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Multifractal products of stochastic processes: construction and some basic properties

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Petteri Mannersalo ,
Ilkka Norros  and
Rudolf H. Riedi
Petteri Mannersalo*
Affiliation:
VTT Technical Research Centre of Finland
Ilkka Norros*
Affiliation:
VTT Technical Research Centre of Finland
Rudolf H. Riedi*
Affiliation:
Rice University
*
∗ Postal address: VTT Information Technology, PO Box 1202, FIN-02044 VTT, Finland.
∗ Postal address: VTT Information Technology, PO Box 1202, FIN-02044 VTT, Finland.
∗∗∗ Postal address: Department of Electrical and Computer Engineering, Digital Signal Processing Group, Rice University, MS 380, Houston, TX 77251-1892, USA.
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Abstract

In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a nonstationary process. To overcome this problem, we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study the ℒ2-convergence, nondegeneracy, and continuity of the limit process. Establishing a power law for its moments, we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.

Keywords

Stochastic processesrandom measuresmultifractalsteletraffic modeling

MSC classification

Primary: 60G57: Random measures
Secondary: 60G30: Continuity and singularity of induced measures
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 888 - 903
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1] Arbeiter, M. (1991). Random recursive construction of self-similar fractal measures. The noncompact case. Prob. Theory Relat. Fields 88, 497–520.Google Scholar
[2] Arbeiter, M. and Patzchke, N. (1996). Random self-similar multifractals. Math. Nachr. 181, 5–42.CrossRefGoogle Scholar
[3] Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Prob. Theory Relat. Fields 113, 535–569.Google Scholar
[4] Burd, G. and Waymire, E. (2000). Independent random cascades on Galton–Watson trees. Proc. Amer. Math. Soc. 128, 2753–2761.Google Scholar
[5] Falconer, K. (1994). The multifractal spectrum of statistically self-similar measures. J. Theoret. Prob. 7, 681–702.Google Scholar
[6] Fan, A. (1997). Sur le chaos de Lévy d'indice 0 < α <1. Ann. Sci. Math. Québec 21, 53–66.Google Scholar
[7] Fan, A. and Kahane, J. (2001). How many intervals cover a point in a random dyadic covering? Portugaliae Math. 58, 59–75.Google Scholar
[8] Fan, A. and Shieh, N. (2000). Multifractal spectrum for some random Gibbs measures. Statist. Prob. Lett. 47, 25–31.Google Scholar
[9] Feldmann, A., Gilbert, A. and Willinger, W. (1998). Data networks as cascades: investigating the multifractal nature of Internet WAN traffic. In Proc. ACM SIGCOMM'98 Conf. (Vancouver, September 1998), ACM, New York, pp. 42–55.Google Scholar
[10] Gupta, V. and Waymire, E. (1990). Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res. Atmospheres 95, 1999–2009.Google Scholar
[11] Holley, R. and Waymire, E. (1992). Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Prob. 2, 819–845.CrossRefGoogle Scholar
[12] Jaffard, S. (1996). Sur la nature multifractale des processus du Lévy. C. R. Acad. Sci. Paris 323, 1059–1064.Google Scholar
[13] Jaffard, S. (1999). The multifractal nature of Lévy processes. Prob. Theory Relat. Fields 114, 207–227.Google Scholar
[14] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9, 105–150.Google Scholar
[15] Kahane, J.-P. (1987). Positive martingales and random measures. Chinese Ann. Math. 8, 1–12.Google Scholar
[16] Kahane, J. (1989). Random multiplications, random coverings and multiplicative chaos. In Analysis at Urbana, Vol. 1 (London Math. Soc. Lecture Notes 137), eds Berkson, E., Peck, N. and Uhl, J., Cambridge University Press, pp. 196–255.Google Scholar
[17] Kahane, J. (2000). Random coverings and multiplicative processes. In Fractal Geometry and Stochastics, Vol. 2 (Prog. Prob. 46), eds Bandt, C., Graf, S. and Zähle, M., Birkhäuser, Basel, pp. 125–146.Google Scholar
[18] Kahane, J.-P. and Peyriére, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145.Google Scholar
[19] Lévy Véhel, J. and Riedi, R. (1997). Fractional Brownian motion and data traffic modeling: the other end of the spectrum. In Fractals in Engineering, eds Lévy Véhel, J., Lutton, E. and Tricot, C., Springer, Berlin, pp. 185–202.Google Scholar
[20] Mandelbrot, B. (1972). Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence (Lecture Notes Phys. 12), eds Rosenblatt, M. and Van Atta, C., Springer, Berlin, pp. 331–351.Google Scholar
[21] Mandelbrot, B. (1974). Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358.Google Scholar
[22] Mannersalo, P. and Norros, I. (1997). Multifractal analysis of real ATM traffic: a first look. Tech. Rep. COST257TD(97)19, VTT Information Technology.Google Scholar
[23] Molchan, G. (1996). Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681–702.Google Scholar
[24] Molchan, G. (2002). Mandelbrot cascade measures independent of branching parameter. J. Statist. Phys. 107, 977–988.Google Scholar
[25] Patzchke, N. (1997). Self-conformal multifractal measures. Adv. Appl. Math. 19, 486–513.Google Scholar
[26] Peyriére, J., (1977). Calculs de dimensions de Hausdorff. Duke Math. J. 44, 591–601.Google Scholar
[27] Riedi, R. and Lévy Véhel, J. (1997). TCP traffic is multifractal: a numerical study. Res. Rep. 3129, INRIA, Rocquencourt.Google Scholar
[28] Riedi, R., Crouse, M., Ribeiro, V. and Baraniuk, R. (1999). A multifractal wavelet model with application to network traffic. IEEE Trans. Inf. Theory 45, 992–1018.Google Scholar
[29] Roberts, J., Mocci, U. and Virtamo, J. (eds) (1997). Broadband Network Teletraffic (Lecture Notes Comput. Sci. 1155). Springer, Berlin.Google Scholar
[30] Waymire, E. and Williams, S. (1995). Multiplicative cascades: dimension spectra and dependence. J. Fourier Anal. Appl. Spec. Issue, 589–609.Google Scholar
[31] Waymire, E. and Williams, S. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348, 585–631.Google Scholar
[32] Waymire, E. and Williams, S. (1997). Markov cascades. In Classical and Modern Branching Processes (IMA Vol. Math. Appl. 84), eds Athreya, K. and Jagers, P., Springer, Berlin, pp. 305–321.Google Scholar