Lithuanian Mathematical Journal

, Volume 55, Issue 2, pp 159–192 | Cite as

Linear Multifractional Stable Motion: Wavelet Estimation of H(·) and α Parameters*

  • Antoine Ayache
  • Julien Hamonier


Stoev and Taqqu introduced a linear multifractional stable motion (LMSM), an extension of a linear fractional stable motion (LFSM) such that the Hurst parameter H becomes a function H(t). The stability parameter α determines tail heaviness of marginal distributions of LMSM. Under some conditions, Stoev and Taqqu showed that H(t 0) is its self-similarity exponent at a t 0 ≠ 0; also, recently, Ayache and Hamonier established that H(t 0)−1/α and min tI H(t)−1/α are its local Hölder exponent at t 0 and uniform Hölder exponent on a compact interval I.

We construct, strongly consistent wavelet estimators of min tI H(t), H(t 0), and α when α ∈ (1, 2) and H(·) is smooth with values in \( \left[\underset{\bar{\mkern6mu}}{H},\overline{H}\right]\subset \left(1/\alpha, 1\right) \).


stable stochastic processes wavelet coefficients Hölder regularity local self-similarity laws of large numbers 


60G22 60G52 60M09 


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  1. 1.
    P. Abry, B. Pesquet-Popescu, and M.S. Taqqu, Estimation ondelette des paramètres de stabilité et d’autosimilarité des processus α-stables autosimilaires, in 17ème Colloque sur le traitement du signal et des images, Vannes, France, 13–17 septembre, 1999, GRETSI, Groupe d’Etudes du Traitement du Signal et des Images, 1999, pp. 933–936.Google Scholar
  2. 2.
    A. Ayache and C. El-Nouty, The small ball behavior of a non stationary increments process: The multifractional Brownian motion, preprint No. 8, CMLA, 2004.Google Scholar
  3. 3.
    A. Ayache and J. Hamonier, Linear fractional stable motion: A wavelet estimator of the α parameter, Stat. Probab. Lett., 82(8):1569–1575, 2012.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    A. Ayache and J. Hamonier, Linear multifractional stable motion: Fine path properties, Rev. Mat. Iberoam., 30(4):1301–1354, 2014.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    A. Ayache and J. Lévy Véhel, Identification of the pointwise Hölder exponent of generalized multifractional Brownian motion, Stochastic Processes Appl., 111(1):119–156, 2004.CrossRefMATHGoogle Scholar
  6. 6.
    J.M. Bardet and D. Surgailis, Nonparametric estimation of the local Hurst function of multifractional processes, Stochastic Processes Appl., 123(3):1004–1045, 2013.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    A. Benassi, S. Cohen, and J. Istas, Identifying the multifractional function of a Gaussian process, Stat. Probab. Lett., 39(4):337–345, 1998.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    A. Benassi, S. Jaffard, and D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoam., 13(1):19–90, 1997.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    J.F. Coeurjolly, Identification of multifractional Brownian motion, Bernoulli, 11(6):987–1008, 2005.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    J.F. Coeurjolly, Erratum: Identification of multifractional Brownian motion, Bernoulli, 12(2):381–382, 2006.CrossRefMathSciNetGoogle Scholar
  11. 11.
    I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Ser. Appl. Math., Vol. 61, SIAM, Philadelphia, 1992.Google Scholar
  12. 12.
    L. Delbeke, Wavelet Based Estimators for the Hurst Parameter of a Self-Similar Process, PhD thesis, KU Leuven, Belgium, 1998.Google Scholar
  13. 13.
    L. Delbeke and P. Abry, Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion, Stochastic Processes Appl., 86(2):177–182, 2000.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    P. Embrechts and M. Maejima, Self-Similar Processes, Princeton Series in Applied Mathematics, Princeton Univ. Press, Princeton, NJ, 2002.Google Scholar
  15. 15.
    K.J. Falconer, Tangent fields and the local structure of random fields, J. Theor. Probab., 15(3):731–750, 2002.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    K.J. Falconer, The local structure of random processes, J. Lond. Math. Soc., 67(2):657–672, 2003.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    C. Lacaux, Real harmonizable multifractional Lévy motions, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B, 40(3):259–277, 2004.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    R. Le Guével, An estimation of the stability and the localisability functions of multistable processes, Electron. J. Stat., 7:1129–1166, 2013.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    R. Lopes, A. Ayache, N. Makni, P. Puech, A. Villers, S. Mordon, and N. Betrouni, Prostate cancer characterization on MR images using fractal features, Medical Physics, 38(1):83–95, 2011.CrossRefGoogle Scholar
  20. 20.
    R.F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, Rapport de recherche de l’INRIA, 2645:239–265, 1995.Google Scholar
  21. 21.
    Q. Peng, Inférence statistique pour des processus multifractionnaires cachés dans un cadre de modèles à volatilité stochastique, PhD thesis, Université Lille 1, 2011.Google Scholar
  22. 22.
    V. Pipiras, M.S. Taqqu, and P. Abry, Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation, Bernoulli, 13(4):1091–1123, 2007.CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.MATHGoogle Scholar
  24. 24.
    G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Variables. Stochastic Models with Infinite Variance, Stochastic Modeling, Chapman & Hall, New York, 1994.Google Scholar
  25. 25.
    S. Stoev, V. Pipiras, and M.S. Taqqu, Estimation of the self-similarity parameter in linear fractional stable motion, Signal Process., 82(12):1873–1901, 2002.CrossRefMATHGoogle Scholar
  26. 26.
    S. Stoev and M.S. Taqqu, Stochastic properties of the linear multifractional stable motion, Adv. Appl. Probab., 36(4):1085–1115, 2004.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.UMR CNRS 8524, Laboratoire Paul Painlevé, Bât. M2, Université Lille 1Villeneuve d’Ascq CedexFrance
  2. 2.EA 2694, Laboratoire de Biomathématiques, Université Lille 2Lille CedexFrance

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