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Renewal of singularity sets of random self-similar measures

Part of: Stochastic processes Limit theorems Classical measure theory

Published online by Cambridge University Press:  01 July 2016

Julien Barral  and
Stéphane Seuret
Julien Barral*
Affiliation:
INRIA Rocquencourt
Stéphane Seuret*
Affiliation:
Université Paris XII
*
∗ Postal address: Equipe Sosso2, Domaine de Voluceau Rocquencourt, 78153 Le Chesnay cedex, France. Email address: julien.barral@inria.fr
∗∗ Postal address: Laboratoire d'Analyse et de Mathématiques Appliquées, Faculté de Sciences et Technologie, Bat. P3, 4ème étage, 61 avenue du Général de Gaulle, 94010 Créteil cedex, France. Email address: seuret@univ-paris12.fr
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Abstract

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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.

Keywords

Random measurelarge deviationsHausdorff dimensionself-similarityfractal

MSC classification

Primary: 60G57: Random measures 60F10: Large deviations 28A78: Hausdorff and packing measures
Secondary: 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 162 - 188
Copyright
Copyright © Applied Probability Trust 2007 

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