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

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

2008-12-24

arxiv.org/abs/0812.4556v3

Annals of Applied Probability 2010, Vol. 20, No. 4, 1205-1218

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP664 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP664

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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