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Renewal of singularity sets of random self-similar measures

Part of: Stochastic processes Limit theorems Classical measure theory

Published online by Cambridge University Press:  01 July 2016

Julien Barral  and
Stéphane Seuret
Julien Barral*
Affiliation:
INRIA Rocquencourt
Stéphane Seuret*
Affiliation:
Université Paris XII
*
∗ Postal address: Equipe Sosso2, Domaine de Voluceau Rocquencourt, 78153 Le Chesnay cedex, France. Email address: [email protected]
∗∗ Postal address: Laboratoire d'Analyse et de Mathématiques Appliquées, Faculté de Sciences et Technologie, Bat. P3, 4ème étage, 61 avenue du Général de Gaulle, 94010 Créteil cedex, France. Email address: [email protected]
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Abstract

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This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

Keywords

Random measurelarge deviationsHausdorff dimensionself-similarityfractal

MSC classification

Primary: 60G57: Random measures 60F10: Large deviations 28A78: Hausdorff and packing measures
Secondary: 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 162 - 188
Copyright
Copyright © Applied Probability Trust 2007 

References

Arbeiter, M. and Patzschke, N. (1996). Random self-similar multifractals. Math. Nachr. 181, 5–42.Google Scholar
Arnold, V. I. (1963). Small denominators and problems of stability of motion in classical and celestial mechanics. Ups. Mat. Nauk. 18, 91–192 (in Russian). English translation: Russian Math. Surveys 18, 85-191.Google Scholar
Bacry, E. and Muzy, J.-F. (2003). Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236, 449–475.CrossRefGoogle Scholar
Barral, J. (1999). Moments, continuité et analyse multifractale des martingales de Mandelbrot. Prob. Theory Relat. Fields 113, 535–569.Google Scholar
Barral, J. (2000). Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Prob. 13, 1027–1060.Google Scholar
Barral, J. and Mandelbrot, B. B. (2002). Multifractal products of cylindrical pulses. Prob. Theory Relat. Fields 124, 409–430.CrossRefGoogle Scholar
Barral, J. and Mandelbrot, B. B. (2004). Random multiplicative multifractal measures. In Fractal Geometry and Applications, eds Lapidus, M. L. and van Frankenhuijsen, M., AMS, Providence, RI, pp. 3–90.Google Scholar
Barral, J. and Seuret, S. (2004). Heterogeneous ubiquitous systems and Hausdorff dimension in R d . Submitted.Google Scholar
Barral, J. and Seuret, S. (2004). Sums of Dirac masses and conditioned ubiquity. C. R. Acad. Sci. Paris I 339, 787–792.Google Scholar
Barral, J. and Seuret, S. (2005). Combining multifractal additive and multiplicative chaos. Commun. Math. Phys. 257, 473–497.Google Scholar
Barral, J. and Seuret, S. (2005). The singularity spectrum of Lévy processes in multifractal time. To appear in Adv. Math. Google Scholar
Barral, J. and Seuret, S. (2005). Inside singularities sets of random Gibbs measures. J. Statist. Phys. 120, 1101–1124.Google Scholar
Barral, J., Ben Nasr, F. and Peyrière, J. (2003). Comparing multifractal formalisms: the neighboring boxes conditions. Asian J. Math. 7, 149–165.Google Scholar
Barral, J., Coppens, M. O. and Mandelbrot, B. B. (2003). Multiperiodic multifractal martingale measures. J. Math. Pure Appl. 82, 1555–1589.CrossRefGoogle Scholar
Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, 137–151.Google Scholar
Billingsley, P. (1965). Ergodic Theory and Information. John Wiley, New York.Google Scholar
Brown, G., Michon, G. and Peyrière, J. (1992). On the multifractal analysis of measures. J. Statist. Phys. 66, 775–790.Google Scholar
Collet, P. and Koukiou, F. (1992). Large deviations for multiplicative chaos. Commun. Math. Phys. 147, 329–342.CrossRefGoogle Scholar
Dodson, M. M. (2002). Exceptional sets in dynamical systems and Diophantine approximation. In Rigidity in Dynamics and Geometry, Springer, Berlin, pp. 77–98.CrossRefGoogle Scholar
