Convergence of complex multiplicative cascades

Julien Barral, Xiong Jin, Benoît Mandelbrot

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1219-1252
Number of pages34
JournalThe Annals of Applied Probability
Issue number4
Publication statusPublished - 2010


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