Mandelbrot cascades on random weighted trees and nonlinear smoothing transforms

TL;DR

This paper studies Mandelbrot cascades on random trees, revealing how different weight conditions lead to either a central limit theorem or convergence to a fixed point of a nonlinear smoothing transformation.

Contribution

It introduces a framework for analyzing Mandelbrot cascades on random trees, extending the understanding of fixed points in nonlinear smoothing transformations under various weight assumptions.

Findings

Conservative weights lead to a functional central limit theorem for the cascade dynamics.

Non-conservative weights cause the cascade to converge to a fixed point of a quadratic smoothing transformation.

The limit laws can be constructed as limits of non-negative martingales.

Abstract

We consider complex Mandelbrot multiplicative cascades on a random weigh\-ted tree. Under suitable assumptions, this yields a dynamics $\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order 2. Moreover, we can exhibit two main behaviors: If the weights are conservative, i.e., sum up to~1 almost surely, we find a domain for the initial law $\mu$ such that a non-standard (functional) central limit theorem is valid for the orbit $(\T^n\mu)_{n\ge 0}$ (this completes in a non trivial way our previous result in the case of non-negative Mandelbrot cascades on a regular tree). If the weights are non conservative, we find a domain for the initial law $\mu$ over which $(\T^n\mu)_{n\ge 0}$ converges to the law of a non trivial random variable whose law turns out to be a fixed point of a quadratic smoothing transformation, which naturally extends the usual notion of (linear) smoothing transformation; moreover, this limit law can be built as the limit of a non-negative martingale. Also, the dynamics can be modified to build fixed points of higher degree smoothing transformations.