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Simulation of a Local Time Fractional Stable Motion

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Séminaire de Probabilités XLIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2006))

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Abstract

The aim of this paper is to simulate sample paths of a class of symmetric α-stable processes. This will be achieved by using the series expansion of the processes seen as shot noise series. In our case, as the general term of the series expansion has to be approximated, a first result is needed in shot noise theory. Then, this will lead to a convergence rate of the approximation towards the Local Time Fractional Stable Motion.

AMS 2000 Subject Classification: Primary 60G18, Secondary 60F25, 60E07, 60G52

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References

  1. Ahrens, J.H., Dieter, U.: Computer methods for sampling from the exponential and normal distributions. Commun. ACM 15, 873–882 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahrens, J.H., Dieter, U.: Extensions of Forsythe’s method for random sampling from the normal distribution. Math. Comp. 27, 927–937 (1973)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Biermé, H., Meerschaert, M.M., Scheffler, H.-P.: Operator scaling stable random fields. Stoch. Process. Appl. 117(3), 312–332 (2007)

    Article  MATH  Google Scholar 

  5. Cohen, S., Lacaux, C., Ledoux, M.: A general framework for simulation of fractional fields. Stoch. Process. Appl. 118(9), 1489–1517 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen, S., Samorodnistky, G.: Random rewards, fractional brownian local times and stable self-similar processes. Ann. Appl. Probab. 16, 1432–1461 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crovella, M.E., Bestavros, A.: Self-similarity in World Wide Web traffic: evidence and possiblecauses. IEEE/ACM Trans. Networking 5(6), 835–846 (1997)

    Article  Google Scholar 

  8. Davies, R.B., Harte, D.S.: Tests for Hurst effect. Biometrika 74(1), 95–101 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8(1), 1–67 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Janicki, A., Kokoszka, P.: Computer investigation of the rate of convergence of Lepage type series to α-stable random variables. Statistics 23(4), 365–373 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaj, I., Taqqu, MS: Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In: Vares, M.E., Sidoravicius, V. (eds.) Brazilian Probability School, 10th anniversary volume (2007)

    Google Scholar 

  12. Kolmogorov, A.N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. CR (Doklady) Acad. URSS (NS) 26(1), 15–1 (1940)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Lacaux, C.: Series representation and simulation of multifractional Levy motions. Adv. Appl. Prob. 36(1), 171–197 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer (1991)

    Google Scholar 

  16. Levy, J.B., Taqqu, M.S.: Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6(1), 23–44 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mandelbrot, B.B., Ness, J.W.V.: Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev. 10(4), 422–437 (1968)

    MATH  Google Scholar 

  18. Mikosch, T., Resnick, S., Rootzen, H., Stegeman, A.: Is network traffic approximated by stable Levy motion or fractional Brownian motion. Ann. Appl. Probab. 12(1), 23–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pitt, L.: Local times for gaussian vector fields. Indiana Univ. Math. J. 27, 309–330 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rosiński, J.: On path properties of certain infinitely divisible Processes. Stoch. Process. Appl. 33(1), 73–87 (1987)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Rosiński, J.: Series representations of Lévy processes from the perspective of point processes. In: Lévy Processes, pp. 401–415. Birkhäuser Boston, Boston, MA (2001)

    Google Scholar 

  23. Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman & Hall (1994)

    Google Scholar 

  24. Wood, A.T.A., Chan, G.: Simulation of stationary Gaussian processes in [0, 1]d. J. Comput. Graph. Stat. 3(4), 409–432 (1994)

    MathSciNet  Google Scholar 

  25. Xiao, Y.: Strong local nondeterminism and the sample path properties of Gaussian random fields. In: Asymptotic Theory in Probability and Statistics with Applications, pp. 136–176. Higher Education Press, Beijing (2007)

    Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062, Toulouse, France

    Matthieu Marouby

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  1. Matthieu Marouby
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Correspondence to Matthieu Marouby .

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Editors and Affiliations

  1. Lab. de Probabilités et Modèles Aléatoir, Université Pierre et Marie Curie, place Jussieu 4, Paris Cedex 05, 75252, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Marouby, M. (2011). Simulation of a Local Time Fractional Stable Motion. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIII. Lecture Notes in Mathematics(), vol 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15217-7_9

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