Moments, continuity and multifractal analysis of Mandelbrot martingales

• 成果类型: Article
• 署名作者: Barral, J
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050217
• 发表日期: 1999
• 页码: 535-569
• 关键词: iterated random multiplications; smoothing transformation; invariant distributions; fixed-points; cascades; decompositions; dimensions; extension; fractals

摘要

Here, a Mandelbrot measure is a statistically self-similar measure mu on the boundary of a c-ary tree, obtained by multiplying random weights indexed by the nodes of the tree. We take a particular interest in the random variable Y = parallel to mu parallel to : we study the existence of finite moments of negative orders for Y, conditionally to Y > 0, and the continuity properties of Y with respect to the weights. Our results on moments make possible to study, with probability one, the existence of a local Holder exponent for mu, almost everywhere with respect to another Mandelbrot measure, as well as to perform the multifractal analysis of mu, under hypotheses that are weaker than those usually assumed.