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February 2000 Limit theorems for Mandelbrot's multiplicative cascades
Quansheng Liu, Alain Rouault
Ann. Appl. Probab. 10(1): 218-239 (February 2000). DOI: 10.1214/aoap/1019737670

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

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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

Information

Published: February 2000
First available in Project Euclid: 25 April 2002

zbMATH: 1161.60316
MathSciNet: MR1765209
Digital Object Identifier: 10.1214/aoap/1019737670

Subjects:
Primary: 60G42
Secondary: 60F05, 60F10

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.10 • No. 1 • February 2000
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