Construction of Signed Multiplicative Cascades

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, one that contains the complex extension of b-adic canonical cascades, compound Poisson cascades, and more generally infinitely divisible cascades. For this class, we provide a general sufficient condition for almost sure uniform convergence to a non-trivial limit. Such limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling.