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.
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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
Cohen, S., Lacaux, C., Ledoux, M.: A general framework for simulation of fractional fields. Stoch. Process. Appl. (2008, in press)
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)
Doukhan, P., Oppenheim, G., Taqqu, M.S.: Theory and Applications of Long-Range Dependence. Birkhäuser, Boston (2003)
Embrechts, P., Maejima, M.: Self-Similar Processes. Princeton University Press, Princeton (2002)
Greenspan, D.: Introduction to Numerical Analysis and Applications. Markham Publishing Company, Chicago (1971)
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)
Katznelson, Y.: An Introduction to Harmonic Analysis. Dover, New York (1976)
Kôno, N., Maejima, M.: Hölder continuity of sample path of some self-similar stable processes. Tokyo J. Math. 14, 93–100 (1991)
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)
Park, K., Willinger, W.: Self-Similar Network Traffic and Performance Evaluation. Wiley, New York (2000)
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)
Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994)
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)
Stoev, S., Pipiras, V., Taqqu, M.S.: Estimation of the self-similarity parameter in linear fractional stable motion. Signal Process. 82, 1873–1901 (2002)
Takashima, K.: Sample path properties of ergodic self-similar processes. Osaka J. Math. 26, 159–189 (1989)
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)
Wu, W.B., Michailidis, G., Zhang, D.: Simulating sample path of linear fractional stable motion. IEEE Trans. Inform. Theory 50(6), 1086–1096 (2004)
Wu, W.B., Michailidis, G., Zhang, D.: Simulation Conference (2002). Proceedings of the Winter 2, 1958–1963 (2004)
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The second author was partially supported by NSF grant DMS-0417869.
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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
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DOI: https://doi.org/10.1007/s00041-008-9011-7