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Uniform convergence for complex [0, 1]-martingales

Abstract : 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.
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Submitted on : Thursday, February 21, 2013 - 2:52:36 PM
Last modification on : Friday, January 21, 2022 - 3:13:44 AM

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Julien Barral, Xiong Jin, Benoît Mandelbrot. Uniform convergence for complex [0, 1]-martingales. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2010, 20 (4), pp.1205-1218. ⟨10.1214/09-AAP664⟩. ⟨hal-00793058⟩

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