Self-similar continuous cascades supported by random Cantor sets. Application to rainfall data

TL;DR

This paper presents a new class of self-similar continuous cascade models supported by random Cantor sets, with potential applications to modeling rainfall data, especially dry period durations.

Contribution

It introduces a novel extension of continuous cascade models supported by fractal sets of arbitrary dimension, linking fractal geometry with stochastic modeling.

Findings

Model can reproduce distribution of dry period durations in rainfall data

Supports arbitrary fractal dimensions in cascade models

Provides mathematical properties and scaling analysis of the new construction

Abstract

We introduce a variant of continuous random cascade models that extends former constructions introduced by Barral-Mandelbrot and Bacry-Muzy in the sense that they can be supported by sets of arbitrary fractal dimension. The so introduced sets are exactly self-similar stationary versions of random Cantor sets formerly introduced by Mandelbrot as "random cutouts". We discuss the main mathematical properties of our construction and compute its scaling properties. We then illustrate our purpose on several numerical examples and we consider a possible application to rainfall data. We notably show that our model allows us to reproduce remarkably the distribution of dry period durations.