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 $r$.
Citation
Quansheng Liu. Alain Rouault. "Limit theorems for Mandelbrot's multiplicative cascades." Ann. Appl. Probab. 10 (1) 218 - 239, February 2000. https://doi.org/10.1214/aoap/1019737670
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