Fractional multiplicative processes

Julien Barral; Benoît Mandelbrot

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 4, page 1116-1129
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Barral, Julien, and Mandelbrot, Benoît. "Fractional multiplicative processes." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 1116-1129. <http://eudml.org/doc/78056>.

@article{Barral2009,
abstract = {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&gt;1/2.},
author = {Barral, Julien, Mandelbrot, Benoît},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random functions; martingales; central limit theorem; brownian motion; laws stable under random weighted mean; fractals; Hausdorff dimension; Brownian motion},
language = {eng},
number = {4},
pages = {1116-1129},
publisher = {Gauthier-Villars},
title = {Fractional multiplicative processes},
url = {http://eudml.org/doc/78056},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Barral, Julien
AU - Mandelbrot, Benoît
TI - Fractional multiplicative processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 1116
EP - 1129
AB - 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&gt;1/2.
LA - eng
KW - random functions; martingales; central limit theorem; brownian motion; laws stable under random weighted mean; fractals; Hausdorff dimension; Brownian motion
UR - http://eudml.org/doc/78056
ER -

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