no. 9
Probability Theory
A class of multifractal semi-stable processes including Lévy subordinators and Mandelbrot multiplicative cascades
[Une classe de processus multifractals semi-stables contenant subordinateurs de Lévy et cascades multiplicatives de Mandelbrot]

Présenté par : Yves Meyer

Barral, Julien1 ; Seuret, Stéphane2
1 Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France
2 Laboratoire d'analyse et de mathématiques appliquées, Université Paris 12 – Val-de-Marne, UFR des sciences et technologie, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France
Comptes Rendus. Mathématique, Tome 341 (2005) no. 9, pp. 579-582.

Nous présentons une classe de processus auto-similaires en loi naturellement associés aux généralisations des lois semi-stables considérées. Cette classe contient en particulier les subordinateurs stables de Lévy ainsi que les cascades multiplicatives de Mandelbrot ; ses éléments sont des cas particuliers des processus de Lévy en temps multifractal étudiés ailleurs. Nous étudions leur nature multifractale.

We exhibit a class of statistically self-similar processes naturally associated with the so-called fixed points of the smoothing transformation. This class includes stable subordinators and Mandelbrot multiplicative cascades. Both these processes are special examples of Lévy processes in multifractal time, which are studied in other works. We describe their multifractal nature.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.09.020
Barral, Julien 1 ; Seuret, Stéphane 2

1 Équipe Sosso2, INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay cedex, France
2 Laboratoire d'analyse et de mathématiques appliquées, Université Paris 12 – Val-de-Marne, UFR des sciences et technologie, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France 
@article{CRMATH_2005__341_9_579_0,
     author = {Barral, Julien and Seuret, St\'ephane},
     title = {A class of multifractal semi-stable processes including {L\'evy} subordinators and {Mandelbrot} multiplicative cascades},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {579--582},
     publisher = {Elsevier},
     volume = {341},
     number = {9},
     year = {2005},
     doi = {10.1016/j.crma.2005.09.020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.09.020/}
}
TY  - JOUR
AU  - Barral, Julien
AU  - Seuret, Stéphane
TI  - A class of multifractal semi-stable processes including Lévy subordinators and Mandelbrot multiplicative cascades
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 579
EP  - 582
VL  - 341
IS  - 9
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2005.09.020/
DO  - 10.1016/j.crma.2005.09.020
LA  - en
ID  - CRMATH_2005__341_9_579_0
ER  - 
%0 Journal Article
%A Barral, Julien
%A Seuret, Stéphane
%T A class of multifractal semi-stable processes including Lévy subordinators and Mandelbrot multiplicative cascades
%J Comptes Rendus. Mathématique
%D 2005
%P 579-582
%V 341
%N 9
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2005.09.020/
%R 10.1016/j.crma.2005.09.020
%G en
%F CRMATH_2005__341_9_579_0
Barral, Julien; Seuret, Stéphane. A class of multifractal semi-stable processes including Lévy subordinators and Mandelbrot multiplicative cascades. Comptes Rendus. Mathématique, Tome 341 (2005) no. 9, pp. 579-582. doi : 10.1016/j.crma.2005.09.020. http://archive.numdam.org/articles/10.1016/j.crma.2005.09.020/

[1] Barral, J. Continuity of the multifractal spectrum of a statistically self-similar measure, J. Theoret. Probab., Volume 13 (2000), pp. 1027-1060

[2] Barral, J.; Seuret, S. Combining multifractal additive and multiplicative chaos, Commun. Math. Phys., Volume 257 (2005) no. 2, pp. 473-497

[3] J. Barral and S. Seuret, Heterogeneous ubiquitous systems in Rd and Hausdorff dimensions, Preprint (2005), | arXiv

[4] J. Barral and S. Seuret, Renewal of singularity sets of statistically self-similar measures, J. Stat. Phys., in press; | arXiv

[5] J. Barral, S. Seuret, The singularity spectrum of Lévy processes in multifractal time, Preprint (2005)

[6] Brown, G.; Michon, G.; Peyrière, J. On the multifractal analysis of measures, J. Stat. Phys., Volume 66 (1992) no. 3–4, pp. 775-790

[7] Durrett, R.; Liggett, T. Fixed points of the smoothing transformation, Z. Wahrsch. verw. Gebiete, Volume 64 (1983), pp. 275-301

[8] Guivarc'h, Y. Sur une extension de la notion de loi semi-stable, Ann. Inst. H. Poincaré, Probab. Statist., Volume 26 (1990), pp. 261-285

[9] Jaffard, S. The multifractal nature of Lévy processes, Probab. Theory Relat. Fields, Volume 114 (1999) no. 2, pp. 207-227

[10] Kahane, J.-P.; Peyrière, J. Sur certaines martingales de Benoît Mandelbrot, Adv. Math., Volume 22 (1976), pp. 131-145

[11] Lévy, P. Théorie des erreurs. La loi de Gauss et les lois exceptionnelles, Bull. Soc. Math., Volume 52 (1924), pp. 49-85

[12] Liu, Q. Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stoch. Proc. Appl., Volume 95 (2001), pp. 83-107

[13] Mandelbrot, B.B. Intermittent turbulence in self-similar cascades: divergence of hight moments and dimension of the carrier, J. Fluid. Mech., Volume 62 (1974), pp. 331-358

[14] B. Mandelbrot, A. Fischer, L. Calvet, A multifractal model of asset returns, Cowles Foundation Discussion Paper #1164 (1997)

[15] Olsen, L. A multifractal formalism, Adv. Math., Volume 116 (1995), pp. 92-195

[16] Riedi, R. Multifractal processes (Doukhan, P.; Oppenheim, G.; Taqqu, M.S., eds.), Long Range Dependence: Theory and Applications, Birkhäuser, Basel, 2002, pp. 625-715

Cité par Sources :