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
Ahrens, J.H., Dieter, U.: Computer methods for sampling from the exponential and normal distributions. Commun. ACM 15, 873–882 (1972)
Ahrens, J.H., Dieter, U.: Extensions of Forsythe’s method for random sampling from the normal distribution. Math. Comp. 27, 927–937 (1973)
Benassi, A., Cohen, S., Istas, J.: Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8(1), 97–115 (2002)
Biermé, H., Meerschaert, M.M., Scheffler, H.-P.: Operator scaling stable random fields. Stoch. Process. Appl. 117(3), 312–332 (2007)
Cohen, S., Lacaux, C., Ledoux, M.: A general framework for simulation of fractional fields. Stoch. Process. Appl. 118(9), 1489–1517 (2008)
Cohen, S., Samorodnistky, G.: Random rewards, fractional brownian local times and stable self-similar processes. Ann. Appl. Probab. 16, 1432–1461 (2006)
Crovella, M.E., Bestavros, A.: Self-similarity in World Wide Web traffic: evidence and possiblecauses. IEEE/ACM Trans. Networking 5(6), 835–846 (1997)
Davies, R.B., Harte, D.S.: Tests for Hurst effect. Biometrika 74(1), 95–101 (1987)
Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8(1), 1–67 (1980)
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)
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)
Kolmogorov, A.N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. CR (Doklady) Acad. URSS (NS) 26(1), 15–1 (1940)
Lacaux, C.: Real harmonizable multifractional Lévy motions. Ann. Inst. H. Poincar Probab. Statist. 40(3), 259–277 (2004)
Lacaux, C.: Series representation and simulation of multifractional Levy motions. Adv. Appl. Prob. 36(1), 171–197 (2004)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer (1991)
Levy, J.B., Taqqu, M.S.: Renewal reward processes with heavy-tailed interrenewal times and heavy-tailed rewards. Bernoulli 6(1), 23–44 (2000)
Mandelbrot, B.B., Ness, J.W.V.: Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev. 10(4), 422–437 (1968)
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)
Pitt, L.: Local times for gaussian vector fields. Indiana Univ. Math. J. 27, 309–330 (1978)
Rosiński, J.: On path properties of certain infinitely divisible Processes. Stoch. Process. Appl. 33(1), 73–87 (1987)
Rosiński, J.: On series representations of infinitely divisible random vectors. Ann. Probab. 18(1), 405–430 (1990)
Rosiński, J.: Series representations of Lévy processes from the perspective of point processes. In: Lévy Processes, pp. 401–415. Birkhäuser Boston, Boston, MA (2001)
Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman & Hall (1994)
Wood, A.T.A., Chan, G.: Simulation of stationary Gaussian processes in [0, 1]d. J. Comput. Graph. Stat. 3(4), 409–432 (1994)
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)
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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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