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

TL;DR

This paper extends the theory of positive T-martingales to complex-valued martingales on [0,1], providing conditions for uniform convergence and introducing new multifractal processes for modeling signals.

Contribution

It introduces a framework for complex T-martingales, establishing uniform convergence criteria and generating new multifractal processes for signal analysis.

Findings

Established a sufficient condition for almost sure uniform convergence.

Constructed new examples of multifractal processes.

Extended martingale theory to complex-valued functions.

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.