Limit theorems for Mandelbrot's multiplicative cascades

• 成果类型: Article
• 署名作者: Liu, QS; Rouault, A
• 署名单位: Universite de Rennes; Universite Paris Saclay
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/aoap/1019737670
• 发表日期: 2000
• 页码: 218-239
• 关键词: iterated random multiplications; smoothing transformation; invariant distributions; fixed-points; extension

摘要

Let W greater than or equal to 0 be a random variable with EW = 1, and let Z((r)) (r greater than or equal to 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 --> infinity: 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.