# Asian Journal of Mathematics

## Volume 22 (2018)

### Mandelbrot cascades on random weighted trees and nonlinear smoothing transforms

Pages: 883 – 918

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n5.a5

#### Authors

Julien Barral (Département de Mathématiques, Institut Galilée, Université Paris, Villetaneuse, France)

Jacques Peyrière (Département de Mathématiques, Faculté des Sciences, Université Paris-Sud, Orsay, France; and Department of Mathematics and Systems Science, Beihang University, Beijing, China)

#### Abstract

We consider complex Mandelbrot multiplicative cascades on a random weighted tree. Under suitable assumptions, this yields a dynamics $\mathsf{T}$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations) and which have a finite moment of order $2$. 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 μ such that a non-standard (functional) central limit theorem is valid for the orbit ${(\mathsf{T}^n \mu)}_{n \geq 0}$. The limit process possesses a structure combining multiplicative and additive cascade (this completes in a non trivial way our previous result in the case of nonnegative Mandelbrot cascades on a regular tree). If the weights are non conservative, we find a domain for the initial law $\mu$ over which ${(\mathsf{T}^n \mu)}_{n \geq 0}$ converges in law to 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 nonnegative martingale. Also, the dynamics can be modified to build fixed points of higher degree smoothing transformations.

#### Keywords

multiplicative cascades, Mandelbrot martingales, smoothing transformations, dynamical systems, central limit theorem, Gaussian processes, Random fractals, Wasserstein distance, Galton–Watson tree

#### 2010 Mathematics Subject Classification

37C99, 60F05, 60F17, 60G15, 60G17, 60G42