Taiwanese Journal of Mathematics

Limit Theorems for Multiplicative Cascades in a Random Environment

Shunli Hao

Full-text: Open access

Abstract

Let $\zeta = (\zeta_{0},\zeta_{1},\ldots)$ be a sequence of independent and identically distributed random variables. For $r \geq 2$, let $\mu_r$ be Mandelbrot's (limit) measure of multiplicative cascades defined with positive weights indexed by nodes of a regular $r$-ary tree, and let $Z^{(r)}$ be the mass of $\mu_r$. We study asymptotic properties of $Z^{(r)}$ and the sequence of random measures $(\mu_r)_{r}$ as $r \to \infty$. We obtain some laws of large numbers and a central limit theorem. The results extend ones established by Liu and Rouault (2000) and by Liu, Rio and Rouault (2003).

Article information

Source
Taiwanese J. Math., Volume 21, Number 4 (2017), 943-959.

Dates
Received: 26 August 2014
Revised: 14 February 2017
Accepted: 16 February 2017
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501120842

Digital Object Identifier
doi:10.11650/tjm/5216

Mathematical Reviews number (MathSciNet)
MR3684394

Zentralblatt MATH identifier
06871353

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 random environment law of large numbers central limit theorem large deviations

Citation

Hao, Shunli. Limit Theorems for Multiplicative Cascades in a Random Environment. Taiwanese J. Math. 21 (2017), no. 4, 943--959. doi:10.11650/tjm/5216. https://projecteuclid.org/euclid.twjm/1501120842


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