Uniform convergence for complex $[\mathbf{0,1}]$-martingales

Julien Barral, Xiong Jin, Benoît Mandelbrot

Published 2008-12-24, updated 2010-10-21Version 3

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.