Convergence of complex multiplicative cascades

Julien Barral, Xiong Jin, Benoît Mandelbrot

    Research output: Contribution to journalArticlepeer-review


    The familiar cascade measures are sequences of random positive measures obtained on [0, 1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time. © Institute of Mathematical Statistics, 2010.
    Original languageEnglish
    Pages (from-to)1219-1252
    Number of pages33
    JournalAnnals of Applied Probability
    Issue number4
    Publication statusPublished - Aug 2010


    • Continuous function-valued martingales
    • Functional central limit theorem
    • Laws stable under random weighted mean
    • Multifractals
    • Multiplicative cascades


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