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.
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REFERENCES
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).
P. H. Cootner, ed., The Random Character of Stock Market Prices (MIT Press, 1964).
H. E. Daniels, Saddlepoint approximations in statistics, Ann. Math. Stat. 25:631–649 (1954).
R. Durrett and T. M. Liggett, Fixed points of the smoothing transformation, Z. Wahr. 64:275–301 (1983).
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.
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).
H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica (Utrecht) 8D:435–444 (1983).
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.
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).
Q. S. Liu, An extension of a functional equation of Poincaré and Mandelbrot, Asian Journal of Mathematics 6:145–168 (2002).
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.
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).
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.
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.
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.
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.
B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1982).
B. B. Mandelbrot, Fractals in physics: squig clusters, diffusions, fractal measures and the unicity of fractal dimension, J. Stat. Phys. 34:895–930 (1984).
B. B. Mandelbrot, Multifractal measures, especially for the geophysicist, Pure and Applied Geophysics 131:5–42 (1989a).
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.
B. B. Mandelbrot, Negative fractal dimensions and multifractals, Phys. A 163:306–315 (1990a).
B. B. Mandelbrot, New "anomalous" multiplicative multifractals: left-sided f (α) and the modeling of DLA, Physica A 168:95–111 (1990b).
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.
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).
B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (Selecta Volume E) (Springer-Verlag, 1997).
B. B. Mandelbrot, Multifractals and 1/f Noise: Wild Self-Affinity in Physics (Selecta Volume N) (Springer-Verlag, 1999a).
B. B. Mandelbrot, A multifractal walk through Wall Street, Scientific American, February issue, 50–53 (1999b).
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).
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).
B. B. Mandelbrot, Scaling in financial prices, III: Cartoon Brownian motions in multifractal time, Quantitative Finance 1:427–440 (2001c).
B. B. Mandelbrot, Scaling in financial prices, IV: Multifractal concentration, Quantitative Finance 1:558–559 (2001d).
B. B. Mandelbrot, Stochastic volatility, power-laws and long memory, Quantitative Finance 1:427–440 (2001e).
B. B. Mandelbrot, Gaussian Self-Affinity and Fractals (Selecta Volume H) (Springer-Verlag, 2002).
B. B. Mandelbrot, 2003, Forthcoming.
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.
A. M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13:77–81 (1962).
L. Telser, (ed.) Classic Futures: Lessons from the Past for the Electronic age (Risk Books, London, 2000).
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Mandelbrot, B.B. Multifractal Power Law Distributions: Negative and Critical Dimensions and Other “Anomalies,” Explained by a Simple Example. Journal of Statistical Physics 110, 739–774 (2003). https://doi.org/10.1023/A:1022159802564
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DOI: https://doi.org/10.1023/A:1022159802564