Abstract
Here, a Mandelbrot measure is a statistically self-similar measure μ 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 = ‖μ‖: 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 Hölder exponent for μ, almost everywhere with respect to another Mandelbrot measure, as well as to perform the multifractal analysis of μ, under hypotheses that are weaker than those usually assumed.
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Received 30 April 1997 / Revised 1 July 1998
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Barral, J. Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab Theory Relat Fields 113, 535–569 (1999). https://doi.org/10.1007/s004400050217
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DOI: https://doi.org/10.1007/s004400050217
- Mathematics Subject Classification (1991): 60G57, 28A75, 60F10, 60D05, 60J99, 60G17, 76F99
- Key words and phrases: Analyse multifractale, martingales de Mandelbrot, cascades multiplicatives