Fractional multiplicative processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, p. 1116-1129

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.

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.

DOI : https://doi.org/10.1214/08-AIHP198
Classification:  60F05,  60F15,  60F17,  60G18,  60G42,  28A78
Keywords: random functions, martingales, central limit theorem, brownian motion, laws stable under random weighted mean, fractals, Hausdorff dimension
@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},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     pages = {1116-1129},
     doi = {10.1214/08-AIHP198},
     zbl = {1201.60035},
     mrnumber = {2572167},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_4_1116_0}
}
Barral, Julien; Mandelbrot, Benoît. Fractional multiplicative processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, pp. 1116-1129. doi : 10.1214/08-AIHP198. http://www.numdam.org/item/AIHPB_2009__45_4_1116_0/

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