Convergence of complex multiplicative cascades

Abstract : 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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Article dans une revue
The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2010, 20 (4), pp.1219-1252. 〈10.1214/09-AAP665〉
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Dernière modification le : jeudi 11 janvier 2018 - 06:12:26

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Julien Barral, Xiong Jin, Benoît B. Mandelbrot. Convergence of complex multiplicative cascades. The Annals of Applied Probability : an official journal of the institute of mathematical statistics, The Institute of Mathematical Statistics, 2010, 20 (4), pp.1219-1252. 〈10.1214/09-AAP665〉. 〈hal-00790894〉

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