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Volumn 110, Issue 3-6, 2003, Pages 739-774

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

Author keywords

Anomalies; Critical dimensions; Multifractals; Negative dimensions; Power law distribution; Two valued canonical measure

Indexed keywords


EID: 0037273511     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022159802564     Document Type: Conference Paper
Times cited : (49)

References (38)
  • 1
    • 0000182415 scopus 로고
    • A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations
    • 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).
    • (1952) Ann. Math. Stat. , vol.23 , pp. 493-507
    • Chernoff, H.1
  • 3
    • 0001367156 scopus 로고
    • Saddlepoint approximations in statistics
    • H. E. Daniels, Saddlepoint approximations in statistics, Ann. Math. Stat. 25:631-649 (1954).
    • (1954) Ann. Math. Stat. , vol.25 , pp. 631-649
    • Daniels, H.E.1
  • 4
    • 51249186049 scopus 로고
    • Fixed points of the smoothing transformation
    • R. Durrett and T. M. Liggett, Fixed points of the smoothing transformation, Z. Wahr. 64:275-301 (1983).
    • (1983) Z. Wahr. , vol.64 , pp. 275-301
    • Durrett, R.1    Liggett, T.M.2
  • 6
    • 3342916075 scopus 로고
    • Fractal measures and their singularities: The characterization of strange sets
    • 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).
    • (1986) Phys. Rev. A , vol.33 , pp. 1141-1151
    • Halsey, T.C.1    Jensen, M.H.2    Kadanoff, L.P.3    Procaccia, I.4    Shraiman, B.I.5
  • 7
    • 0346372923 scopus 로고
    • The infinite number of generalized dimensions of fractals and strange attractors
    • H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica (Utrecht) 8D:435-444 (1983).
    • (1983) Physica (Utrecht) , vol.8 D , pp. 435-444
    • Hentschel, H.G.E.1    Procaccia, I.2
  • 8
    • 0003329153 scopus 로고
    • Sur certaines martingales de B. Mandelbrot
    • Translated in Mandelbrot 1999a as Chapter N17
    • 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.
    • (1976) Adv. in Math , vol.22 , pp. 131-145
    • Kahane, J.P.1    Peyrière, J.2
  • 9
    • 84958440725 scopus 로고
    • A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number
    • 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).
    • (1962) J. Fluid Mech. , vol.13 , pp. 82-85
    • Kolmogorov, A.N.1
  • 10
    • 0347234098 scopus 로고    scopus 로고
    • An extension of a functional equation of Poincaré and Mandelbrot
    • Q. S. Liu, An extension of a functional equation of Poincaré and Mandelbrot, Asian Journal of Mathematics 6:145-168 (2002).
    • (2002) Asian Journal of Mathematics , vol.6 , pp. 145-168
    • Liu, Q.S.1
  • 11
    • 0001504360 scopus 로고
    • The variation of certain speculative prices
    • 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, 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.
    • (1963) The Journal of Business (Chicago) , vol.36 , pp. 394-419
    • Mandelbrot, B.B.1
  • 12
    • 0040133826 scopus 로고
    • Une classe de processus stochastiques homothétiques à soi; application à la loi climatologique de H. E. Hurst
    • Translated as Chapter H9 of Mandelbrot (2002)
    • B. B. Mandelbrot, Une 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).
    • (1965) Comptes Rendus (Paris) , vol.260 , pp. 3274-3277
    • Mandelbrot, B.B.1
  • 13
    • 0000036811 scopus 로고
    • The variation of some other speculative prices
    • Reprinted as Chapter E14 of Mandelbrot 1997, 419-443, in Telser 2000, and several other collections of papers on finance
    • 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.
    • (1967) Journal of Business (Chicago) , vol.40 , pp. 393-413
    • Mandelbrot, B.B.1
  • 14
    • 0008354399 scopus 로고
    • Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence
    • M. Rosenblatt and C. Van Atta, eds. (Springer-Verlag, New York). Reprinted in Mandelbrot 1999a as Chapter N14
    • 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.
    • (1972) Statistical Models and Turbulence , pp. 333-351
    • Mandelbrot, B.B.1
  • 15
    • 0015969022 scopus 로고
    • Intermittent turbulence in self similar cascades; divergence of high moments and dimension of the carrier
    • Reprinted in Mandelbrot 1999a as Chapter N15
    • 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.
    • (1974) J. Fluid Mech. , vol.62 , pp. 331-358
    • Mandelbrot, B.B.1
  • 16
    • 0001287197 scopus 로고
    • Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire
    • Reprinted in Mandelbrot, 1999a as Chapter N16
    • B. B. Mandelbrot, Multiplications 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.
    • (1974) Comptes Rendus (Paris) , vol.278 A , pp. 289-292
    • Mandelbrot, B.B.1
  • 18
    • 0000718295 scopus 로고
    • Fractals in physics: Squig clusters, diffusions, fractal measures and the unicity of fractal dimension
    • B. B. Mandelbrot, Fractals in physics: squig clusters, diffusions, fractal measures and the unicity of fractal dimension, J. Stat. Phys. 34: 895-930 (1984).
