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An Introduction to Mandelbrot Cascades

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New Trends in Applied Harmonic Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

In this course, we propose an elementary and self-contained introduction to canonical Mandelbrot random cascades. The multiplicative construction is explained and the necessary and sufficient condition of non-degeneracy is proved. Then, we discuss the problem of the existence of moments and the link with non-degeneracy. We also calculate the almost sure dimension of the measures. Finally, we give an outline on multifractal analysis of Mandelbrot cascades. This course was delivered in September 2013 during a meeting of the “Multifractal Analysis GDR” (GDR \( \mbox{ n}^{\mbox{ o}} \) 3475 of the french CNRS).

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Acknowledgements

I would like to thank Stéphane Jaffard and Stéphane Seuret who offered me the opportunity to deliver this course during the GDR meeting at Porquerolles Island (September 22–27, 2013)

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Authors and Affiliations

  1. Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 CLERMONT-FERRAND - CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177, Aubiére, France

    Yanick Heurteaux

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  1. Yanick Heurteaux
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Correspondence to Yanick Heurteaux .

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Editors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Akram Aldroubi

  2. Depto. de Matemática, Univ. de Buenos Ai, IMAS - UBA/CONICET, Buenos Aires, Argentina

    Carlos Cabrelli

  3. UFR de Sciences et Technologie, Universite Paris Est Creteil, Creteil Cedex, Paris, France

    Stephane Jaffard

  4. Depto. de Matemática, Univ. de Buenos Ai, IMAS, UBA/CONICET, Buenos Aires, Argentina

    Ursula Molter

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Heurteaux, Y. (2016). An Introduction to Mandelbrot Cascades. In: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U. (eds) New Trends in Applied Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27873-5_3

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