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Random Wavelet Series Based on a Tree-Indexed Markov Chain

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Abstract

We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Hölder exponent form a set with large intersection.

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References

  1. Arneodo A., Bacry E., Jaffard S., Muzy J.-F.: Singularity spectrum of multifractal functions involving oscillating singularities. J. Fourier Anal. Appl. 4(2), 159–174 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubry J.-M., Jaffard S.: Random wavelet series. Commun. Math. Phys. 227(3), 483–514 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Barral J., Seuret S.: From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11(5), 589–614 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barral J., Seuret S.: The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 14(1), 437–468 (2007)

    Article  MathSciNet  Google Scholar 

  5. Basseville M., Benveniste A., Chou K.C., Golden S.A., Nikoukah R., Willsky A.S.: Modeling and estimation of multiresolution stochastic processes. IEEE Trans. Inform. Theory 38, 766–784 (1992)

    Article  Google Scholar 

  6. Brouste, A.: Étude d’un processus bifractal et application statistique en géologie. Ph.D. thesis, Université Joseph Fourier, Grenoble, 2006

  7. Brouste A., Renard F., Gratier J.-P., Schmittbuhl J.: Variety of stylolites’ morphologies and statistical characterization of the amount of heterogeneities in the rock. J. Struct. Geol. 29(3), 422–434 (2007)

    Article  ADS  Google Scholar 

  8. Chipman H.A., Kolaczyk E.D., McCulloch R.E.: Adaptive Bayesian wavelet shrinkage. J. Amer. Statist. Assoc. 92, 1413–1421 (1997)

    Article  MATH  Google Scholar 

  9. Choi H., Baraniuk R.G.: Multiscale image segmentation using wavelet-domain hidden Markov models. IEEE Trans. Image Process. 10(9), 1309–1321 (2001)

    Article  MathSciNet  Google Scholar 

  10. Crouse M.S., Nowak R.D., Baraniuk R.G.: Wavelet-based statistical processing using hidden Markov models. IEEE Trans. Signal Process. 46(4), 886–902 (1998)

    Article  MathSciNet  Google Scholar 

  11. Diligenti, M., Frasconi, P., Gori, M.: Image document categorization using hidden tree Markov models and structured representations. In: Proc. Int. Conf. on Applications of Pattern Recognition S. Singh, N. Murshed, W. Kropatsch, eds., Lecture Notes in Computer Science Vol. 2013, London: Springer Verlag, 2001

  12. Donoho D.L., Johnstone I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90(432), 1200–1224 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Durand, A.: Random fractals and tree-indexed Markov chains. http://arxiv.org/abs/0709.3598, 2007

  14. Durand A.: Sets with large intersection and ubiquity. Math. Proc. Cambridge Philos. Soc. 144, 119–144 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Durand, A.: Singularity sets of Lévy processes. Probab. Theory Related Fields, to appear, doi:10.1007/s00440-007-0134-6, 2008

  16. Durand A.: Ubiquitous systems and metric number theory. Adv. Math. 218(2), 368–394 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Falconer K.J.: Sets with large intersection properties. J. London Math. Soc. (2) 49(2), 267–280 (1994)

    MATH  MathSciNet  Google Scholar 

  18. Falconer K.J.: Fractal geometry: Mathematical foundations and applications. Second ed. John Wiley & Sons Inc., New York (2003)

    MATH  Google Scholar 

  19. Flandrin P.: Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inform. Theory 38(2), 910–917 (1992)

    Article  MathSciNet  Google Scholar 

  20. Grossmann A., Morlet J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723–736 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Holschneider, M., Kronland-Martinet, R., Morlet, J., Tchamitchian, P.: A real-time algorithm for signal analysis with the help of the wavelet transform. Wavelets, Time-Frequency Methods and Phase Space, J.-M. Combes, A. Grossmann, P. Tchamitchian, eds., Berlin: Springer-Verlag, 1989, pp. 286–297

  22. Jaffard S.: The multifractal nature of Lévy processes. Probab. Theory Related Fields 114, 207–227 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jaffard S.: On lacunary wavelet series. Ann. Appl. Probab. 10(1), 313–329 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jaffard S.: Multifractal functions: recent advances and open problems. Bull. Soc. Roy. Sci. Liège 73(2-3), 129–153 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Jaffard S.: Wavelet techniques in multifractal analysis. Proc. Sympos. Pure Math. 72(2), 91–151 (2004)

    MathSciNet  Google Scholar 

  26. Jaffard, S., Meyer, Y.: Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Amer. Math. Soc. 123(587) (1996)

  27. Kahane J.-P.: Some random series of functions. Second edn. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  28. Kronland-Martinet R., Morlet J., Grossmann A.: Analysis of sound patterns through wavelet transforms. Int. J. Pattern Recogn. Artif. Intell. 1(2), 273–301 (1988)

    Article  Google Scholar 

  29. Lee, N., Huynh, Q., Schwarz, S.: New methods of linear time-frequency analysis for signal detection. Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, New Brunswick, NJ: IEEE, 1996, pp. 13–16

  30. Lemarié-Rieusset P.G., Meyer Y.: Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2(1–2), 1–18 (1986)

    MathSciNet  Google Scholar 

  31. Lévy-Véhel, J., Riedi, R.: Fractional brownian motion and data traffic modeling: The other end of the spectrum. In: Fractals in Engineering, J. Lévy-Véhel, E. Lutton, C. Tricot, eds., Berlin Heidelberg New York: Springer-Verlag, 1997, pp. 185–202

  32. Mallat S., Hwang W.L.: Singularity detection and processing with wavelets. IEEE Trans. Inform. Theory 38(2), 617–643 (1992)

    Article  MathSciNet  Google Scholar 

  33. Mallat S., Zhong S.: Characterization of signals from multiscale edges. IEEE Trans. Pattern Anal. Mach. Intell. 14(7), 710–732 (1992)

    Article  Google Scholar 

  34. Meyer Y.: Ondelettes et opérateurs. Hermann, Paris (1990)

    Google Scholar 

  35. Rogers C.A.: Hausdorff measures. Cambridge University Press, Cambridge (1970)

    MATH  Google Scholar 

  36. Shapiro J.M.: Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans. Signal Process. 41(12), 3445–3465 (1993)

    Article  MATH  Google Scholar 

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Author notes

  1. Arnaud Durand

    Present address: Applied and Computational Mathematics – MC 217-50, California Institute of Technology, Pasadena, CA, 91125, USA

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  1. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Arnaud Durand

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  1. Arnaud Durand
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Correspondence to Arnaud Durand.

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Communicated by M.B. Ruskai

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Durand, A. Random Wavelet Series Based on a Tree-Indexed Markov Chain. Commun. Math. Phys. 283, 451–477 (2008). https://doi.org/10.1007/s00220-008-0504-7

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  • DOI: https://doi.org/10.1007/s00220-008-0504-7

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