An introduction to Mandelbrot cascades

TL;DR

This paper provides an elementary, self-contained introduction to Mandelbrot cascades, covering their construction, non-degeneracy conditions, moment existence, measure dimension, and multifractal analysis.

Contribution

It offers a comprehensive, accessible overview of Mandelbrot cascades, including new proofs and insights into their properties and multifractal structure.

Findings

Established necessary and sufficient conditions for non-degeneracy

Calculated the almost sure dimension of Mandelbrot measures

Outlined the multifractal analysis framework for cascades

Abstract

In this course, we propose an elementary and self-contained introduction to canonical Mandelbrot random cascades. The multiplicative construction is explained and the necessary and sufficient condition of non-degeneracy is proved. Then, we discuss the problem of the existence of moments and the link with nondegeneracy. We also calculate the almost sure dimension of the measures. Finally, we give an outline on multifractal analysis of Mandelbrot cascades. This course was delivered in september 2013 during a meeting of the "Multifractal Analysis GDR" (GDR no 3475 of the french CNRS).