Abstract
We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abry, P., Flandrin, P. Taqqu, M.S., and Veitch, D. (1998). Wavelets for the analysis, estimation and synthesis of scaling data. In Park, K. and Willinger, W., Eds.,Self-Similar Network Traffic and Performance Evaluation, Wiley (Interscience Division), New York. To appear.
Abry, P. and Sellan, F. (1996). The wavelet-based synthesis for the fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation,Appl. Comp. Harmonic Anal.,3(4), 377–383.
Abry, P. and Veitch, D. (1997). Wavelet analysis of long range dependent traffic,IEEE Trans. Info. Theor., To appear.
Beran, J. (1994).Statistics for Long-Memory Processes, Chapman & Hall, New York.
Billingsley, P. (1968).Convergence of Probability Measures, John Wiley & Sons, New York.
Brockwell, P.J. and Davis, R.A. (1987).Time Series: Theory and Methods, Springer-Verlag, New York.
Chow, Y.S. and Teicher, H. (1988).Probability Theory: Independence, Interchangeability, Martingales, 2nd ed., Springer-Verlag, New York.
Cohen, A. and Ryan, R.D. (1995).Wavelets and Multiscale Signal Processing, Chapman & Hall, London.
Elliott. F.W. and Majda, A.J. (1994). A wavelet Monte Carlo method for turbulent diffusion with many spatial scales,J. Comp. Phys.,113(1), 82–111.
Flandrin, P. (1992). Wavelet analysis and synthesis of fractional Brownian motion,IEEE Trans. Info. Theor.,IT-38(2), 910–917.
Gelfand, I.M. and Shilov, G.E. (1964).Generalized Functions: Properties and Operations, vol. 1, Academic Press, New York.
Gelfand, I.M. and Vilenkin, N.Ya. (1964).Generalized Functions: Applications of Harmonic Analysis, vol.4, Academic Press, New York.
Heneghan, C., Lowen, S.B., and Teich, M.C. (1996). Two-dimensional fractional Brownian motion: wavelet analysis and synthesis,Proc. IEEE Southwest Symp. Image Anal. Interpretation, San Antonio.
Hernández, E. and Weiss, G. (1996).A First Course on Wavelets, CRC Press, Boca Raton, FL.
Houdré, C. (1993). Wavelets, probability and statistics: some bridges, In Benedetto, J.J. and Frazier, M.W., Eds.,Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 361–394.
Jaffard, S. and Meyer, Y. (1996).Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions, vol. 587, American Mathematical Society, Providence, RI.
Kwapień, S. and Woyczyński, N.A. (1992).Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston.
Lau, W.-C., Erramilli, A., Wang, J.L., and Willinger, W. (1995). Self-similar traffic generation: The random midpoint displacement algorithm and its properties, InProc. ICC '95, Seattle, WA, 466–472.
Leland, W.E., Taqqu, M.S., Willinger, W., and Wilson, D.V. (1994). On the self-similar nature of Ethernet traffic (Extended version),IEEE/ACM Trans. Networking,2, 1–15.
Lévy, P. (1948).Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1st ed., 2nd éd., 1965.
Mandelbrot, B.B. (1982).The Fractal Geometry of Nature. Freeman, W.W., and Co., Eds., New York.
Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications,SIAM Rev.,10, 422–437.
Masry, E. (1993). The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion,IEEE Trans. Info. Theor.,39(1), 260–264.
Meyer, Y. (1992).Ondelettes, Hermann, Paris.
Meyer, Y. (1992).Wavelets and Operators, vol. 37. Cambridge University Press, Cambridge, UK.
Meyer, Y. (1993).Wavelets—Algorithms and Applications, SIAM Society for Industrial and Applied Mathematics, Philadelphia, PA.
Meyer, Y. and Coifman, R. (1997).Wavelets: Calderón-Zygmund and Multilinear Operators, vol. 48. Cambridge University Press, Cambridge, UK.
Norros, I. (1995). On the use of fractional Brownian motion in the theory of connectionless networks,IEEE J. Selected Areas Comm.,13, 953–962.
Pipiras, V. and Taqqu, M.S. (1998). Convergence of the Weierstrass-Mandelbrot process to fractional Brownian motion, preprint.
Ramanathan, J. and Zeitouni, O. (1991). On the wavelet transform of fractional Brownian motion,IEEE Trans. Info. Theor.,IT-37(4), 1156–1158.
Samorodnitsky, G. and Taqqu, M.S. (1994).Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York.
Sellan, F. (1995). Synthèse de mouvements browniens fractionnaires à l'aide de la transformation par ondelettes,Comptes Rendus de l'Académie des Sciences de Paris,321, 351–358, Série I.
Sinai, Ya.G. (1976). Self-similar probability distributions,Theor. Prob. Appl.,21, 64–80.
Wornell, G. (1996).Signal Processing with Fractals: A Wavelet-Based Approach, Prentice-Hall, Upper Saddle River, NJ.
Zygmund, A. (1979).Trigonometric Series, Vols. I and II, Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Additional information
Communicated by John J. Benedetto
Rights and permissions
About this article
Cite this article
Meyer, Y., Sellan, F. & Taqqu, M.S. Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion. The Journal of Fourier Analysis and Applications 5, 465–494 (1999). https://doi.org/10.1007/BF01261639
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01261639
Keywords and Phrases
- Fractional ARIMA
- midpoint displacement technique
- fractional Gaussian noise
- fractional derivative
- generalized functions
- self-similarity