Advances in Applied Probability

Renewal of singularity sets of random self-similar measures

Julien Barral and Stéphane Seuret

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This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

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Adv. in Appl. Probab. Volume 39, Number 1 (2007), 162-188.

First available in Project Euclid: 30 March 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G57: Random measures 60F10: Large deviations 28A78: Hausdorff and packing measures
Secondary: 28A80: Fractals [See also 37Fxx]

Random measure large deviations Hausdorff dimension self-similarity fractal


Barral, Julien; Seuret, Stéphane. Renewal of singularity sets of random self-similar measures. Adv. in Appl. Probab. 39 (2007), no. 1, 162--188. doi:10.1239/aap/1175266474.

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