Journal of Statistical Physics

, Volume 110, Issue 3–6, pp 739–774

Multifractal Power Law Distributions: Negative and Critical Dimensions and Other “Anomalies,” Explained by a Simple Example

  • Benoit B. Mandelbrot
Article

Abstract

“Divergence of high moments and dimension of the carrier” is the subtitle of Mandelbrot's 1974 seed paper on random multifractals. The key words “divergence” and “dimension” met very different fates. “Dimension” expanded into a multifractal formalism based on an exponent α and a function f(α). An excellent exposition in Halsey et al. 1986 helped this formalism flourish. But it does not allow divergent high moments and the related inequalities f(α)<0 and α<0. As a result, those possibilities did not flourish. Now their time has come for diverse reasons. The broad 1974 definitions of α and f allow α<0 and f(α)<0, but the original presentation demanded to be both developed and simplified. This paper shows that both multifractal anomalies occur in a very simple example, which has been crafted for this purpose. This example predicts the power law distribution. It generalizes α and f(α) beyond their usual roles of being a Hölder exponent and a Hausdorff dimension. The effect is to allow either f or both f and α to be negative, and the apparent anomalies are made into sources of new important information. In addition, this paper substantially clarifies the subtle way in which randomness manifests itself in multifractals.

Multifractals power-law distribution negative dimensions critical dimensions anomalies two-valued canonical measure 

