The singularity spectrum of Lévy processes in multifractal time

Advances in Mathematics

Volumn 214, Issue 1, 2007, Pages 437-468

  Barral, Julien ^a   Seuret, Stéphane ^b  

^a INRIA   (France)

^b UFR des Sciences et Technologie   (France)

Author keywords

Hausdorff measures and dimension; Multifractal functions and measures; Self similar measures; Stochastic processes

Indexed keywords

EID: 34250328016     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2007.02.007     Document Type: Article

References (37)

1. Arbeiter M., and Patzschke N. Random self-similar multifractals. Math. Nachr. 181 (1996) 5-42
2. Ayache A., and Lévy Véhel J. Generalized multifractional Brownian motion: Definition and preliminary results. Fractals-Theory and Applications in Engineering (1999), Springer, London 17-32
3. Bacry E., and Muzy J.-F. Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 (2003) 449-475
4. Barral J. Continuity of the multifractal spectrum of a statistically self-similar measure. J. Theoret. Probab. 13 (2000) 1027-1060
5. Barral J., and Mandelbrot B. Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 (2002) 409-430
6. Barral J., and Mandelbrot B. Random multiplicative multifractal measures, Parts I, II, III. Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proc. Sympos. Pure Math. vol. 72, 2 (2004), Amer. Math. Soc., Providence, RI 3-90
7. Barral J., and Seuret S. Sums of Dirac masses and conditioned ubiquity. C. R. Acad. Sci. Paris Sér. I 339 (2004) 787-792
8. Barral J., and Seuret S. From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11 (2005) 589-614
9. Barral J., and Seuret S. Combining multifractal additive and multiplicative chaos. Comm. Math. Phys. 257 (2005) 473-497
10. Barral J., and Seuret S. Inside singularity sets of random Gibbs measures. J. Stat. Phys. 120 (2005) 1101-1124
11. Barral J., and Seuret S. Renewal of singularity sets of random self-similar measures. preprint (2005). http://fr.arxiv.org/abs/math.PR/0503421 Adv. in Appl. Probab., in press
12. d and Hausdorff dimensions. preprint (2005). http://fr.arxiv.org/abs/math.GM/0503419
13. Brown G., Michon G., and Peyrière J. On the multifractal analysis of measures. J. Stat. Phys. 66 3-4 (1992) 775-790
14. Chainais P., Riedi R., and Abry P. On scale invariant infinitely divisible cascades. IEEE Trans. Inform. Theory 51 3 (March 2005)
15. Durrett R., and Liggett T. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983) 275-301
16. Falconer K.J. The multifractal spectrum of statistically self-similar measures. J. Theoret. Prob. 7 (1994) 681-702
17. Fan A.H. Multifractal analysis of infinite products. J. Stat. Phys. 86 5/6 (1997) 1313-1336
18. Guivarc'h Y. Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990) 261-285
19. Holley R., and Waymire E.C. Multifractal dimensions and scaling exponents for strongly bounded random fractals. Ann. Appl. Probab. 2 (1992) 819-845
20. Jaffard S. Exposants de Hölder en des points donnés et coefficients d'ondelettes. C. R. Acad. Sci. Paris Sér. I 308 (1989) 79-81
21. Jaffard S. Old friends revisited: The multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 1 (1997) 1-22
22. Jaffard S. The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 2 (1999) 207-227
23. Jaffard S. On lacunary wavelet series. Ann. Appl. Probab. 10 1 (2000) 313-329
24. Kahane J.-P. Positive martingales and random measures. Chinese Ann. Math. Ser. B 8 (1987) 1-12
25. Kahane J.-P. Produits de poids aléatoires et indépendants et applications. In: Bélair J., and Dubuc S. (Eds). Fractal Geometry and Analysis (1991) 277-324
26. Kahane J.-P., and Peyrière J. Sur certaines martingales de Benoît Mandelbrot. Adv. Math. 22 (1976) 131-145
27. Kifer Y. Fractals via random iterated function systems and random geometric constructions. Fractal Geometry and Stochastics. Finsterbergen, 1994. Progr. Probab. vol. 37 (1995), Birkhäuser, Basel 145-164
28. Liu Q. Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 (2001) 83-107
29. Mandelbrot B.B. Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris 278 (1974) 289-292 and 355-358
30. Mandelbrot B.B. Intermittent turbulence in self-similar cascades: Divergence of hight moments and dimension of the carrier. J. Fluid. Mech. 62 (1974) 331-358
31. B. Mandelbrot, A. Fischer, L. Calvet, A multifractal model of asset returns, Cowles Foundation Discussion Paper #1164 (1997)
32. Mattila P. Geometry of Sets and Measures in Euclidean Spaces (1995), Cambridge Univ. Press
33. Molchan G.M. Scaling exponents and multifractal dimensions for independent random cascades. Comm. Math. Phys. 179 (1996) 681-702
34. Olsen L. A multifractal formalism. Adv. Math. 116 (1995) 92-195
35. Peyrière J. Turbulence et dimension de Hausdorff. C. R. Acad. Sci. Paris Sér. A 278 (1974) 567-569
36. Riedi R. Multifractal processes. In: Doukhan P., Oppenheim G., and Taqqu M.S. (Eds). Long Range Dependence: Theory and Applications (2002), Birkhäuser 625-715
37. Shepp L.A. Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 23 (1972) 163-170