Mandelbrot's cascades: a legendary destiny

Abstract

The original density is 1 for t ∈ (0, 1), b is an integer base (b ≥ 2), and p ∈ (0, 1) is a parameter. The first construction stage divides the unit interval into b subintervals and multiplies the density in each subinterval by either 1 or −1 with the respective frequencies of 1 2 + p 2 and 1 2 − p 2 . It is shown that the resulting density can be renormalized so that, as n → ∞ (n being the number of iterations) the signed measure converges in some sense to a nondegenerate limit. If H = 1 + log b p > 1/2, hence p > b −1/2 , renormalization creates a martingale, the convergence is strong, and the limit shares the Hölder and Hausdorff properties of the fractional Brownian motion of exponent H. If H ≤ 1/2, hence p ≤ b −1/2 , this martingale does not converge. However, a different normalization can be applied, for H ≤ 1 2 to the martingale itself and for H > 1 2 to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of H and of b. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.

An Introduction to Mandelbrot Cascades

“…Taking the limit in the equality (5), the total mass Y ∞ = m([0, 1] is also a solution in law of the equation…”

Explicit fixed points of the smoothing transformation

“…We see that u satisfies the following differential equation 2 , ω is bounded in a neighborhood of 1. Indeed, by continuity,…”

“…The Mandelbrot martingales. To get a solution to Equation (2) one way is to use the Mandelbrot construction [7]. We set…”

“…Since then this model and Equation (2) have been extensively studied and generalized (for instance [5,3,4,6]). See [1,2] for a survey.…”