Durrett, R. and Liggett, T. (1983). Fixed points of the smoothing transformation. Z. Wahrscheinlichkeitsth. 64, 275–301.CrossRefGoogle Scholar
Falconer, K. J. (1994). The multifractal spectrum of statistically self-similar measures. J. Theoret. Prob. 7, 681–702.Google Scholar
Falconer, K. J. (2000). Representation of families of sets by measures, dimension spectra and Diophantine approximation. Math. Proc. Camb. Phil. Soc. 128, 111–121.Google Scholar
Fan, A. H. (1997). Multifractal analysis of infinite products. J. Statist. Phys. 86, 1313–1336.Google Scholar
Frisch, U. and Parisi, G. (1985). Fully developed turbulence and intermittency. In Proc. Internat. Summer School Phys., North-Holland, Amsterdam, pp. 84–88.Google Scholar
Graf, S., Mauldin, R. D. and Williams, S. C. (1988). The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc. 71, 381.Google Scholar
Halsey, T. C. et al. (1986). Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33, 1141–1151. (Correction: 34 (1986), 1601.)Google Scholar
Holley, R. and Waymire, E. C. (1992). Multifractal dimensions and scaling exponents for strongly bounded random fractals. Ann. Appl. Prob. 2, 819–845.CrossRefGoogle Scholar
Jaffard, S. (1999). The multifractal nature of Lévy processes. Prob. Theory Relat. Fields 114, 207–227.Google Scholar
Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9, 105–150.Google Scholar
Kahane, J.-P. (1987). Positive martingales and random measures. Chinese Ann. Math. B 8, 1–12.Google Scholar
Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoıcirc;t Mandelbrot. Adv. Math. 22, 131–145.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Khanin, K. and Kifer, Y. (1996). Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. 171, 107–140.Google Scholar
Kifer, Y. (1995). Fractals via random iterated function systems and random geometric constructions. In Fractal Geometry and Stochastics (Finsterbergen, 1994; Progress Prob. 37), Birkhäuser, Basel, pp. 145–164.Google Scholar
Lévy Véhel, J. and Riedi, R. H. (1997). TCP traffic is multifractal: a numerical study. Res. Rep. RR-3129, INRIA.Google Scholar
Liu, Q. (1996). The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In Trees (Progress Prob. 40), Birkhäuser, Basel, pp. 51–80.CrossRefGoogle Scholar
Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.Google Scholar
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Process. Appl. 95, 83–107.CrossRefGoogle Scholar
Mandelbrot, B. 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
Mandelbrot, B. B. (1997). Fractals and Scaling in Finance. Discontinuity, Concentration, Risk. Springer, New York.CrossRefGoogle Scholar
Mandelbrot, B. B. (2001). Scaling in financial prices. III. Cartoon Brownian motions in multifractal time. Quant. Finance 1, 427–440.CrossRefGoogle Scholar
Mandelbrot, B. B., Fisher, L. and Calvet, A. (1997). A multifractal model for asset returns. CFDP 1164. Available at http://cowles.econ.yale.edu/P/cd/dy1997.htm.Google Scholar
Molchan, G. M. (1996). Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681–702.Google Scholar
Muzy, J.-F. and Bacry, E. (2002). Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws. Phys. Rev. E 66, 056121.Google Scholar
Olsen, L. (1994). Random Geometrically Graph Directed Self-similar Multifractals (Pitman Res. Notes Math. Ser. 307). John Wiley, New York.Google Scholar
Olsen, L. (1995). A multifractal formalism. Adv. Math. 116, 82–196.CrossRefGoogle Scholar
Ossiander, M. and Waymire, E. C. (2000). Statistical estimation for multiplicative cascades. Ann. Statist. 28, 1533–1560.Google Scholar
Parry, W. and Pollicott, M. (1990). Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics (Astérisque 187– 188). Société Mathématique de France, Paris.Google Scholar
Riedi, R. (1995). An improved multifractal formalism and self-similar measures. J. Math. Anal. Appl. 189, 462–490.Google Scholar
Riedi, R. (2003). Multifractal processes. In Theory and Applications of Long-Range Dependence, Birkhäuser, Boston, MA, pp. 625–716.Google Scholar