Singularity Spectrum of Generic α-Hölder Regular Functions After Time Subordination

  • Zoltán BuczolichEmail author
  • Stéphane Seuret


A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by \({\mathcal {B}}_{1}^{\alpha}\) the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function \(g\in {\mathcal {B}}_{1}^{\alpha}\) one obtains a time subordination of g simply by considering the composite function Z=gf, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space \(\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha}\) equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of gf are investigated for a generic (typical) element (f,g)∈ℰ α . In particular we determine the generic Hölder singularity spectrum of gf.


Hölder regularity Time subordination Hausdorff measure Hausdorff dimension 

Mathematics Subject Classification (2000)



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  1. 1.
    Bacry, E., Delour, J., Muzy, J.F.: Multifractal random walks. Phys. Rev. E 64 (2001) Google Scholar
  2. 2.
    Barral, J., Seuret, S.: The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 14(1), 437–468 (2007) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buczolich, Z., Nagy, J.: Hölder spectrum of typical monotone continuous functions. Real Anal. Exch. 26, 133–156 (2000/01) MathSciNetGoogle Scholar
  4. 4.
    Falconer, K.J.: Fractal Geometry. Wiley, New York (1990) zbMATHGoogle Scholar
  5. 5.
    Falconer, K.J.: Techniques in Fractal Geometry. Wiley, New York (1997) zbMATHGoogle Scholar
  6. 6.
    Hurd, T., Kuznetsov, A.: On the first passage time for Brownian motion subordinated by a Lévy process. J. Appl. Probab. 46(1), 181–198 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Jaffard, S.: On the Frisch-Parisi conjecture. J. Math. Pures Appl. 79(6), 525–552 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Jaffard, S.: Wavelet techniques in multifractal analysis. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Proc. Symposia in Pure Mathematics. AMS, Providence (2004) Google Scholar
  9. 9.
    Jaffard, S., Mandelbrot, B.: Local regularity of nonsmooth wavelet expansions and application to the Polya function. Adv. Math. 120(2), 265–282 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ludena, C.: L p-variations for multifractal fractional random walks. Ann. Appl. Probab. 18(3), 1138–1163 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Mandelbrot, B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974) zbMATHCrossRefGoogle Scholar
  12. 12.
    Mandelbrot, B.: Fractals and Scaling in Finance. Springer, Berlin (1998) Google Scholar
  13. 13.
    Mandelbrot, B., Fischer, A., Calvet, L.: A multifractal model of asset returns. Cowles Foundation Discussion Paper #1164 (1997) Google Scholar
  14. 14.
    Muzy, J.-F., Bacry, E.: Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws. Phys. Rev. E 66 (2002) Google Scholar
  15. 15.
    Riedi, R.: Multifractal processes In: Theory and Applications of Long-Range Dependence, pp. 625–716. Birkhäuser, Boston (2003) Google Scholar
  16. 16.
    Seuret, S.: On multifractality and time subordination. Adv. Math. 220, 936–963 (2009) MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.LAMA, CNRS UMR 8050Université Paris-EstCréteil CedexFrance

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