Limit Theorems for Multiplicative Cascades in a Random Environment

Abstract

Let $\zeta = (\zeta_{0},\zeta_{1},\ldots)$ be a sequence of independent and identically distributed random variables. For $r \geq 2$, let $\mu_r$ be Mandelbrot's (limit) measure of multiplicative cascades defined with positive weights indexed by nodes of a regular $r$-ary tree, and let $Z^{(r)}$ be the mass of $\mu_r$. We study asymptotic properties of $Z^{(r)}$ and the sequence of random measures $(\mu_r)_{r}$ as $r \to \infty$. We obtain some laws of large numbers and a central limit theorem. The results extend ones established by Liu and Rouault (2000) and by Liu, Rio and Rouault (2003).