  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.
