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Hölderian random functions

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Fractals in Engineering

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Summary

Hölder regularity which plays a key rôle in fractal geometry raises an increasing interest in probability and statistics. In this paper we discuss various aspects of local and global regularity for stochastic processes and random fields. As a main result we show the invariability of the pointwise Hölder exponent of a continuous and nowhere differentiable random field which has stationary increments and satisfies a zero-one law. We also survey some recent uses of Hölder spaces in limit theorems for stochastic processes and statistics.

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References

  1. Ayache A (2001) Du mouvement brownien fractionnaire au mouvement brownien multifractionnaire. Technique et Science Informatiques. 209:1133–1152

    Google Scholar 

  2. Ayache A, Lévy Véhel J (1999) Generalized multifractional Brownian motion: definition and preliminary results. In Dekking M, Lévy Véhel J, Lutton E, Tricot C (eds) Fractals: Theory and Applications in Engineering. Springer 17–32

    Google Scholar 

  3. Ayache A, Lévy Véhel J (2000) The Generalized multifractional Brownian motion. Statistical Inference for Stochastic Processes 3:7–18

    Article  MathSciNet  Google Scholar 

  4. Ayache A, Lévy Véhel J (2004) Identification of the pointwise Hölder exponent of Generalized Multifractional Brownian Motion. Stochastic Processes and their Applications 111:119–156

    Article  MathSciNet  Google Scholar 

  5. Ayache A, Léger S (2000) The multifractional Brownian sheet. To appear in Ann Mat Blaise Pascal

    Google Scholar 

  6. Ayache A, Taqqu MS (2003) Multifractional Processes with Random Exponent. Preprint to appear in Publicacions Matemàtiques

    Google Scholar 

  7. Benassi A, Jaffard S, Roux D (1997) Elliptic Gaussian random processes. Rev Mat Iberoamericana 131:19–89

    MathSciNet  Google Scholar 

  8. Barral J, Lévy Véhel J (2004) Multifractal analysis of a class of additive processes with correlated nonstationary increments. Electronic Journal of Probability 9:508–543

    MathSciNet  Google Scholar 

  9. Ciesielski Z (1960) On the isomorphisms of the spaces Hα and m. Bull Acad Pol Sci Ser Sci Math Phys 8:217–222

    MATH  MathSciNet  Google Scholar 

  10. Davies S, Hall P (1999) Fractal analysis of surface roughness by using spatial data. J R Statist B 611:3–37

    Article  MathSciNet  Google Scholar 

  11. Doukhan P, Oppenheim G, Taqqu MS, (eds) (2002) Theory and Applications of Long-range Dependence. Birkhäuser, Boston

    Google Scholar 

  12. Falconer K (1990) Fractal Geometry: Mathematical Fondations and Applications. John Wiley and Sons, New York

    Google Scholar 

  13. Erickson RV (1981) Lipschitz smoothness and convergence with applications to the central limit theorem for summation processes. Annals of Probability 9:831–851

    MATH  MathSciNet  Google Scholar 

  14. Hamadouche D (1998) Weak convergence of smoothed empirical process in Hölder spaces. Stat Probab Letters 36:393–400

    Article  MATH  MathSciNet  Google Scholar 

  15. Hamadouche D (2000) Invariance principles in Hölder spaces. Portugaliae Mathematica 57:127–151

    MATH  MathSciNet  Google Scholar 

  16. Hamadouche D, Suquet Ch (1999) Weak Hölder convergence of stochastic processes with application to the perturbed empirical process. Applicationes Math 26:63–83

    MathSciNet  Google Scholar 

  17. Hamadouche D, Suquet Ch (2004) Smoothed quantile processes in Hölder spaces. Pub IRMA Lille (Preprint) 62 IV

    Google Scholar 

  18. Jaffard S, Meyer Y, Ryan RD (2001) Wavelets Tools for Science & Technology. SIAM Philadelphia.

    Google Scholar 

  19. Kerkyacharian G, Roynette B (1991) Une démonstration simple des théorèmes de Kolmogorov, Donsker et Ito-Nisio. Comptes Rendus de l'Académie des Sciences Paris, Série I 312:877–882

    MathSciNet  Google Scholar 

  20. Kolmogorov AN (1940) Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen raum. Comptes Rendus (Doklady) de l'Académie des Sciences de l' URSS (NS) 26:115–118

    MATH  Google Scholar 

  21. Kuelbs J (1973) The invariance principle for Banach space valued random variables. Journal of Multivariate Analysis 3:161–172

    Article  MATH  MathSciNet  Google Scholar 

  22. Lamperti J (1962) On convergence of stochastic processes. Transactions of the American Mathematical Society 104:430–435

    Article  MATH  MathSciNet  Google Scholar 

  23. Laukaitis A, Račkauskas A (2002) Testing changes in Hilbert space autoregressive models. Lietuvos Matematikos Rinkinys 42:434–447

