Limit theorems for Mandelbrot's multiplicative cascades

Let W.0 be a random variable with EW=1, and let Z(r)(r.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..: 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.