Fractional multiplicative processes
Barral, Julien ; Mandelbrot, Benoît
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 1116-1129 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Les mesures sur [0, 1] auto-similaires en loi sont limites de processus multiplicatifs construits à partir de poids aléatoires distribués sur les sous-intervalles b-adiques de [0, 1]. Ces poids sont i.i.d., positifs et d'espérance 1/b. Il est naturel d'étendre la construction à des poids prenant des valeurs négatives. On obtient alors des martingales à valeurs dans les fonctions continues sur [0, 1]. Nous nous intéressons, pour H∈(0, 1), à la martingale (Bn)n≥1 de ce type construite en prenant des poids à valeurs dans {-b-H, b-H}, afin que Bn converge presque sûrement uniformément vers une fonction B auto-similaire en loi dont la régularité Höldérienne et les propriétés fractales soient comparables à celles du mouvement brownien fractionnaire d'exposant H. C'est bien le cas lorsque H∈(1/2, 1), et la construction fournit alors un nouvel exemple de loi invariante par moyenne pondérée aléatoire. Cette loi satisfait la même équation fonctionnelle qu'une loi stable symétrique usuelle d'indice 1/H. Si H∈(0, 1/2], Bn diverge presque sûrement, mais il existe une normalisation naturelle par une suite (an)n≥1 telle que la marche aléatoire corrélée normalisée Bn/an converge en loi vers la restriction à [0, 1] du mouvement brownien standard. Des théorèmes limites sont également associés au cas H>1/2.

Statistically self-similar measures on [0, 1] are limit of multiplicative cascades of random weights distributed on the b-adic subintervals of [0, 1]. These weights are i.i.d., positive, and of expectation 1/b. We extend these cascades naturally by allowing the random weights to take negative values. This yields martingales taking values in the space of continuous functions on [0, 1]. Specifically, we consider for each H∈(0, 1) the martingale (Bn)n≥1 obtained when the weights take the values -b-H and b-H, in order to get Bn converging almost surely uniformly to a statistically self-similar function B whose Hölder regularity and fractal properties are comparable with that of the fractional brownian motion of exponent H. This indeed holds when H∈(1/2, 1). Also the construction introduces a new kind of law, one that it is stable under random weighted averaging and satisfies the same functional equation as the standard symmetric stable law of index 1/H. When H∈(0, 1/2], to the contrary, Bn diverges almost surely. However, a natural normalization factor an makes the normalized correlated random walk Bn/an converge in law, as n tends to ∞, to the restriction to [0, 1] of the standard brownian motion. Limit theorems are also associated with the case H>1/2.

Publié le : 2009-01-01
DOI : https://doi.org/10.1214/08-AIHP198
Classification:  60F05,  60F15,  60F17,  60G18,  60G42,  28A78 
@article{AIHPB_2009__45_4_1116_0,
     author = {Barral, Julien and Mandelbrot, Beno\^\i t},
     title = {Fractional multiplicative processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {45},
     year = {2009},
     pages = {1116-1129},
     doi = {10.1214/08-AIHP198},
     mrnumber = {2572167},
     zbl = {1201.60035},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2009__45_4_1116_0}
}

Barral, Julien; Mandelbrot, Benoît. Fractional multiplicative processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) pp. 1116-1129. doi : 10.1214/08-AIHP198. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_4_1116_0/

[1] M. Arbeiter and N. Patzschke. Random self-similar multifractals. Math. Nachr. 181 (1996) 5-42. | MR 1409071 | Zbl 0873.28003

[2] J. Barral. Continuity of the multifractal spectrum of a statistically self-similar measure. J. Theoret. Probab. 13 (2000) 1027-1060. | MR 1820501 | Zbl 0977.37024

[3] J. Barral and B. B. Mandelbrot. Random multiplicative multifractal measures. Proc. Sympos. Pure Math. 72 3-90. AMS, Providence, RI, 2004. | MR 2112119 | Zbl 1117.28006

[4] J. Barral and S. Seuret. The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 214 (2007) 437-468. | MR 2348038 | Zbl 1131.60039

[5] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0861.60003

[6] P. Billingsley. Convergence of Probability Measures, 2nd edition. Probability and Statistics. Wiley, New York, 1999. | MR 1700749 | Zbl 0944.60003

[7] P. Collet and F. Koukiou. Large deviations for multiplicative chaos. Comm. Math. Phys. 147 (1992) 329-342. | MR 1174416 | Zbl 0755.60022

[8] J. Dedecker, P. Doukhan, G. Lang, J. R. León, S. Louhichi and C. Prieur. Weak Dependence: With Examples and Applications. Lecture Notes in Statistics 190. Springer, New York, 2007. | MR 2338725 | Zbl 1165.62001

[9] R. Durrett and T. Liggett. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983) 275-301. | MR 716487 | Zbl 0506.60097

[10] A. Dvoretsky, P. Erdös and S. Kakutani. Nonincrease everywhere of the Brownian motion process. Proc. 4th Berkeley Sympos. Math. Stat. Prob. II (1961) 103-116. | MR 132608 | Zbl 0111.15002

[11] N. Enriquez. A simple construction of the fractional Brownian motion. Stochastic Process Appl. 109 (2004) 203-223. | MR 2031768 | Zbl 1075.60019

[12] K. J. Falconer. The multifractal spectrum of statistically self-similar measures. J. Theoret. Probab. 7 (1994) 681-702. | MR 1284660 | Zbl 0805.60034

[13] K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications, 2nd edition. Wiley, New Jersey, 2003. | MR 2118797 | Zbl 1060.28005

[14] Y. Guivarc'H. Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990) 261-285. | Numdam | MR 1063751 | Zbl 0703.60012

[15] R. Holley and E. C. Waymire. Multifractal dimensions and scaling exponents for strongly bounded random fractals. Ann. Appl. Probab. 2 (1992) 819-845. | MR 1189419 | Zbl 0786.60064

[16] B. R. Hunt. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc. 126 (1998) 791-800. | MR 1452806 | Zbl 0897.28004

[17] S. Jaffard. The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 (1999) 207-227. | MR 1701520 | Zbl 0947.60039

[18] J.-P. Kahane. Multiplications aléaroires et dimensions de Hausdorff. Ann. Inst. H. Poincaré Probab Statist. 23 (1987) 289-296. | Numdam | MR 898497 | Zbl 0619.60005

[19] J.-P. Kahane. J. Peyrière. Sur certaines martingales de Benoît Mandelbrot. Adv. Math. 22 (1976) 131-145. | MR 431355 | Zbl 0349.60051

[20] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1988. | MR 917065 | Zbl 0638.60065

[21] A. N. Kolmogorov. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. URSS (N.S.) 26 (1940) 115-118. | JFM 66.0552.03 | MR 3441

[22] Q. Liu. Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 (2001) 83-107. | MR 1847093 | Zbl 1058.60068

[23] B. B. Mandelbrot and J. W. Van Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review 10 (1968) 422-437. | MR 242239 | Zbl 0179.47801

[24] B. B. Mandelbrot. Multiplications aléatoire itérées et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris 278 (1974) 289-292, 355-358. | Zbl 0276.60096

[25] B. B. Mandelbrot. Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62 (1974) 331-358. | Zbl 0289.76031

[26] G. M. Molchan. Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179 (1996) 681-702. | MR 1400758 | Zbl 0853.76032

[27] M. Ossiander and E. C. Waymire. Statistical estimation for multiplicative cascades. Ann. Statist. 28 (2000) 1533-1560. | MR 1835030 | Zbl 1105.60305

[28] M. Taqqu. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 (1975) 287-302. | MR 400329 | Zbl 0303.60033