The Annals of Applied Probability

Convergence of complex multiplicative cascades

Julien Barral, Xiong Jin, and Benoît Mandelbrot

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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.

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Ann. Appl. Probab., Volume 20, Number 4 (2010), 1219-1252.

First available in Project Euclid: 20 July 2010

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles 60G18: Self-similar processes 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter
Secondary: 28A78: Hausdorff and packing measures

Multiplicative cascades continuous function-valued martingales functional central limit theorem laws stable under random weighted mean multifractals


Barral, Julien; Jin, Xiong; Mandelbrot, Benoît. Convergence of complex multiplicative cascades. Ann. Appl. Probab. 20 (2010), no. 4, 1219--1252. doi:10.1214/09-AAP665.

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