Asian Journal of Mathematics

Volume 22 (2018)

Number 5

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

Full Text (PDF format)

Received 4 March 2016

Accepted 23 November 2016

Published 9 November 2018