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Computer simulation of Lévy α-stable variables and processes

  • Aleksander Weron
  • Rafal Weron
Part I: Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 457)

Abstract

The aim of this paper is to demonstrate how the appropriate numerical, statistical and computer techniques can be successfully applied to the modeling of some physical systems. We propose to use a fast and accurate method of computer generation of Lévy α-stable random variates.

Keywords

Computer Generation Stable Distribution Infinite Variance Random Length Random Variable Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    L.M. Berliner, Statistics, probability and chaos, Stat. Science 7, 69–90 (1992).Google Scholar
  2. 2.
    S. Chatterjee and M. R. Yilmaz, Chaos, fractals and statistics, Stat. Science 7, 49–68 (1992).Google Scholar
  3. 3.
    J. M. Chambers, C. L. Mallows and B. Stuck, A method for simulating stable random variables, J. Amer. Statist. Assoc. 71, 340–344 (1976).Google Scholar
  4. 4.
    C. Daniel, Applications of Statistics to Industrial Experimentation, Wiley & Sons, New York, (1976).Google Scholar
  5. 5.
    L. Devroye, A Course in Density Estimation, Birkhäuser, Boston, (1987).Google Scholar
  6. 6.
    P. Embrechts, C. Kluppelberg and T. Mikosch, Modelling Extremal Events in Insurance and Finance, (1995), in preparation.Google Scholar
  7. 7.
    A. Janicki, Z. Michna and A. Weron Approximation of stochastic differential equations driven by α-stable Lévy motion, preprint, (1994).Google Scholar
  8. 8.
    A. Janicki and A. Weron Simulation and Chaotic Behavior of α-Stable Stochastic Processes, (1994), Marcel Dekker, New York.Google Scholar
  9. 9.
    A. Janicki and A. Weron, Can one see α-stable variables and processes ?, Stat. Science, 9, 109–126 (1994).Google Scholar
  10. 10.
    M. Kanter, Stable densities under change of scale and total variation inequalities, Ann. Probab. D31, 697–707 (1975).Google Scholar
  11. 11.
    J. Klafter, G. Zumofen and M.F. Shlesinger, Fractal description of anomalous diffusion in dynamical systems, Fractals, 1, 389–404 (1993).Google Scholar
  12. 12.
    I.A. Koutrouvelis, Regression-type estimation of the parameters of stable laws, J. Amer. Statist. Assoc., 75, 918–928 (1980).Google Scholar
  13. 13.
    A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, (1994), Springer, New York.Google Scholar
  14. 14.
    B. Mandelbrot, The variation of certain speculative prices, J. Business, 36, 394–419 (1963).Google Scholar
  15. 15.
    B. Mandelbrot and J. W. Van Nes, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10, 422–437 (1968).Google Scholar
  16. 16.
    R.M. Mantegna, Fast, accurate algorithm for numerical simulation of Lévy stable stochastic process, Phys. Rev. E49, 4677–4683 (1994).Google Scholar
  17. 17.
    G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes, (1994) Chapman & Hall, New York.Google Scholar
  18. 18.
    M. Shao and C. L. Nikias, Signal processing with fractional lower order moments: stable processes and their applications, Proc. IEEE, 81, 986–1010 (1993).Google Scholar
  19. 19.
    A. Weron, Stable processes and measures: A survey, in Probability Theory on Vector Spaces III (D.Szynal, A. Weron, eds.) 306–364, Lecture Notes in Mathematics 1080, (1984) Springer, New York.Google Scholar
  20. 20.
    R. Weron, On the Chambers-Mallows-Stuck method for simulating skewed stable random variables, Statist. Probab. Lett., (1995), submitted.Google Scholar
  21. 21.
    R. Weron, Performance of the estimators of stable law parameters, Appl. Math., (1995) submitted.Google Scholar
  22. 22.
    B.J. West and V. Seshadri, Linear systems with Lévy fluctuations, Physica, A113, 203–216 (1982).Google Scholar
  23. 23.
    G. Zumofen, A. Blumen, J. Klafter and M.F. Schlesinger, Lévy walks for turbulence: A numerical study, J. Stat. Phys., 54, 1519–1528 (1989).Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Aleksander Weron
    • 1
  • Rafal Weron
    • 1
  1. 1.The Hugo Steinhaus Center for Stochastic MethodsTechnical University of WroclawWroclawPoland

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