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REFERENCES

  1. H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Stat. 23:493–507 (1952).Google Scholar
  2. P. H. Cootner, ed., The Random Character of Stock Market Prices (MIT Press, 1964).Google Scholar
  3. H. E. Daniels, Saddlepoint approximations in statistics, Ann. Math. Stat. 25:631–649 (1954).Google Scholar
  4. R. Durrett and T. M. Liggett, Fixed points of the smoothing transformation, Z. Wahr. 64:275–301 (1983).Google Scholar
  5. U. Frisch and G. Parisi, Fully developed turbulence and intermittency, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, ed. (North-Holland, 1985), pp. 84–86, Excerpted in Mandelbrot 1999a.Google Scholar
  6. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33:1141–1151 (1986).Google Scholar
  7. H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica (Utrecht) 8D:435–444 (1983).Google Scholar
  8. J. P. Kahane and J. Peyrière, Sur certaines martingales de B. Mandelbrot, Adv. in Math 22:131–145 (1976). Translated in Mandelbrot 1999a as Chapter N17.Google Scholar
  9. A. N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13:82–85 (1962).Google Scholar
  10. Q. S. Liu, An extension of a functional equation of Poincaré and Mandelbrot, Asian Journal of Mathematics 6:145–168 (2002).Google Scholar
  11. B. B. Mandelbrot, The variation of certain speculative prices, The Journal of Business (Chicago) 36:394–419, (1963). Reprinted in Cootner 1964, as Chapter E 14 of Mandelbrot 1997, in Telser 2000, and several other collections of papers on finance.Google Scholar
  12. B. B. Mandelbrot, FRUne classe de processus stochastiques homothétiques à soi; application à la loi climatologique de H. E. Hurst, Comptes Rendus (Paris) 260:3274–3277 (1965). Translated as Chapter H9 of Mandelbrot (2002).Google Scholar
  13. B. B. Mandelbrot, The variation of some other speculative prices, Journal of Business (Chicago) 40:393–413, (1967). Reprinted as Chapter E14 of Mandelbrot 1997, 419–443, in Telser 2000, and several other collections of papers on finance.Google Scholar
  14. B. B. Mandelbrot, Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, in Statistical Models and Turbulence, M. Rosenblatt and C. Van Atta, eds. (Springer-Verlag, New York, 1972), pp. 333–351. Reprinted in Mandelbrot 1999a as Chapter N14.Google Scholar
  15. B. B. Mandelbrot, Intermittent turbulence in self similar cascades; divergence of high moments and dimension of the carrier, J. Fluid Mech. 62:331–358 (1974a). Reprinted in Mandelbrot 1999a as Chapter N15.Google Scholar
  16. B. B. Mandelbrot, FrMultiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire, Comptes Rendus (Paris) 278A:289–292 and 355–358 (1974b). Reprinted in Mandelbrot, 1999a as Chapter N16.Google Scholar
  17. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1982).Google Scholar
  18. B. B. Mandelbrot, Fractals in physics: squig clusters, diffusions, fractal measures and the unicity of fractal dimension, J. Stat. Phys. 34:895–930 (1984).Google Scholar
  19. B. B. Mandelbrot, Multifractal measures, especially for the geophysicist, Pure and Applied Geophysics 131:5–42 (1989a).Google Scholar
  20. B. B. Mandelbrot, A class of multinomial multifractal measures with negative (latent) values for the "dimension" f(α), in Fractals' Physical Origin and Properties, L. Pietronero, ed. (Plenum, New York, 1989b), pp. 3–29.Google Scholar
  21. B. B. Mandelbrot, Negative fractal dimensions and multifractals, Phys. A 163:306–315 (1990a).Google Scholar
  22. B. B. Mandelbrot, New "anomalous" multiplicative multifractals: left-sided f (α) and the modeling of DLA, Physica A 168:95–111 (1990b).Google Scholar
  23. B. B. Mandelbrot, Limit lognormal multifractal measures, in Frontiers of Physics: Landau Memorial Conference, E. A. Gotsman, Y. Ne'eman, and A. Voronel, eds. (Pergamon, New York, 1990c), pp. 309–340.Google Scholar
  24. B. B. Mandelbrot, Negative dimensions and Hölders, multifractals and their Hölder spectra, and the role of lateral preasymptotics in science, in J. P. Kahane's meeting (Paris, 1993). A. Bonami and J. Peyrière, eds., The Journal of Fourier Analysis and Applications special issue, 409–432 (1995).Google Scholar
  25. B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (Selecta Volume E) (Springer-Verlag, 1997).Google Scholar
  26. B. B. Mandelbrot, Multifractals and 1/f Noise: Wild Self-Affinity in Physics (Selecta Volume N) (Springer-Verlag, 1999a).Google Scholar
  27. B. B. Mandelbrot, A multifractal walk through Wall Street, Scientific American, February issue, 50–53 (1999b).Google Scholar
  28. B. B. Mandelbrot, Scaling in financial prices, I: Tails and dependence, Quantitative Finance 1:113–124, (2001a). Reprint: Beyond Efficiency and Equilibrium, Doyne Farmer and John Geanakoplos, eds. (Oxford UK, The University Press, 2002).Google Scholar
  29. B. B. Mandelbrot, Scaling in financial prices, II: Multifractals and the star equation, Quantitative Finance 1:124–130, (2001b). Reprint: Beyond Efficiency and Equilibrium, Doyne Farmer and John Geanakoplos, eds. (Oxford UK, The University Press, 2002).Google Scholar
  30. B. B. Mandelbrot, Scaling in financial prices, III: Cartoon Brownian motions in multifractal time, Quantitative Finance 1:427–440 (2001c).Google Scholar
  31. B. B. Mandelbrot, Scaling in financial prices, IV: Multifractal concentration, Quantitative Finance 1:558–559 (2001d).Google Scholar
  32. B. B. Mandelbrot, Stochastic volatility, power-laws and long memory, Quantitative Finance 1:427–440 (2001e).Google Scholar
  33. B. B. Mandelbrot, Gaussian Self-Affinity and Fractals (Selecta Volume H) (Springer-Verlag, 2002).Google Scholar
  34. B. B. Mandelbrot, 2003, Forthcoming.Google Scholar
  35. B. B. Mandelbrot, L. Calvet, and A. Fisher, The Multifractal Model of Asset Returns. Large Deviations and the Distribution of Price Changes. The Multifractality of the Deutschmark/US Dollar Exchange Rate. Discussion Papers numbers 1164, 1165, and 1166 of the Cowles Foundation for Economics at Yale University, New Haven, CT, 1997. Available on the web: http://papers.ssrn.com/sol3/paper.taf? ABSTRACT_ID=78588. http://papers.ssrn.com/sol3/paper.taf? ABSTRACT_ID=78606. http://papers.ssrn.com/sol3/paper.taf? ABSTRACT_ID=78628.Google Scholar
  36. A. M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13:77–81 (1962).Google Scholar
  37. L. Telser, (ed.) Classic Futures: Lessons from the Past for the Electronic age (Risk Books, London, 2000).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
  1. 1.Sterling Professor of Mathematical SciencesYale University New Haven

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