications of Long-Range Dependence. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  5. Embrechts, P., Maejima, M.: Self-Similar Processes. Princeton University Press, Princeton (2002)

    Google Scholar 

  6. Greenspan, D.: Introduction to Numerical Analysis and Applications. Markham Publishing Company, Chicago (1971)

    MATH  Google Scholar 

  7. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)

    MATH  Google Scholar 

  8. Katznelson, Y.: An Introduction to Harmonic Analysis. Dover, New York (1976)

    MATH  Google Scholar 

  9. Kôno, N., Maejima, M.: Hölder continuity of sample path of some self-similar stable processes. Tokyo J. Math. 14, 93–100 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V.: On the self-similar nature of ethernet traffic. IEEE/ACM Transactions on Networking 2, 1–15 (1994)

    Article  Google Scholar 

  11. Park, K., Willinger, W.: Self-Similar Network Traffic and Performance Evaluation. Wiley, New York (2000)

    Google Scholar 

  12. Perrin, E., Harba, R., Jennane, R., Iribarren, I.: Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises. IEEE Signal Process. Lett. 9(11), 382–384 (2002)

    Article  Google Scholar 

  13. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Stoev, S., Pipiras, V., Taqqu, M.S.: Estimation of the self-similarity parameter in linear fractional stable motion. Signal Process. 82, 1873–1901 (2002)

    Article  MATH  Google Scholar 

  17. Takashima, K.: Sample path properties of ergodic self-similar processes. Osaka J. Math. 26, 159–189 (1989)

    MATH  MathSciNet  Google Scholar 

  18. Willinger, W., Paxson, V., Taqqu, M.S.: Self-similarity and heavy tails: structural modeling of network traffic. In: Adler, R., Feldman, R., Taqqu, M. (eds.) A Practical Guide to Heavy Tails (Statistical) Techniques, and Applications, pp. 27–53. Birkhäuser, Boston (1998)

    Google Scholar 

  19. Wu, W.B., Michailidis, G., Zhang, D.: Simulating sample path of linear fractional stable motion. IEEE Trans. Inform. Theory 50(6), 1086–1096 (2004)

    Article  MathSciNet  Google Scholar 

  20. Wu, W.B., Michailidis, G., Zhang, D.: Simulation Conference (2002). Proceedings of the Winter 2, 1958–1963 (2004)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hans-Peter Scheffler.

Additional information

The second author was partially supported by NSF grant DMS-0417869.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Biermé, H., Scheffler, HP. Fourier Series Approximation of Linear Fractional Stable Motion. J Fourier Anal Appl 14, 180–202 (2008). https://doi.org/10.1007/s00041-008-9011-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-008-9011-7

Keywords

Mathematics Subject Classification (2000)

Navigation