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 caseH >1/2. © Association des Publications de l'Institut Henri Poincaré, 2009.