The Annals of Applied Probability

Limit theorems for Mandelbrot's multiplicative cascades

Quansheng Liu and Alain Rouault

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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 $r$.

Article information

Source
Ann. Appl. Probab., Volume 10, Number 1 (2000), 218-239.

Dates
First available in Project Euclid: 25 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019737670

Digital Object Identifier
doi:10.1214/aoap/1019737670

Mathematical Reviews number (MathSciNet)
MR1765209

Zentralblatt MATH identifier
1161.60316

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

Keywords
Self-similar cascades Mandelbrot's martingales law of large numbers central limit theorem convergence of moment generating function large deviations

Citation

Liu, Quansheng; Rouault, Alain. Limit theorems for Mandelbrot's multiplicative cascades. Ann. Appl. Probab. 10 (2000), no. 1, 218--239. doi:10.1214/aoap/1019737670. https://projecteuclid.org/euclid.aoap/1019737670


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