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

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