Mandelbrot cascades

Mandelbrot cascades are random measures on $[0,1]^2$ built by repeatedly splitting squares. Start with mass $1$ on the unit square. At each level, split every square into $4$ and multiply the mass in each new square by an independent copy of a random variable $W \geq 0$ with $\mathbb{E}(W) = 1$.

Here $W = 0$ or $W = 2$ each with probability $p$, and $W = 1$ otherwise. The case $p = 0.1$ is the example in Chapter 5 of my PhD thesis. Darker squares carry more mass.

Almost surely, conditioned on the cascade not dying out.