Uniform convergence for complex [0,1]-martingales

Julien Barral, Xiong Jin, Benoît Mandelbrot

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    Positive T-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval T = [0, 1] and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling. © Institute of Mathematical Statistics, 2010.
    Original languageEnglish
    Pages (from-to)1205-1218
    Number of pages13
    JournalAnnals of Applied Probability
    Issue number4
    Publication statusPublished - Aug 2010


    • Continuous function-valued martingales
    • Multifractals
    • Multiplicative cascades
    • T-martingales


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