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Fractional multiplicative processes

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>1/2.
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Submitted on : Thursday, February 21, 2013 - 4:16:23 PM
Last modification on : Friday, April 23, 2021 - 3:28:04 PM

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Julien Barral, Benoît Mandelbrot. Fractional multiplicative processes. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2009, 45 (4), pp.1116-1129. ⟨10.1214/08-AIHP198⟩. ⟨hal-00793123⟩

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