    Google Scholar 

  24. Ledoux M, Talagrand M (1991) Probability in Banach Spaces. Springer-Verlag, Berlin Heidelberg.

    Google Scholar 

  25. Lévy P (1937) Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris, Second Edition (1954)

    Google Scholar 

  26. Lévy Véhel J (1998) Introduction to the Multifractal Analysis of Images. In Fisher Y (ed) Fractal Image Encoding and Analysis. Springer

    Google Scholar 

  27. Lévy Véhel J (2002) Signal enhancement based on Hölder regularity analysis. IMA Volumes in Mathematics and its Applications 132:197–209

    MATH  Google Scholar 

  28. Peltier RF, Lévy Véhel J (1995) Multifractional Brownian Motion: definition and preliminary results. Rapport de recherche de l'INRIA, No 2645

    Google Scholar 

  29. Lifshits MA (1995) Gaussian Random Functions. Kluwer Academic Publishers, Dordrecht Boston London

    Google Scholar 

  30. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10:422–437

    Article  MathSciNet  Google Scholar 

  31. Račkauskas A, Suquet Ch (1999) Central limit theorem in Hölder spaces. Probability and Mathematical Statistics 19:155–174

    Google Scholar 

  32. Račkauskas A, Suquet Ch (1999) Random fields and central limit theorem in some generalized Hölder spaces. In: Grigelionis B et al (eds) Prob Theory and Math Statist. Proceedings of the 7th Vilnius Conference (1998) TEV Vilnius and VSP Utrecht 599–616

    Google Scholar 

  33. Račkauskas A, Suquet Ch (2001) Invariance principles for adaptive self-normalized partial sums processes. Stochastic Processes and their Applications 95:63–81

    Article  MathSciNet  Google Scholar 

  34. Račkauskas A, Suquet Ch (2001) Hölder versions of Banach spaces valued random fields. Georgian Mathematical Journal 82:347–362

    MathSciNet  Google Scholar 

  35. Račkauskas A, Suquet Ch (2002) Hölder convergences of multivariate empirical characteristic functions. Mathematical Methods of Statistics vol. 11:3:341–357

    MathSciNet  Google Scholar 

  36. Račkauskas A, Suquet Ch (2004) Necessary and suffcient condition for the Hölderian functional central limit theorem. Journal of Theoretical Probability 171:221–243

    Article  MathSciNet  Google Scholar 

  37. Račkauskas A, Suquet Ch (2002) On the Hölderian functional central limit theorem for i.i.d. random elements in Banach space. In: Berkes I, Csáki E, Csörgő M (eds) Limit Theorems in Probability and Statistics. Balatonlelle (1999) (János Bolyai Mathematical Society, Budapest 2:485–498

    Google Scholar 

  38. Račkauskas A, Suquet Ch (2003) Necessary and suffcient condition for the Lamperti invariance principle. Theory of Probability and Mathematical Statistics 68:115–124

    Google Scholar 

  39. Račkauskas A, Suquet Ch (2003) Invariance principle under self-normalization for nonidentically distributed random variables. Acta Applicandae Mathematicae 791–2:83–103

    Article  MathSciNet  Google Scholar 

  40. Račkauskas A, Suquet Ch (2004) Central limit theorems in Hölder topologies for Banach space valued random fields. Theory of Probability and its Applications 491:109–125

    Google Scholar 

  41. Račkauskas A, Suquet Ch (2004) Hölder norm test statistics for epidemic change. Journal of Statistical Planning and Inference 1262:495–520

    Article  MathSciNet  Google Scholar 

  42. Račkauskas A, Suquet Ch (2003) Testing epidemic change of infinite dimensional parameters. Preprint to appear in Statistical Inference for Stochastic Processes

    Google Scholar 

  43. Rosinski J, Samorodnitsky G (1996) Symmetrization and concentration inequalities for multilinear forms with applications to zero-one laws for Lévy chaos. Ann Probab 241:422–437

    Article  MathSciNet  Google Scholar 

  44. Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes. Chapman & Hall.

    Google Scholar 

  45. Tricot C (1993) Curves and Fractal dimensions. Springer-Verlag.

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire P. Painlevé, CNRS UMR 8524, Université Lille 1, 59655, Villeneuve d'Ascq cedex, France

    Antoine Ayache, Philippe Heinrich, Laurence Marsalle & Charles Suquet

Authors
  1. Antoine Ayache
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  2. Philippe Heinrich
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  3. Laurence Marsalle
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  4. Charles Suquet
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Editor information

Editors and Affiliations

  1. INRIA Rocquencourt, Domaine de Voluceau-Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Jacques Lévy-Véhel  & Evelyne Lutton  & 

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© 2005 Springer-Verlag London Limited

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Ayache, A., Heinrich, P., Marsalle, L., Suquet, C. (2005). Hölderian random functions. In: Lévy-Véhel, J., Lutton, E. (eds) Fractals in Engineering. Springer, London. https://doi.org/10.1007/1-84628-048-6_3

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  • DOI: https://doi.org/10.1007/1-84628-048-6_3

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84628-047-4

  • Online ISBN: 978-1-84628-048-1

  • eBook Packages: EngineeringEngineering (R0)

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