Dynamics of Mandelbrot Cascades

TL;DR

This paper investigates the dynamics of Mandelbrot multiplicative cascades, revealing their connection to fixed points of smoothing transformations and establishing a central limit theorem with a Gaussian process limit.

Contribution

It introduces a detailed analysis of the cascade dynamics, linking them to fixed points and deriving a central limit theorem for the system.

Findings

Trajectories converge to fixed points of smoothing transformations.

A central limit theorem is established for the cascade system.

The Gaussian process limit is characterized as an additive cascade of normal variables.

Abstract

Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an `additive cascade' of independent normal variables.