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 $r$.
Publié le : 2000-02-14
Classification:
Self-similar cascades,
Mandelbrot's martingales,
law of large numbers,
central limit theorem,
convergence of moment generating function,
large deviations,
60G42,
60F05,
60F10
@article{1019737670,
author = {Liu, Quansheng and Rouault, Alain},
title = {Limit theorems for Mandelbrot's multiplicative cascades},
journal = {Ann. Appl. Probab.},
volume = {10},
number = {2},
year = {2000},
pages = { 218-239},
language = {en},
url = {http://dml.mathdoc.fr/item/1019737670}
}
Liu, Quansheng; Rouault, Alain. Limit theorems for Mandelbrot's multiplicative cascades. Ann. Appl. Probab., Tome 10 (2000) no. 2, pp. 218-239. http://gdmltest.u-ga.fr/item/1019737670/