    • (1984) J. Stat. Phys. , vol.34 , pp. 895-930
    • Mandelbrot, B.B.1
  • 19
    • 0024483599 scopus 로고
    • Multifractal measures, especially for the geophysicist
    • B. B. Mandelbrot, Multifractal measures, especially for the geophysicist, Pure and Applied Geophysics 131:5-42 (1989a).
    • (1989) Pure and Applied Geophysics , vol.131 , pp. 5-42
    • Mandelbrot, B.B.1
  • 20
    • 0011001056 scopus 로고
    • A class of multinomial multifractal measures with negative (latent) values for the "dimension" f(α)
    • L. Pietronero, ed. (Plenum, New York)
    • 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.
    • (1989) Fractals' Physical Origin and Properties , pp. 3-29
    • Mandelbrot, B.B.1
  • 21
    • 45149136697 scopus 로고
    • Negative fractal dimensions and multifractals
    • B. B. Mandelbrot, Negative fractal dimensions and multifractals, Phys. A 163:306-315 (1990a).
    • (1990) Phys. A , vol.163 , pp. 306-315
    • Mandelbrot, B.B.1
  • 22
    • 44949288993 scopus 로고
    • New "anomalous" multiplicative multifractals: Left-sided f(α) and the modeling of DLA
    • B. B. Mandelbrot, New "anomalous" multiplicative multifractals: left-sided f(α) and the modeling of DLA, Physica A 168:95-111 (1990b).
    • (1990) Physica A , vol.168 , pp. 95-111
    • Mandelbrot, B.B.1
  • 23
    • 0039180252 scopus 로고
    • Limit lognormal multifractal measures
    • E. A. Gotsman, Y. Ne'eman, and A. Voronel, eds. (Pergamon, New York)
    • 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.
    • (1990) Frontiers of Physics: Landau Memorial Conference , pp. 309-340
    • Mandelbrot, B.B.1
  • 24
    • 85033091503 scopus 로고
    • Negative dimensions and Hölders, multifractals and their Hölder spectra, and the role of lateral preasymptotics in science
    • Paris
    • 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).
    • (1993) J. P. Kahane's Meeting
    • Mandelbrot, B.B.1
  • 25
    • 84862033054 scopus 로고
    • special issue
    • 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).
    • (1995) The Journal of Fourier Analysis and Applications , pp. 409-432
    • Bonami, A.1    Peyrière, J.2
  • 28
    • 0002812609 scopus 로고    scopus 로고
    • A multifractal walk through Wall Street
    • February issue
    • B. B. Mandelbrot, A multifractal walk through Wall Street, Scientific American, February issue, 50-53 (1999b).
    • (1999) Scientific American , pp. 50-53
    • Mandelbrot, B.B.1
  • 29
    • 85013894409 scopus 로고    scopus 로고
    • Scaling in financial prices, I: Tails and dependence
    • Reprint: Beyond Efficiency and Equilibrium, Doyne Farmer and John Geanakoplos, eds. (Oxford UK, The University Press, 2002)
    • 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).
    • (2001) Quantitative Finance , vol.1 , pp. 113-124
    • Mandelbrot, B.B.1
  • 30
    • 0013335095 scopus 로고    scopus 로고
    • Scaling in financial prices, II: Multifractals and the star equation
    • 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).
    • (2001) Quantitative Finance , vol.1 , pp. 124-130
    • Mandelbrot, B.B.1
  • 31
    • 85008857289 scopus 로고    scopus 로고
    • Scaling in financial prices, III: Cartoon Brownian motions in multifractal time
    • B. B. Mandelbrot, Scaling in financial prices, III: Cartoon Brownian motions in multifractal time, Quantitative Finance 1:427-440 (2001 c).
    • (2001) Quantitative Finance , vol.1 , pp. 427-440
    • Mandelbrot, B.B.1
  • 32
    • 85012476771 scopus 로고    scopus 로고
    • Scaling in financial prices, IV: Multifractal concentration
    • B. B. Mandelbrot, Scaling in financial prices, IV: Multifractal concentration, Quantitative Finance 1:558-559 (2001d).
    • (2001) Quantitative Finance , vol.1 , pp. 558-559
    • Mandelbrot, B.B.1
  • 33
    • 85008770448 scopus 로고    scopus 로고
    • Stochastic volatility, power-laws and long memory
    • B. B. Mandelbrot, Stochastic volatility, power-laws and long memory, Quantitative Finance 1:427-440 (2001e).
    • (2001) Quantitative Finance , vol.1 , pp. 427-440
    • Mandelbrot, B.B.1
  • 36
    • 0347234100 scopus 로고    scopus 로고
    • Discussion Papers numbers 1164, 1165, and 1166 of the Cowles Foundation for Economics at Yale University, New Haven, CT. Available on the web; ABSTRACT_ID = 78588. http://papers.ssrn.com/sol3/paper.taf? ABSTRACT_ID = 78606. http://papers.ssrn.com/sol3/paper.taf? ABSTRACT_ID = 78628
    • 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.
    • (1997) The Multifractal Model of Asset Returns. Large Deviations and the Distribution of Price Changes. The Multifractality of the Deutschmark/US Dollar Exchange Rate
    • Mandelbrot, B.B.1    Calvet, L.2    Fisher, A.3
  • 37
    • 84956238589 scopus 로고
    • Some specific features of atmospheric turbulence
    • A. M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13:77-81 (1962).
    • (1962) J. Fluid Mech. , vol.13 , pp. 77-81
    • Obukhov, A.M.1


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