Limit theorems for Mandelbrot's multiplicative cascades

Abstract

Let $W \geq 0$ be a random variable with $EW = 1$, and let $Z^{(r)} (r \geq 2)$ be the limit of a Mandelbrot’s martingale, defined as sums of product of independent random weights having the same distribution as $W$, indexed by nodes of a homogeneous $r$-ary tree. We study asymptotic properties of $Z^{(r)}$ as $r \to \infty$: we obtain a law of large numbers, a central limit theorem, a result for convergence of moment generating functions and a theorem of large deviations. Some results are extended to the case where the number of branches is a random variable whose distribution depends on a parameter $r$.